Mathematische Zeitschrift

, Volume 276, Issue 1, pp 571–610

Bridgeland’s stabilities on abelian surfaces

Authors

  • Shintarou Yanagida
    • Research Institute for Mathematical SciencesKyoto University
    • Department of Mathematics, Faculty of ScienceKobe University
Article

DOI: 10.1007/s00209-013-1214-1

Cite this article as:
Yanagida, S. & Yoshioka, K. Math. Z. (2014) 276: 571. doi:10.1007/s00209-013-1214-1

Abstract

In this paper, we shall study the structure of walls for Bridgeland’s stability conditions on abelian surfaces. In particular, we shall study the structure of walls for the moduli spaces of rank 1 complexes on an abelian surface with the Picard number 1.

Mathematics Subject Classification (1991)

14D20

1 Introduction

Let \(X\) be an abelian surface over a field \(\mathfrak{k }\). Denote by \(\mathrm{Coh }(X)\) the category of coherent sheaves on \(X\), by \(\mathbf D (X)\) the bounded derived category of \(\mathrm{Coh }(X)\) and by \(K(X)\) the Grothendieck group of \(\mathbf D (X)\). The (algebraic) Mukai lattice of \(X\) consists of \(H^*(X,\mathbb{Z })_{\mathrm{alg }}:=\mathbb{Z } \oplus \mathrm{NS }(X) \oplus \mathbb{Z }\) and an integral bilinear form \(\langle \;\;,\;\; \rangle \) on \(H^*(X,\mathbb{Z })_{\mathrm{alg }}\):
$$\begin{aligned} \langle x,y \rangle := (x_1,y_1)-x_0 y_2-x_2 y_0 \in \mathbb{Z }, \end{aligned}$$
where \(x=(x_0,x_1,x_2), y=(y_0,y_1,y_2) \in \mathbb{Z } \oplus \mathrm{NS }(X) \oplus \mathbb{Z }\) and \((x_1,y_1)\) is the intersection pairing. We also denote \(x=(x_0,x_1,x_2) \in H^*(X,\mathbb{Z })_{\mathrm{alg }}\) by \(x=x_0+x_1+x_2 \varrho _X\). Thus \(\varrho _X\) corresponds to the fundamental class of \(X\).
For \(\beta \in \mathrm{NS }(X)_\mathbb Q \) and an ample divisor \(\omega \in \mathrm{Amp }(X)_\mathbb{Q }\), Bridgeland [3] constructed a stability condition \(\sigma _{\beta ,\omega }=(\mathfrak A _{(\beta ,\omega )},Z_{(\beta ,\omega )})\) on \(\mathbf D (X)\). Here \(\mathfrak A _{(\beta ,\omega )}\) is a tilting of \(\mathrm{Coh }(X)\), and \(Z_{(\beta ,\omega )}:K(X) \rightarrow \mathbb C \) is a group homomorphism called the stability function. In terms of the Mukai lattice \((H^*(X,\mathbb Z )_{\mathrm{alg }}, \langle \cdot ,\cdot \rangle ), Z_{(\beta ,\omega )}\) is given by
$$\begin{aligned} Z_{(\beta ,\omega )}(E)=\left\langle e^{\beta +\sqrt{-1}\omega },v(E) \right\rangle , \quad E \in K(X). \end{aligned}$$
Here \(v(E):=\mathrm{ch }(E)\) is the Mukai vector of \(E\). Hereafter for an object \(E \in \mathbf D (X)\), we simply write \(Z_{(\beta ,\omega )}(E) := Z_{(\beta ,\omega )}([E])\), where \([E]\) is the class of \(E\) in \(K(X)\). For abelian surfaces, these kind of stability conditions form a connected component of the space of stability conditions up to the action of the universal cover \(\widetilde{\mathrm{GL }}^+(2,\mathbb{R })\) of \(\mathrm{GL }^+(2,\mathbb{R })\) as stated in [3, sect. 15].
Let \(\phi _{(\beta ,\omega )} :\mathfrak A _{(\beta ,\omega )} {{\setminus }} \{0\} \rightarrow (0,1]\) be the phase function, which is defined by
$$\begin{aligned} Z_{(\beta ,\omega )}(E)=|Z_{(\beta ,\omega )}(E)| e^{ \sqrt{-1}\pi \phi _{(\beta ,\omega )}(E)}, \end{aligned}$$
for \(0 \ne E \in \mathfrak{A }_{(\beta ,\omega )}\). Let \(M_{(\beta ,\omega )}(v)\) be the moduli space of \(\sigma _{(\beta ,\omega )}\)-semi-stable objects \(E\) with \(v(E)=v\). It is a projective scheme if \((\beta ,\omega )\) is general [8, Thm. 1.2]. The stability of objects depends only on a chamber of the parameter space \(\mathrm{NS }(X)_\mathbb{R } \times \mathrm{Amp }(X)_\mathbb{R }\). For a special chamber, the stability coincides with Gieseker stability [2, 3, 7, 12]. For the analysis of Gieseker stability, Fourier–Mukai transforms [11] are very useful tool. However Fourier–Mukai transforms do not preserve Gieseker stability in general, since the category of coherent sheaves is not preserved. On the other hand, Fourier–Mukai transforms induce isomorphisms of the moduli spaces of Bridgeland stable objects as a consequence of [3, Prop. 10.3]. So it is very natural to study the moduli of Bridgeland stable objects and its dependence on the parameters even for the study of Gieseker stability.

For abelian surfaces, the studies on the dependence of the parameters, i.e., those of wall and chamber structures, are started by two groups Maciocia and Meachan [5], and Minamide et al. [7, 8]. In this paper, we continue our study on the wall and chamber structures. In particular, we shall study the behavior of the structures under Fourier–Mukai transforms. We are mainly interested in the most simple case, that is, the case where \(\mathrm{NS }(X)=\mathbb{Z }H\). In this case, the parameter space is the upper half plane, the action of Fourier–Mukai transforms can be described by an arithmetic subgroup of \(\mathrm{SL }(2,\mathbb{R })\) and the action on the upper half plane is the natural one. So we can study the structure in detail.

Let us explain the contents of this paper. We first define walls and chambers by using the characterization of walls in [8, Prop. 5.7]. Then we present a few basic properties of walls and chambers. As we explained, by the paper of Bridgeland [3, Prop. 10.3], Fourier–Mukai transforms preserve the stability of objects, that is, they induce isomorphisms of the moduli spaces. We shall show that the taking dual functor \(\mathcal{D }_X\) also preserves the stability of objects (Theorem 3.8). These are done in Sects. 2 and 3

In [14], we introduced a useful notion semi-homogeneous presentations. It is a presentation of a coherent sheaf as the kernel or the cokernel of a homomorphism \(V_{-1} \rightarrow V_0\) of semi-homogeneous sheaves \(V_{-1}, V_0\) with some numerical conditions. Extracting the numerical conditions from \(V_{-1}\) and \(V_0\), we also introduced the notion numerical solutions and constructed moduli spaces of simple complexes \(V_{-1} \rightarrow V_0\) associated to numerical solutions. In [7], we found a relation between these moduli spaces and the wall crossing behavior for Bridgeland stability conditions. In this paper, we give a supplement of this relation. We shall relate a particular wall called a codimension 0 wall for every numerical solution. If \((\beta ,\omega )\) belongs to a codimension 0 wall, then for any neighborhood \(U\) of \((\beta ,\omega )\), there is no \(\sigma _{(\beta ,\omega )}\)-semi-stable object which is \(\sigma _{(\beta ',\omega ')}\)-semi-stable for all \((\beta ',\omega ') \in U\).

In Sect. 5, we fix an ample divisor \(H\) and study special stability conditions \(\sigma _{(\beta +sH,tH)}\) parametrized by the upper half plane \(\{ (s,t) \mid s,t\in \mathbb{R }, s>0 \}\). In the \((s,t)\)-plane, the equations of walls are very simple. They define circles or lines forming a pencil of circles passing through imaginary points (Remark 5.4). Thus they do not intersect each other. For a principally polarized abelian surface with Picard number 1, these kind of results are obtained by Maciocia and Meachan [5]. Thus our results are generalization of theirs.

We also explain that there are one or two unbounded chambers which parametrize Gieseker semi-stable sheaves. In [17], one of the authors showed that Gieseker’s stability is preserved under the Fourier–Mukai transform, if the degree of the stable sheaf is sufficiently large. We shall explain the result as an application of this section (Proposition 5.25).

In Sect. 6, we assume that \(\mathrm{NS }(X)=\mathbb{Z }H\). We are mainly interested in the Mukai vector \(v\) which is a Fourier–Mukai transform of \(1-\ell \varrho _X\). So we may assume that \(v=1-\ell \varrho _X\).

In [14], we described the algebraic part of the Mukai lattice \(H^*(X,\mathbb{Z })_{\mathrm{alg }}\) as a lattice in the vector space of quadratic forms of two variables. Then we can describe the action of Fourier–Mukai transforms as a natural \(\mathrm{GL }(2,\mathbb{R })\)-action. By using these results, we shall study the structure of walls. In particular, we shall classify codimension 0 walls by using our description of Mukai lattice. We set \(n:=(H^2)/2\). If \(\sqrt{\ell / n} \in \mathbb{Q }\), then there is a unique wall of codimension 0, since there is one numerical solution. Assume that \(\sqrt{\ell / n} \not \in \mathbb{Q }\). In this case, these are infinitely many numerical solutions, thus we have infinitely many codimension 0 walls. Let \(G_{n,\ell } \subset \mathrm{GL }(2,\mathbb{R })\) be the subgroup generated by the cohomological action of (covariant or contravariant) Fourier–Mukai transforms preserving \(\pm v\). \(G_{n,\ell }\) acts on the set of walls. We show that there are finitely many \(G_{n,\ell }\)-orbits of walls and the set of codimension 0 walls forms an orbit of \(G_{n,\ell }\). Each orbit has two accumulation points \((\pm \sqrt{\ell /n},0)\) in the \((s,t)\)-plane. An observant reader will see that the main part of Sect. 6 essentially appeared in [13] without using Bridgeland stability condition.

In Sect. 7, we shall study the structure of walls for \(v=1-\ell \varrho _X, \ell \le 4\) on a principally polarized abelian surface \(X\). As an application, we shall classify \(M_H(v)\) for a primitive Mukai vector with \(\langle v^2 \rangle /2 \le 4\).

In [9] and [10, Thm. 3], Mukai announced that \(M_H(v) \cong X \times \mathrm{Hilb }^{\langle v^2 \rangle /2}(X)\) for a primitive Mukai vector with \(\langle v^2 \rangle /2=1,2,3\). Moreover he determined the Fourier–Mukai transform which induces the isomorphism. By using the structure of walls, we give an explanation of Mukai’s results for \(\langle v^2 \rangle /2=2,3\). It is quite surprising that Mukai discovered his results 30 years ago without using Bridgeland’s stability conditions.

In appendix, we continue to assume that \(X\) is an abelian surface with \(\mathrm{NS }(X)=\mathbb{Z }H\) and \((H^2)/2=n\). We shall identify the period space with the upper half plane. Then we show that the action of auto-equivalences coincides with the action of the modular group \(\Gamma _0(n)\) on the upper half plain.

Finally we would like to mention related works which appeared during our preparation of this manuscript. We note that the examples of Bridgeland stability conditions in this paper are generalized to an arbitrary projective surfaces by Arcara and Bertram [1]. For these stability conditions, Maciocia [4] studied the structure of walls. In particular, he proved similar results to Sect. 5 in a much more general context. For the stability conditions on principally polarized abelian surfaces, Meachan [6] studied the structure of walls in detail. In particular he independently found examples of walls with accumulation points.

2 Preliminaries on Bridgeland’s stability condition

As in the introduction, let \(X\) be an abelian surface over a field \(\mathfrak k \), and fix an ample divisor \(H\) on \(X\).

2.1 Notations

  1. (i)
    Mukai lattice The (algebraic) Mukai lattice of \(X\) consists of \(H^*(X,\mathbb{Z })_{\mathrm{alg }}:=\mathbb{Z } \oplus \mathrm{NS }(X) \oplus \mathbb{Z }\) and an integral bilinear form \(\langle \;\;,\;\; \rangle \) on \(H^*(X,\mathbb{Z })_{\mathrm{alg }}\):
    $$\begin{aligned} \langle x,y \rangle := (x_1,y_1)-x_0 y_2-x_2 y_0 \in \mathbb{Z }, \end{aligned}$$
    where \(x=(x_0,x_1,x_2), y=(y_0,y_1,y_2) \in \mathbb{Z } \oplus \mathrm{NS }(X) \oplus \mathbb{Z }\) and \((x_1,y_1)\) is the intersection pairing. We also denote \(x=(x_0,x_1,x_2)\) by \(x=x_0+x_1+x_2 \varrho _X\).
     
  2. (ii)
    Mukai vector The Mukai vector \(v(E)\in H^*(X,\mathbb Z )_{\mathrm{alg }}\) of \(E \in \mathrm{Coh }(X)\) is defined by
    $$\begin{aligned} v(E):=\mathrm{ch }(E) \sqrt{\mathrm{td }_X} =\mathrm{ch }(E) =\mathrm{rk }E+c_1(E)+ \chi (E) \varrho _X. \end{aligned}$$
    We also use the vectorial notation
    $$\begin{aligned} v(E)=(\mathrm{rk }E,c_1(E),\chi (E)). \end{aligned}$$
    For an object \(E\) of \(\mathbf{D }(X), v(E)\) is defined by \(\sum \nolimits _{k}(-1)^k v(E^k)\), where \((E^k)=(\cdots \rightarrow E^{-1} \rightarrow E^0 \rightarrow E^1 \rightarrow \cdots )\) is the bounded complex representing the object \(E\).
     
  3. (iii)

    Positivity A Mukai vector \(v=(r,\xi ,a) \ne 0\) is positive, if (i) \(r>0\) or (ii) \(r=0\) and \(\xi \) is effective, or (iii) \(r=\xi =0\) and \(a>0\). We denote a positive Mukai vector \(v\) by \(v>0\). A Mukai vector \(v\) is called isotropic if \(\langle v^2 \rangle =0\).

     
  4. (iv)
    An expression of Mukai vector We take an ample \(\mathbb{Q }\)-divisor \(H\). For \(v \in H^*(X,\mathbb Z )_{\mathrm{alg }}\) and \(\beta \in \mathrm{NS }(X)_\mathbb Q \), we set
    $$\begin{aligned} r_\beta (v):= \mathrm{rk }(v)\!=\!-\langle v,\varrho _X \rangle ,\quad a_\beta (v):=-\langle v,e^\beta \rangle ,\quad d_{\beta ,H}(v):=\frac{\langle v,H\!+\!(H,\beta )\varrho _X \rangle }{(H^2)}.\nonumber \\ \end{aligned}$$
    (2.1)
    If the choice of \(H\) is clear, then we write \(d_\beta (v):=d_{\beta ,H}(v)\) for simplicity. By using (2.1), we have
    $$\begin{aligned} v&= e^\beta \left\{ r_\beta (v)+a_\beta (v) \varrho _X + (d_{\beta ,H}(v) H+D_\beta (v))\right\} \nonumber \\&= r_\beta (v)e^\beta +a_\beta (v) \varrho _X + (d_{\beta ,H}(v) H+D_\beta (v))+(d_{\beta ,H}(v) H+D_\beta (v),\beta )\varrho _X.\nonumber \\ \end{aligned}$$
    (2.2)
    where \(D_\beta (v) \in H^{\perp } \cap \mathrm{NS }(X)_\mathbb Q \). We also have
    $$\begin{aligned} \langle v^2 \rangle =d_{\beta ,H}(v)^2 (H^2)-2r_\beta (v) a_\beta (v) +(D_\beta (v)^2). \end{aligned}$$
    (2.3)
     
  5. (v)

    Moduli spaces of Gieseker semi-stable sheaves For \(\beta \in \mathrm{NS }(X)_\mathbb Q \), we define the \(\beta \)-twisted semi-stability replacing the usual Hilbert polynomial \(\chi (E(nH))\) by \(\chi (E(-\beta +nH))\). Then \(v\) is positive if and only if \(\chi (E(-\beta +nH)) >0\) for \(E \in \mathbf{D}(X)\) with \(v(E)=v\) and \(n \gg 0\). For a positive Mukai vector \(v, \mathcal{M }_H^\beta (v)^{ss}\) denotes the moduli stack of \(\beta \)-twisted semi-stable sheaves \(E\) on \(X\) with \(v(E)=v\). \(\overline{M}_H^\beta (v)\) denotes the moduli scheme of \(S\)-equivalence classes of \(\beta \)-twisted semi-stable sheaves \(E\) on \(X\) with \(v(E)=v\) and \(M_H^\beta (v)\) denotes the open subscheme consisting of \(\beta \)-twisted stable sheaves. If \(\beta =0\), then we write \(\overline{M}_H(v):=\overline{M}_H^\beta (v)\).

     
  6. (vi)
    Fourier–Mukai transforms For a proper morphism \(f:Z_1 \rightarrow Z_2\), we denote the derived pull-back \(\mathbf{L}f^*\) and the derived direct image \(\mathbf{R}f_*\) by \(f^*\) and \(f_*\) respectively. For \(\mathbf{E} \in \mathbf{D}(X \times Y), \Phi _{X \rightarrow Y}^{\mathbf{E}}:\mathbf{D}(X) \rightarrow \mathbf{D}(Y)\) denotes the integral functor whose kernel is \(\mathbf{E}\):
    $$\begin{aligned} \Phi _{X \rightarrow Y}^{\mathbf{E}}(E)=p_{Y*}(p_X^*(E) \otimes \mathbf{E}),\; E \in \mathbf{D}(X), \end{aligned}$$
    (2.4)
    where \(p_X\) and \(p_Y\) are projections from \(X \times Y\) to \(X\) and \(Y\) respectively. If \(\Phi _{X \rightarrow Y}^\mathbf{E}\) is an equivalence, it is called a Fourier–Mukai transform. If a Fourier–Mukai transform \(\Phi _{X \rightarrow Y}^{\mathbf{E}}\) exists and \(X\) is an abelian surface, then \(Y\) is also an abelian surface and \(\Phi _{X \rightarrow Y}^\mathbf{E}\) induces an isometry of Mukai lattices \(H^*(X,\mathbb{Z })_{\mathrm{alg }} \rightarrow H^*(Y,\mathbb{Z })_{\mathrm{alg }}\). We also denote this isometry by \(\Phi _{X \rightarrow Y}^\mathbf{E}\). \(\mathcal{D }_X(*):=\mathbf{R}\mathcal{H }om_{\mathcal{O }_X}(*,\mathcal{O }_X)\) denotes the taking dual functor. It is a contravariant functor from \(\mathbf{D}(X)\) to \(\mathbf{D}(X)\). A contravariant Fourier–Mukai transform is a composite of a Fourier–Mukai functor and \(\mathcal{D }_X\). If \(X\) is an abelian surface, then it is of the form \(\Phi _{X \rightarrow Y}^{\mathbf{E}[2]} \circ \mathcal{D }_X=\mathcal{D }_Y \circ \Phi _{X \rightarrow Y}^{\mathbf{E}^{\vee }}\) with \(\mathbf{E}^{\vee } := \mathcal{D }_{X \times Y}(\mathbf{E})\). Let \(\alpha \) be a representative of a 2-cocycle of \(H^2_\mathrm{et }(X,\mathcal{O }_X^{\times })\). We have the notion of \(\alpha \)-twisted sheaves on \(X\). Let \(\mathrm{Coh }^\alpha (X)\) be the category of \(\alpha \)-twisted sheaves and \(\mathbf{D}^\alpha (X)\) the bounded derived category of \(\mathrm{Coh }^\alpha (X)\). Then the integral functor \(\Phi _{X \rightarrow Y}^{\mathbf{E}}\) in (2.4) is generalized to this case, where \(\mathbf{E} \in \mathbf{D}^{\alpha ^{-1} \times \beta }(X \times Y)\) and \(\Phi _{X \rightarrow Y}^{\mathbf{E}}:\mathbf{D}^\alpha (X) \rightarrow \mathbf{D}^\beta (Y)\).
     

2.2 Stability conditions and wall/chamber structure

Let us recall the stability conditions given in [3] and [7, §1]. Let
$$\begin{aligned} \mathrm{Amp }(X)_\mathbb{R }:=\{x \in \mathrm{NS }(X)_\mathbb{R } \mid (x^2)>0,(x,D)>0 \}. \end{aligned}$$
be the ample cone of \(X\), where \(D\) is an effective divisor. We take \((\beta ,\omega ) \in \mathrm{NS }(X)_\mathbb{R } \times \mathrm{Amp }(X)_\mathbb{R }\) and \(H \in \mathbb{R }_{>0}\omega \). For \(E \in K(X)\) with \(v=v(E)\) expressed as (2.2), we have
$$\begin{aligned} Z_{(\beta ,\omega )}(E)&= \langle e^{\beta +\sqrt{-1}\omega },v(E) \rangle \\&= -a_\beta (v(E))+\frac{(\omega ^2)}{2}r_\beta (v(E)) +d_{\beta ,H}(v(E))(H,\omega )\sqrt{-1}. \end{aligned}$$
Assume that \((\beta ,\omega ) \in \mathrm{NS }(X)_\mathbb{Q } \times \mathrm{Amp }(X)_\mathbb{Q }\). Let \(\mathfrak A _{(\beta ,\omega )}\) be the tilt of \(\mathrm{Coh }(X)\) with respect to the torsion pair \((\mathfrak T _{(\beta ,\omega )},\mathfrak F _{(\beta ,\omega )})\) defined by
  1. (i)

    \(\mathfrak T _{(\beta ,\omega )}\) is generated by \(\beta \)-twisted stable sheaves with \(Z_{(\beta ,\omega )}(E) \in \mathbb H \cup \mathbb R _{<0}\).

     
  2. (ii)

    \(\mathfrak F _{(\beta ,\omega )}\) is generated by \(\beta \)-twisted stable sheaves with \(-Z_{(\beta ,\omega )}(E) \in \mathbb H \cup \mathbb R _{<0}\),

     
where \(\mathbb H :=\{z \in \mathbb C \mid \mathrm Im z>0 \}\) is the upper half plane. \(\mathfrak A _{(\beta ,\omega )}\) is the abelian category introduced in [3] and it depends only on \(\beta \) and the ray \(\mathbb{Q }_{>0}\omega \).
Then the pair \(\sigma _{(\beta ,\omega )}=(\mathfrak A _{(\beta ,\omega )},Z_{(\beta ,\omega )})\) satisfies the requirement of the stability condition on \(\mathbf D (X)\) [3]. We set \(\mathfrak{A }_{(\beta ,\omega )}^*:= \mathfrak{A }_{(\beta ,\omega )} {\setminus } \{ 0 \}\). Let \(\phi _{(\beta ,\omega )}:\mathfrak{A }_{(\beta ,\omega )}^* \rightarrow (0,1]\) be the phase function of \(\sigma _{(\beta ,\omega )}\):
$$\begin{aligned} Z_{(\beta ,\omega )}(E) = |Z_{(\beta ,\omega )}(E)|e^{\pi \sqrt{-1}\phi _{(\beta ,\omega )}(E)}. \end{aligned}$$
(2.5)
Then the (semi-)stability of an object in \(\mathfrak A _{(\beta ,\omega )}\) with respect to \(Z_{(\beta ,\omega )}\) is defined by using \(\phi _{(\beta ,\omega )}\) as a slope function. More generally, we define \(\phi _{(\beta ,\omega )}(E)\) and the semi-stability as follows.

Definition 2.1

  1. (1)

    We set \(\phi _{(\beta ,\omega )}(E[n]):=\phi _{(\beta ,\omega )}(E)+n\) for \(E \in \mathfrak{A }_{(\beta ,\omega )}^*\).

     
  2. (2)

    \(F \in \mathbf D (X)\) is called semi-stable of phase \(\phi \), if there is an integer \(n\) such that \(E:=F[-n]\) is a semi-stable object of \(\mathfrak A _{(\beta ,\omega )}\) with \(\phi _{(\beta ,\omega )}(E)=\phi -n\). If we want to emphasize the dependence on the stability condition, we say that \(F\) is \(\sigma _{(\beta ,\omega )}\)-semi-stable.

     

Definition 2.2

For a Mukai vector \(v, \mathcal{M }_{(\beta ,\omega )}(v)\) denotes the moduli stack of \(\sigma _{(\beta ,\omega )}\)-semi-stable objects \(E\) of \(\mathfrak A _{(\beta ,\omega )}\) with \(v(E)=v\). \(M_{(\beta ,\omega )}(v)\) denotes the moduli scheme of the \(S\)-equivalence classes of \(\sigma _{(\beta ,\omega )}\)-semi-stable objects \(E\) of \(\mathfrak A _{(\beta ,\omega )}\) with \(v(E)=v\), if it exists.

This definition is a little bit ambiguous, since the phase of \(E \in \mathcal{M }_{(\beta ,\omega )}(v)\) cannot be specified by \(v\). In this paper, we introduce the following definition for the phase of \(v\).

Definition 2.3

For a non-zero Mukai vector \(v \in H^*(X,\mathbb Z )_{\mathrm{alg }}\), we define \(Z_{(\beta ,\omega )}(v) \in \mathbb C \) and \(\phi _{(\beta ,\omega )}(v) \in (-1,1]\) by
$$\begin{aligned} Z_{(\beta ,\omega )}(v)= \langle e^{\beta +\sqrt{-1} \omega },v \rangle = |Z_{(\beta ,\omega )}(v)|e^{\pi \sqrt{-1}\phi _{(\beta ,\omega )}(v)}. \end{aligned}$$
Then
$$\begin{aligned} \phi _{(\beta ,\omega )}(v(E)) =\phi _{(\beta ,\omega )}(E) \end{aligned}$$
for \(0 \ne E \in \mathfrak A _{(\beta ,\omega )} \cup \mathfrak A _{(\beta ,\omega )}[-1]\).

We give several remarks for the definition of \(\phi _{(\beta ,\omega )}(v)\).

Remark 2.4

  1. (1)
    If \(Z_{(\beta ,\omega )}(-v) \in \mathbb{H } \cup \mathbb{R }_{<0}\), then we have
    $$\begin{aligned} M_{(\beta ,\omega )}(v)=\{E[-1] \mid E \in M_{(\beta ,\omega )}(-v) \}, \end{aligned}$$
    (2.6)
    since \(\phi _{(\beta ,\omega )}(v) \in (-1,0]\).
     
  2. (2)
    If we take the phase of \(v\) with \(Z_{(\beta ,\omega )}(-v) \in \mathbb{H } \cup \mathbb{R }_{<0}\) as \(\phi _{(\beta ,\omega )}(v) \in (1,2]\), then
    $$\begin{aligned} M_{(\beta ,\omega )}(v)=\{E[1] \mid E \in M_{(\beta ,\omega )}(-v) \}. \end{aligned}$$
    (2.7)
    More generally, if we take the phase of \(v\) as \(\phi _{(\beta ,\omega )}(v) \in (n,n+1]\), then we have
    $$\begin{aligned} M_{(\beta ,\omega )}(v)=\{E[n] \mid E \in M_{(\beta ,\omega )}((-1)^n v) \}. \end{aligned}$$
    (2.8)
     

Remark 2.5

We take \(v \in H^*(X,\mathbb{Z })_{\mathrm{alg }}\) with \(\langle v^2 \rangle \ge 0\) and \(\mathrm{rk }v \ne 0\). If \(d_{\beta ,\omega }(v)=0\), then we get \(-r_\beta (v) a_\beta (v) \ge 0\) by (2.3). Hence we have the following.
  1. (1)

    If \(\mathrm{rk }v>0\), then \(Z_{(\beta ,\omega )}(v) \in \mathbb{C } {\setminus } \mathbb{R }_{\le 0}\) for all \((\beta ,\omega )\).

     
  2. (2)

    If \(\mathrm{rk }v<0\), then \(Z_{(\beta ,\omega )}(v) \in \mathbb{C } {{\setminus }} \mathbb{R }_{\ge 0}\) for all \((\beta ,\omega )\).

     
Therefore \(\phi _{(\beta ,\omega )}(v)\) in Definition 2.3 is a continuous function of \((\beta ,\omega )\), if \(\mathrm{rk }v>0\). The continuity will be useful if we study the wall/chamber structure on the space of stability conditions. On the other hand, if \(\mathrm{rk }v<0\), then the continuity requires us to define \(\phi _{(\beta ,\omega )}(v)\) as in Remark 2.4 (2), that is, we require \(\phi _{(\beta ,\omega )}(v) \in (0,2)\). In this paper, we mainly treat the case where \(\mathrm{rk }v>0\). So Definition 2.3 is reasonable.

For \(E \in \mathbf{D}(X)\), we set \(S(E):= \{ i \mid H^i(E) \ne 0 \}\). The following claim shows that the phase of a semi-stable object \(E \in \mathbf{D}(X)\) is determined by the cohomology sheaves \(H^i(E) \in \mathrm{Coh }(X)\).

Lemma 2.6

Let \(E\) be a \(\sigma _{(\beta ,\omega )}\)-semi-stable object. Then \(\phi _{(\beta ,\omega )}(E)\) is uniquely determined by \(Z_{(\beta ,\omega )}(E)\) and the set \(S(E)\).

Proof

Since \(E\) is \(\sigma _{(\beta ,\omega )}\)-semi-stable, there is an integer \(n\) such that \(E[-n] \in \mathfrak{A }_{(\beta ,\omega )}\). Then \(S(E) \subset \{-n-1,-n\}\) and \(Z_{(\beta ,\omega )}(E) \in (-1)^n (\mathbb{H } \cup \mathbb{R }_{<0})\). Since \(\phi _{(\beta ,\omega )}(E) \in (n,n+1]\), it is enough to determine \(n\) to prove the statement. If \(\# S(E)=2\), then \(n\) is uniquely determined. Assume that \(S(E)=\{ k \}\). Then \(k=-n-1\) or \(k=-n\). If \(Z_{(\beta ,\omega )}(E) \in (-1)^k (\mathbb{H } \cup \mathbb{R }_{<0})\), then \(n=-k\). If \(Z_{(\beta ,\omega )}(E) \in (-1)^{k+1} (\mathbb{H } \cup \mathbb{R }_{<0})\), then \(n=-k-1\). \(\square \)

Definition 2.7

(cf. [8, Prop. 5.7]) Let \(v_1 \not \in \mathbb{Q }v\) be a Mukai vector with \(\langle v_1^2 \rangle \ge 0, \langle (v-v_1)^2 \rangle \ge 0\) and \(\langle v_1,v-v_1 \rangle >0\). We define a wall for\(v\) by
$$\begin{aligned} W_{v_1}:=\{(\beta ,\omega ) \in \mathrm{NS }(X)_\mathbb{R } \times \mathrm{Amp }(X)_\mathbb{R } \mid \mathbb{R }Z_{(\beta ,\omega )}(v_1)= \mathbb{R }Z_{(\beta ,\omega )}(v) \}. \end{aligned}$$
A connected component of \(\mathrm{NS }(X)_\mathbb{R } \times \mathrm{Amp }(X)_\mathbb{R }{\setminus }\cup _{v_1} W_{v_1}\) is called a chamber for\(v\).

Replacing \(v\) by \(-v\) if necessary, we may assume that \(Z_{(\beta ,\omega )}(v) \in \mathbb{H } \cup \mathbb{R }_{<0}\). Assume that \((\beta ,\omega ) \in W_{v_1}\).

By [8, Prop. 5.7], the conditions for \(v_1\) are equivalent to the existence of \(\sigma _{(\beta ,\omega )}\)-semi-stable objects \(E_1, E_2\) with \(v(E_1)=v_1, v(E_2)=v-v_1\) such that \(\mathbb{R }_{>0}Z_{(\beta ,\omega )}(E_1)= \mathbb{R }_{>0}Z_{(\beta ,\omega )}(E_2)\). In particular, there is a properly \(\sigma _{(\beta ,\omega )}\)-semi-stable object with the Mukai vector \(v\).

For an ample divisor \(H\) and \(b \in \mathbb{Q }\), we set
$$\begin{aligned} \mathfrak{H }_\mathbb{R }^b:= \{ (bH+\eta ,tH) \mid \eta \in \mathrm{NS }(X)_\mathbb{R } \cap H^\perp , t>0 \}. \end{aligned}$$
(2.9)
If \(d_{\beta ,H}(v) \ne 0\), then \(W_{v_1} \cap \mathfrak{H }_\mathbb{R }^b\) is a proper subspace for any \(v_1\) by [7, sect. 3.1]. Since \(\mathfrak{A }_{(\beta ,\omega )}\) is independent of the choice of \((\beta ,\omega )\) as far as \((\beta ,\omega ) \in \mathfrak{H }_\mathbb{R }^b\), we have a wall/chamber structure on \(\mathfrak{H }_\mathbb{R }^b\) in a similar way to define the wall/chamber structure for \(\mu \)-semi-stability of torsion free sheaves (cf. [15, sect. 2]). In particular, \(\sigma _{(\beta ,\omega )}\)-semi-stability is constant on a chamber in \(\mathfrak{H }_\mathbb{R }^b\).

Remark 2.8

Let \(\mathcal{C }\) be a chamber and take a point \((\beta _0,\omega _0)\) in an adjacent wall \(W\). By [8, Cor. 5.17 (2), Rem. 5.20] or its proof, there is a family of \(\sigma _{(\beta ,\omega )}\)-stable objects \(\{\mathcal{E }_t \}_{t \in \mathbb{P }^1}\)\(((\beta ,\omega ) \in \mathcal{C }\)) parametrized by \(\mathbb{P }^1\) and the \(S\)-equivalence class of \(\mathcal{E }_t\) with respect to \(\sigma _{(\beta _0,\omega _0)}\)-semi-stability is independent of \(t\). Thus the \(\sigma _{(\beta ,\omega )}\)-semi-stability changes if \((\beta ,\omega )\) crosses \(W\).

Proposition 2.9

[3, Prop. 9.3] Let \(v\) be a Mukai vector with \(\langle v^2 \rangle >0\). Let \(\mathcal{C }\) be a chamber for \(v\). Then \(\mathcal{M }_{(\beta ,\omega )}(v)\) is independent of \((\beta ,\omega ) \in \mathcal{C }\).

Remark 2.10

For \((\beta _0,\omega _0) \in \mathcal{C }\) and \(E \in \mathcal{M }_{(\beta _0,\omega _0)}(v), \phi _{(\beta ,\omega )}(E)\) is a continuous function on \(\mathcal{C }\) by this proposition. By Lemma 2.6, it is uniquely determined by \(E\).

Proposition 2.11

Let \(\varphi :H^*(X,\mathbb{Z })_{\mathrm{alg }} \rightarrow H^*(Y,\mathbb{Z })_{\mathrm{alg }}\) be an isometry of Mukai lattices. Let \(W_u\) be a wall for \(v\) defined by \(u\). Then \(\varphi (u)\) defines a wall for \(\varphi (v)\).

Proof

We set \(w:=v-u\). Then \(u\) defines a wall for \(v\) if and only if \(\langle u^2 \rangle , \langle w^2 \rangle \ge 0\) and \(\langle u,w \rangle >0\). This condition is preserved under \(\varphi \). So the claim holds. \(\square \)

Lemma 2.12

Let \(v=(r,\xi ,a) \in H^*(X,\mathbb{Z })_{\mathrm{alg }}\) be a Mukai vector with \(\langle v^2 \rangle >0\) and \(r \ne 0\). Let \(v_1=(r_1,\xi _1,a_1)\) be a Mukai vector such that \(r \xi _1-r_1 \xi =0\). Then \(\mathbb{R }Z_{(\beta ,\omega )}(v)=\mathbb{R }Z_{(\beta ,\omega )}(v_1)\) if and only if \(d_{\beta ,\omega }(v)=0\). In particular, if \(v_1\) defines a wall, then the defining equation of the wall \(W_{v_1}\) for \(v\) is \(d_{\beta ,\omega }(v)=0\).

Proof

We have \(r v_1-r_1 v=b \varrho _X, 0 \ne b \in \mathbb{Z }\). Hence \(rZ_{(\beta ,\omega )}(v_1)-r_1 Z_{(\beta ,\omega )}(v)= -b \in \mathbb{R }\). Then we have \(Z_{(\beta ,\omega )}(v_1)=\frac{r_1}{r}Z_{(\beta ,\omega )}(v)-\frac{b}{r}\). Hence the condition is \(d_\beta (v)=0\). \(\square \)

Definition 2.13

For Mukai vectors \(v,v_1\), we set
$$\begin{aligned}&\Sigma _{(\beta ,\omega )}(v,v_1):\\&\quad = (\omega ,H) \left( (r_\beta (v)d_\beta (v_1)-r_\beta (v_1)d_\beta (v)) \frac{(\omega ^2)}{2} -(a_\beta (v)d_\beta (v_1)-a_\beta (v_1)d_\beta (v))\right) , \end{aligned}$$
where \(\omega \in \mathbb{R }_{>0}H\).

We quote the following results which are frequently used in this paper.

Lemma 2.14

[7, Rem. 1.3.5, Lem. 2.1.7]
(1)

\(\phi _{(\beta ,\omega )}(v_1)-\phi _{(\beta ,\omega )}(v) \mod 2\mathbb{Z }\) satisfies \(1>\phi _{(\beta ,\omega )}(v_1)-\phi _{(\beta ,\omega )}(v)>0\) if and only if \(\Sigma _{(\beta ,\omega )}(v,v_1)>0\).

(2)
\(\mathbb{R }Z_{(\beta ,\omega )}(v)=\mathbb{R }Z_{(\beta ,\omega )}(v_1)\) if and only if
$$\begin{aligned} (r_\beta (v_1)d_\beta (v)-r_\beta (v)d_\beta (v_1))\frac{(\omega ^2)}{2} =(a_\beta (v_1)d_\beta (v)-a_\beta (v)d_\beta (v_1)). \end{aligned}$$
(2.10)

3 Fourier–Mukai transforms

3.1 Stability conditions and Fourier–Mukai transforms

We shall recall the (twisted) Fourier–Mukai transform of \(\sigma _{(\beta ,\omega )}\) and its relation to Bridgeland’s stability conditions explained in [8]. Let \(\Phi :\mathbf{D}(X) \rightarrow \mathbf{D}^{\alpha _1}(X_1)\) be a twisted Fourier–Mukai transform such that \(\Phi (r_1 e^\gamma )=-\varrho _{X_1}\) and \(\Phi (\varrho _X)=-r_1 e^{\gamma '}\), where \(\alpha _1\) is a representative of a suitable Brauer class. Then we can describe the cohomological Fourier–Mukai transform as
$$\begin{aligned} \Phi (r e^\gamma +a \varrho _X+\xi +(\xi ,\gamma )\varrho _X) =-\frac{r}{r_1} \varrho _{X_1}-r_1 a e^{\gamma '}+ \frac{r_1}{|r_1|} (\widehat{\xi }+(\widehat{\xi },\gamma ')\varrho _{X_1}), \end{aligned}$$
where \(\xi \in \mathrm{NS }(X)_\mathbb{Q }\) and \( \widehat{\xi }:= \frac{r_1}{|r_1|} c_1(\Phi (\xi +(\xi ,\gamma )\varrho _X)) \in \mathrm{NS }(X_1)_\mathbb{Q }\).

Remark 3.1

By taking a locally free \(\alpha _1\)-twisted stable sheaf \(G\) with \(\chi (G,G)=0\), we have a notion of Mukai vector, thus, we have a map [7, Rem. 1.2.10]:
$$\begin{aligned} v_G:\mathbf{D}^{\alpha _1}(X_1) \rightarrow H^*(X_1,\mathbb{Q })_{\mathrm{alg }}. \end{aligned}$$
We set
$$\begin{aligned} \widetilde{\omega }:&= -\frac{1}{|r_1|} \frac{\frac{((\beta -\gamma )^2)-(\omega ^2)}{2}\widehat{\omega }- (\beta -\gamma ,\omega )(\widehat{\beta }-\widehat{\gamma })}{\left( \frac{((\beta -\gamma )^2)-(\omega ^2)}{2} \right) ^2 +(\beta -\gamma ,\omega )^2},\nonumber \\ \widetilde{\beta }:&= \gamma '-\frac{1}{|r_1|} \frac{\frac{((\beta -\gamma )^2)-(\omega ^2)}{2}(\widehat{\beta }-\widehat{\gamma }) -(\beta -\gamma ,\omega ) \widehat{\omega }}{\left( \frac{((\beta -\gamma )^2)-(\omega ^2)}{2} \right) ^2+(\beta -\gamma ,\omega )^2}. \end{aligned}$$
(3.1)
By [8, sect. 5.1], we get the following commutative diagram:
https://static-content.springer.com/image/art%3A10.1007%2Fs00209-013-1214-1/MediaObjects/209_2013_1214_Equ12_HTML.gif
(3.2)
where
$$\begin{aligned} \zeta =-r_1 \left( \frac{((\gamma -\beta )^2)-(\omega ^2)}{2} +\sqrt{-1}(\beta -\gamma ,\omega ) \right) . \end{aligned}$$
(3.3)

Theorem 3.2

(cf. [8, Thm. 3.8]) Assume that \(\mathbb{R }Z_{(\beta ,\omega )}(v)= \mathbb{R }Z_{(\beta ,\omega )}(e^{\gamma })\) and \(d_\beta (v)>0\). If \(r_1 e^\gamma \) does not define a wall, then \(\Phi \) induces an isomorphism
$$\begin{aligned} \mathcal{M }_{(\beta ,\omega )}(v) \rightarrow \mathcal{M }_{\widetilde{\omega }}^{\gamma '}(u)^{ss}, \end{aligned}$$
where \(u:=\Phi (v)\).

Remark 3.3

Since \(r_1 e^\gamma \) does not define a wall, \(\mathcal{M }_{\widetilde{\omega }}^{\gamma '}(u)^{ss}\) does not depend on the choice of the parameter \(\gamma '\). Thus we may denote \(\mathcal{M }_{\widetilde{\omega }}^{\gamma '}(u)^{ss}\) by \(\mathcal{M }_{\widetilde{\omega }}^{\alpha _1}(u)^{ss}\) or by \(\mathcal{M }_{\widetilde{\omega }}(u)^{ss}\).

As an application of Theorem 3.2, we give a different proof of Proposition 2.9.

Proof

We set \(v:=(r,\xi ,a)\). Assume that \((\beta _0,\omega _0) \in \mathcal{C }, \beta _0 \in \mathrm{NS }(X)_\mathbb{Q }\) and \(\omega _0 \in \mathbb{R }_{>0} H\) with \(H \in \mathrm{Amp }(X)_\mathbb{Q }\). We shall show that \(\sigma _{(\beta ,\omega )}\)-semi-stability is independent of \((\beta ,\omega )\) in a neighborhood of \((\beta _0,\omega _0)\). We first assume that \(d_{\beta _0,H}(v) \ne 0\). Replacing \(v\) by \(-v\), we may assume that \(d_{\beta _0,H}(v)> 0\). We take \(\delta \in \mathrm{NS }(X)_\mathbb{Q }\) such that \((\delta ,H)=1\). Then \(\{ L \in \mathrm{Amp }(X)_\mathbb{R } \mid (L,\delta )>0 \}\) is an open neighborhood of \(H\). We set \(H(\delta ):=\{L \in \mathrm{Amp }(X)_\mathbb{R } \mid (\delta ,L)=1 \}\). Then we have a decomposition
$$\begin{aligned} \{ L \in \mathrm{Amp }(X)_\mathbb{R } \mid (L,\delta )>0 \}= H(\delta ) \times \mathbb{R }_{>0} \end{aligned}$$
by the correspondence \(L \mapsto ((L,\delta )^{-1}L, (L,\delta )) \in H(\delta ) \times \mathbb{R }_{>0}\). For a fixed \(L \in H(\delta )_\mathbb{Q }\) and a fixed \(b \in \mathbb{Q }\), the semi-stability is independent of \((\beta ,\omega ) \in \mathcal{C } \cap \mathfrak{H }_\mathbb{R }^b\), where \((\beta ,\omega )=(bL+\eta ,tL)\) with \(\eta \in L^\perp \) and \(t>0\). In order to prove Proposition 2.9, it is essential to study the deformation of the parameter \((\beta ,\omega )\) in \(\mathrm{NS }(X)_\mathbb{R } \times H(\delta )\).

We can take \(\gamma \in \mathbb{R }_{>0} H+\beta _0 \subset \mathrm{NS }(X)_\mathbb{R }\) such that \(\mathbb{R } Z_{(\beta _0,\omega _0)}(v)= \mathbb{R } Z_{(\beta _0,\omega _0)}(e^\gamma )\) and \((r \gamma -\xi ,H) \ne 0\) [8, sect. 5.1]. Replacing \(\omega _0\) if necessary, we may assume that \(\gamma \in \mathrm{NS }(X)_\mathbb{Q }\). We take a small open neighborhood \(U\) of \(H(\delta )\) and \(I \subset \mathbb{R }_{>0}\) such that \(\omega _0 \in \{tL \mid L \in U,\; t \in I \}\).

Replacing \(U\) if necessary, we find an open neighborhood \(V\) of \(\beta _0\) in \(\mathrm{NS }(X)_\mathbb{R }\) such that
$$\begin{aligned} \frac{(\langle e^\gamma ,e^\beta \rangle (\xi -r \beta )- \langle v,e^\beta \rangle (\gamma -\beta ),L)}{ (r \gamma - \xi ,L)}>0 \end{aligned}$$
(3.4)
for all \((\beta ,L) \in V \times U\). Then we have a continuous function \(t:=t(\beta ,L)\) on \(V \times U\) satisfying \(t>0\) and
$$\begin{aligned} t^2 \frac{(L^2)}{2}= \frac{(\langle e^\gamma ,e^\beta \rangle (\xi -r \beta )- \langle v,e^\beta \rangle (\gamma -\beta ),L)}{ (r \gamma - \xi ,L)}. \end{aligned}$$
We set \(\omega :=tL\). \(\omega \) is a function on \(V \times U\) and we have \(\mathbb{R }Z_{(\beta ,\omega )}(v) =\mathbb{R }Z_{(\beta ,\omega )}(e^\gamma )\) by Lemma 2.14. For the proof of our claim, it is sufficient to show the independence of \(\sigma _{(\beta ,\omega )}\)-semi-stability, where \((\beta ,L) \in V \times U\).

We set \(X_1:=M_{H}(r_1 e^\gamma )\). For a universal family \(\mathbf{E}\) on \(X \times X_1\) as a twisted object, we consider the Fourier–Mukai transform \(\Phi _{X \rightarrow X_1}^{\mathbf{E}^{\vee }[1]}\). Then we have an isomorphism \(\mathcal{M }_{(\beta ,\omega )}(v) \rightarrow \mathcal{M }_{\widetilde{\omega }}(u)^{ss}\), where \(u:=\Phi _{X \rightarrow X_1}^{\mathbf{E}^{\vee }[1]}(v)\). Since \((c_1(u e^{-\widetilde{\beta }}),\widetilde{\omega })=0\) for all \((\beta ,L) \in V \times U\), if \(F \in \mathcal{M }_{\widetilde{\omega }}(u)^{ss}\) contains a subsheaf \(F_1\) with \((c_1(F_1(-\widetilde{\beta })),\widetilde{\omega })=0\) and \(v(F_1) \not \in \mathbb{Q }u\), then \(\Phi _{X_1 \rightarrow X}^{\mathbf{E}[1]}(F_1)\) defines a wall for \(v\). Therefore for any subsheaf \(F_1\) of \(F \in \mathcal{M }_{\widetilde{\omega }}(u)^{ss}, (c_1(F_1(-\widetilde{\beta })),\widetilde{\omega }) \ne 0\) for any \((\beta ,L) \in V \times U\) or we have \(v(F_1) \in \mathbb{Q }u\). For \((\beta _i,L_i) \in V \times U\) (\(i=1,2\)), we set \(\omega _i:=t(\beta _i,L_i)L_i\). For \(F \in \mathcal{M }_{\widetilde{\omega _1}}(u)^{ss} {\setminus } \mathcal{M }_{\widetilde{\omega _2}}(u)^{ss}\), there is a subsheaf \(F_1\) such that \((c_1(F_1(-\widetilde{\beta })),\widetilde{\omega _2}) \ge 0\) and \((c_1(F_1(-\widetilde{\beta })),\widetilde{\omega _1}) \le 0\). Since \((c_1(F_1(-\widetilde{\beta })),\widetilde{\omega })\) is a continuous function on \(V \times U\), it is a contradiction. Therefore \(\mathcal{M }_{\widetilde{\omega }}(u)^{ss}\) is independent of \((\beta ,L)\). Then we see that \(\mathcal{M }_{(\beta ,\omega )}(v)\) is independent of \((\beta ,\omega )\).

We next assume that \(d_{\beta _0,H}(v)=0\). Replacing \(v\) by \(-v\), we may assume that \(\mathrm{rk }v>0\). Since \(Z_{(\beta _0,\omega _0)}(v) \in \mathbb{R }_{>0}\), we may set \(\phi _{(\beta _0,\omega _0)}(v)=0\). Since \((\beta _0,\omega _0)\) is in a chamber, \(\mathcal{M }_{(\beta _0,\omega _0)}(v)=\mathcal{M }_{\omega _0}^{\beta _0}(v)^{ss}\) such that \(\omega _0\) is a general polarization in the sense of \(\mu \)-semi-stability of torsion free sheaves and \(\mathcal{M }_{\omega _0}^{\beta _0}(v)^{ss}\) consists of vector bundles. We write \((\beta ,\omega )=(sL+\eta ,tL)\), where \(L \in H(\delta )\) and \(\eta \in L^\perp \). Similarly we write \((\beta _0,\omega _0)=(s_0 L_0+\eta _0,t_0 L_0)\). We define a continuous function \(s(L):=\frac{(\xi ,L)}{r(L,L)}\) on \(H(\delta )\). Then \((\xi -r\beta ,\omega )=0\) if and only if \(s=s(L)\). We take an open neighborhood \(U_1\) of \(s_0\) in \(\mathbb{R }\) and an open neighborhood \(U_2\) of \((L_0,\eta _0)\) in \(\{(L,\eta ) \mid L \in H(\delta ), \eta \in H^\perp \}\) such that \((sL+\eta ,t_0 L) \in \mathcal{C }\) for \((s,L,\eta ) \in U_1 \times U_2\). We may assume that \(U_1\) is connected and \(s(L) \in U_1\) for \((L,\eta ) \in U_2\). Then \((sL+\eta ,t L) \in \mathcal{C }\) for all \((s,L,\eta ) \in U_1 \times U_2\) and \(t \ge t_0\). Indeed if \((s' L+\eta ,t' L) \in W_{v_1}\) for a \(t' > t_0\), then we can show that \(C:=W_{v_1} \cap \{(sL+\eta ,t L) \mid s \in \mathbb{R }, t \ge 0 \}\) is a semi-circle which does not intersect with \(\{(sL+\eta ,t L) \mid s=s(L) \}\) (see Proposition 5.6 and Lemma 5.3). Then \(C\) intersects with the line \(\{(s L+\eta ,t_0 L) \mid s \in \mathbb{R } \}\) between \(s'\) and \(s(L)\), which is a contradiction. Hence for \(E \in \mathcal{M }_{(sL+\eta ,t L)}(v), E\) or \(E^{\vee }\) is Gieseker semi-stable with respect to \(L\) by [7, Cor. 2.2.9]. Since \(L\) is a general polarization in a neighborhood of \(L_0, \mathcal{M }_{(sL+\eta ,t L)}(v)= \mathcal{M }_{L}(v)^{ss}= \mathcal{M }_{\omega _0}^{\beta _0}(v)^{ss}\) in a neighborhood of \((\beta _0,\omega _0)\).\(\square \)

If \(d_{\beta _0,\omega _0}(v)=0\), then we showed that a \(\sigma _{(\beta ,\omega )}\)-semi-stable object \(E\) with \(v(E)=v\) is \(\beta \)-twisted semi-stable with respect to \(\omega _0\) in a neighborhood of \((\beta _0,\omega _0)\). We can show a weaker result by a more direct method.

Lemma 3.4

Let \(\mathcal{C }\) be a chamber and set \(\mathcal{C }^+:=\{(\beta ,\omega ) \in \mathcal{C } \mid d_{\beta ,\omega }(v) >0 \}\). Assume that \((\beta _0,\omega _0) \in \overline{\mathcal{C }^+}\) and \(d_{\beta _0,\omega _0}(v)=0\). Let \(E\) be a \(\sigma _{(\beta ,\omega )}\)-semi-stable object with \(v(E)=v\) and \(\phi _{(\beta ,\omega )}(E) \in (0,1)\) for a point \((\beta ,\omega ) \in \mathcal{C }^+\). Then
(1)

If \(\mathrm{rk }v>0\), then \(E\) is a \(\mu \)-semi-stable sheaf with respect to \(\omega _0\).

(2)

If \(\mathrm{rk }v<0\), then \(E^{\vee }[1]\) is a \(\mu \)-semi-stable sheaf with respect to \(\omega _0\).

Proof

Since \(E\) is \(\sigma _{(\beta ,\omega )}\)-semi-stable with \(\phi _{(\beta ,\omega )}(E) \in (0,1)\) for all \((\beta ,\omega ) \in \mathcal{C }^+, H^{-1}(E)\) is locally free (see [7, Lem. 2.2.3]). If \(H^{-1}(E)\) contains a \(\mu \)-stable subsheaf \(E_1\) such that \((c_1(E_1(-\beta _0)),\omega _0)/\mathrm{rk }E_1>0\), then there is \((\beta ,\omega ) \in \mathcal{C }^+\) such that \((c_1(E_1(\!-\!\beta )),\omega )/\mathrm{rk }E_1\!>\!0\), which implies \(H^{-1}(E) \not \in \mathfrak{F }_{(\beta ,\omega )}\). Therefore \(H^{-1}(E) \in \mathfrak{F }_{(\beta _0,\omega _0)}\).

If \(H^0(E)\) contains a \(\mu \)-stable quotient sheaf \(E_1\) such that \((c_1(E_1(-\beta _0)),\omega _0)/\mathrm{rk }E_1 < 0\), then there is \((\beta ,\omega ) \in \mathcal{C }^+\) such that \((c_1(E_1(-\beta )),\omega )/\mathrm{rk }E_1< 0\), which implies \(H^0(E) \not \in \mathfrak{T }_{(\beta ,\omega )}\). Therefore \(H^0(E)\) is generated by objects in \(\mathfrak{T }_{(\beta _0,\omega _0)}\) and \(\mu \)-semi-stable sheaves \(E_1\) with \((c_1(E_1(-\beta _0)),\omega _0)=0\). Since \(d_{\beta _0,\omega _0}(E) =d_{\beta _0,\omega _0}(H^{-1}(E)[1])+d_{\beta _0,\omega _0}(H^0(E))\), we have \(d_{\beta _0,\omega _0}(H^{-1}(E)[1]) =d_{\beta _0,\omega _0}(H^0(E))=0, H^{-1}(E)\) is a \(\mu \)-semi-stable locally free sheaf with \((c_1(H^{-1}(E)(-\beta _0)),\omega _0)=0\) and \(H^0(E)\) is an extension of a \(\mu \)-semi-stable torsion free sheaf \(F_2\) with \((c_1(F_2(-\beta _0)),\omega _0)=0\) by a 0-dimensional sheaf \(T\).

We have an exact sequence
$$\begin{aligned} 0 \rightarrow F_1 \rightarrow E \rightarrow F_2 \rightarrow 0 \end{aligned}$$
(3.5)
such that \(F_1\) fits in an exact sequence
$$\begin{aligned} 0 \rightarrow H^{-1}(E)[1] \rightarrow F_1 \rightarrow T \rightarrow 0. \end{aligned}$$
(3.6)
Since \(Z_{(\beta _0,\omega _0)}(F_1) \in \mathbb{R }_{<0}\) and \(Z_{(\beta _0,\omega _0)}(F_2) \in \mathbb{R }_{>0}, \sigma _{(\beta ,\omega )}\)-semi-stability of \(E\) on \(\mathcal{C }^+\) implies that \(F_1=0\) or \(F_2=0\). If \(F_1=0\), then \(\mathrm{rk }E>0\) and the claim (1) holds. If \(F_2=0\), then \(\mathrm{rk }E<0\) and \(E^{\vee }[1]\) is a \(\mu \)-semi-stable torsion free sheaf. \(\square \)

We also have a similar claim for \(\mathcal{C }^-:=\{(\beta ,\omega ) \in \mathcal{C } \mid d_{\beta ,\omega }(v) <0 \}\).

Definition 3.5

Let \(\sigma _{(\beta ,\omega )}\) be a stability condition. For the contravariant Fourier–Mukai transform \(\Phi \circ \mathcal{D }_X\), we set \((\beta ',\omega '):=(-\beta ,\omega )\) and attach the stability condition \(\sigma _{(\widetilde{\beta '},\widetilde{\omega '})}\) associated to \(Z_{(\widetilde{\beta '},\widetilde{\omega '})}\). We call \(\sigma _{(\widetilde{\beta '},\widetilde{\omega '})}\) the stability condition induced by \(\Phi \circ \mathcal{D }_X\).

Lemma 3.6

\(\phi _{(-\beta ,\omega )}(E^{\vee }[1])=-\phi _{(\beta ,\omega )}(E)+1\).

Proof

For a non-zero object \(E \in \mathbf{D}(X)\), we have
$$\begin{aligned} Z_{(-\beta ,\omega )}(E^{\vee }[1])&= -\langle e^{-\beta +\omega \sqrt{-1}},v(E^{\vee }) \rangle \nonumber \\&= -\overline{\langle e^{\beta +\omega \sqrt{-1}},v(E) \rangle } =|Z_{(\beta ,\omega )}(v)|e^{\pi \sqrt{-1} (1-\phi _{(\beta ,\omega )}(E))}. \end{aligned}$$
(3.7)
Here \(\overline{x}\) means the complex conjugate of \(x \in \mathbb{C }\). Hence \(\phi _{(-\beta ,\omega )}(E^{\vee }[1])=-\phi _{(\beta ,\omega )}(E)+1 \mod 2\mathbb{Z }\). Since \(\phi _{(-\beta ,\omega )}((E[n])^{\vee }[1])= \phi _{(-\beta ,\omega )}(E^{\vee }[1])-n\), we shall show that \(\phi _{(-\beta ,\omega )}(E^{\vee }[1]) \in [0,1)\) for \(E \in \mathfrak{A }_{(\beta ,\omega )}\).
We note that \(\mathfrak{A }_{(\beta ,\omega )}\) is generated by (i) 0-dimensional object \(T\), (ii) \(F[1]\) where \(F\) is a locally free \(\mu \)-semi-stable sheaf with \(d_\beta (F) \le 0\), (iii) \(\mu \)-semi-stable sheaf \(E\) with \(d_\beta (E)>0\) and (iv) purely 1-dimensional sheaf \(E\).
  1. (i)

    For a 0-dimensional sheaf \(T, T^{\vee }[1] \in \mathfrak{A }_{(-\beta ,\omega )}[-1]\). Thus \(\phi _{(-\beta ,\omega )}(T^{\vee }[1])=0\). (ii) For a locally free \(\mu \)-semi-stable sheaf \(F\) with \(d_\beta (F) \le 0\), \((F[1])^{\vee }[1]=F^{\vee }\) is a \(\mu \)-semi-stable sheaf with \(d_{-\beta }((F[1])^{\vee }[1]) \ge 0\). Hence \(\phi _{(-\beta ,\omega )}((F[1])^{\vee }[1]) \in [0,1)\). (iii) Let \(E\) be a \(\mu \)-semi-stable sheaf of \(\mathrm{rk }E>0\) and \(d_\beta (E)>0\). Let \(E^{**}\) be the reflexive hull of \(E\) and set \(T:=E^{**}/E\). Then we have an exact triangle

     
$$\begin{aligned} T^{\vee }[1] \rightarrow (E^{**})^{\vee }[1] \rightarrow E^{\vee }[1] \rightarrow T^{\vee }[2]. \end{aligned}$$
Since \((E^{**})^{\vee }\) is a locally free \(\mu \)-semi-stable sheaf with \(d_{-\beta }((E^{**})^{\vee })=-d_\beta (E)<0\) and \(T^{\vee }[2]\) is a 0-dimensional sheaf, \(E^{\vee }[1] \in \mathfrak{A }_{(-\beta ,\omega )}\). (iv) If \(E\) is a purely 1-dimensional sheaf, then \(E^{\vee }[1]\) is a purely 1-dimensional sheaf, which implies \(E^{\vee }[1] \in \mathfrak{A }_{(-\beta ,\omega )}\). Therefore the claim holds. \(\square \)

3.2 Relations of moduli spaces under Fourier–Mukai transforms

We shall say that a pair \((\beta ,\omega )\) is general with respect to\(v\) if \((\beta ,\omega )\) is not on any wall \(W_{v_1}\) for \(v\).

Let \(\Phi :\mathbf{D}(X) \rightarrow \mathbf{D}(Y)\) be a Fourier–Mukai transform. The following is due to Bridgeland.

Proposition 3.7

(cf. [3, Prop. 10.3]) We have a subgroup \(\mathbb{C } \subset \widetilde{\mathrm{GL }}^+(2,\mathbb{R })\). By using the action of \(\mathbb{C }\) on the space of stability conditions, we have
$$\begin{aligned} \sigma _{(\widetilde{\beta },\widetilde{\omega })} \circ \Phi =\lambda \cdot \sigma _{(\beta ,\omega )}, \end{aligned}$$
where \(\lambda \in \mathbb{C }\) satisfies \(\exp (-\pi \sqrt{-1} \lambda )=\zeta ^{-1}\) in (3.2) and \(\phi _{(\widetilde{\beta },\widetilde{\omega })}(\Phi (E))= \phi _{(\beta ,\omega )}(E)-\mathrm Re (\lambda )\).

Theorem 3.8

Let \(v\) be a Mukai vector with \(\langle v^2 \rangle >0\). Assume that \((\beta ,\omega )\) is general with respect to \(v\).
(1)
Any Fourier–Mukai transform \(\Phi \) preserves Bridgeland’s stability condition. Thus \(\Phi \) induces an isomorphism
$$\begin{aligned} M_{(\beta ,\omega )}(v) \rightarrow M_{(\widetilde{\beta },\widetilde{\omega })}(u), \end{aligned}$$
(3.8)
where \(u=\Phi (v)\).
(2)
Any contravariant Fourier–Mukai transform \(\Psi :=[1] \circ \Phi \circ \mathcal{D }_X\) preserves Bridgeland’s stability condition. Thus \(\Psi \) induces an isomorphism
$$\begin{aligned} M_{(\beta ,\omega )}(v) \rightarrow M_{(-\beta ,\omega )}(-v^{\vee }) \rightarrow M_{(\widetilde{\beta '},\widetilde{\omega '})}(u), \end{aligned}$$
(3.9)
where \(u=\Psi (v)\) and \((\beta ',\omega ')=(-\beta ,\omega )\).

The first claim is a consequence of Proposition 3.7. For the proof of (2), it is sufficient to prove the following claim.

Proposition 3.9

Let \(v\) be a Mukai vector with \(\langle v^2 \rangle >0\). Assume that \((\beta ,\omega )\) is general with respect to \(v\). If \(Z_{(\beta ,\omega )}(v) \in \mathbb{H } \cup \mathbb{R }_{<0}\), then we have an isomorphism
$$\begin{aligned} M_{(\beta ,\omega )}(v)&\rightarrow M_{(-\beta ,\omega )}(-v^{\vee })\\ E&\mapsto E^{\vee }[1]. \end{aligned}$$

Proof

Let \(\mathcal{C }\) be a chamber containing \((\beta ,\omega )\). We may assume that \(d_{\beta ,\omega }(v)>0\). Replacing \((\beta ,\omega ) \in \mathcal{C }\) if necessary, we can take \(\gamma \in \mathrm{NS }(X)_\mathbb{Q }\) such that \(\mathbb{R }Z_{(\beta ,\omega )}(e^\gamma )=\mathbb{R }Z_{(\beta ,\omega )}(v)\). We take a primitive vector \(r_1 e^\gamma \) such that \(d_{\beta ,\omega }(r_1 e^\gamma )>0\). We set \(X_1:=M_{(\beta ,\omega )}(r_1 e^{\gamma })\). Let \(\mathbf{E}\) be the universal object on \(X \times X_1\) as a complex of twisted sheaves. We set \(w:=\Phi _{X \rightarrow X_1}^{\mathbf{E}^{\vee }[1]}(v)\). By Theorem 3.2, \(M_{\widetilde{\omega }}^{\alpha _1}(w)\) consists of \(\mu \)-semi-stable locally free sheaves, \(\widetilde{\omega }\) is general with respect to \(w\) and we have an isomorphism
$$\begin{aligned} \Phi _{X \rightarrow X_1}^{\mathbf{E}^{\vee }[1]}: M_{(\beta ,\omega )}(v) \rightarrow M_{\widetilde{\omega }}^{\alpha _1}(w), \end{aligned}$$
where \(w\) satisfies \((c_1(w e^{-\widetilde{\beta }}),\widetilde{\omega })=0\). By taking the dual, we have an isomorphism
$$\begin{aligned} M_{\widetilde{\omega }}^{\alpha _1}(w) \rightarrow M_{\widetilde{\omega }}^{-\alpha _1}(w^{\vee }). \end{aligned}$$
We note that \(\mathbf{F}:=\mathbf{E}^{\vee }[1]\) is a family of stable objects with \(v(\mathbf{F}_{|X \times \{x_1 \}})\!=\!-r_1 e^{-\gamma }\). For \(\Phi _{X \rightarrow X_1}^{\mathbf{F}^{\vee }[1]}\) we define \((\widetilde{-\beta },\widetilde{\omega })\) by (3.1). Thus we substitute \((-\gamma ,-\gamma ',-\beta )\) in (3.1) instead of \((\gamma ,\gamma ',\beta )\) for the definition of \((\widetilde{-\beta },\widetilde{\omega })\). Then we have \( (\widetilde{-\beta },\widetilde{\omega })= (-\widetilde{\beta },\widetilde{\omega })\). Since \(\mathbb{R }Z_{(-\beta ,\omega )}(-r_1 e^{-\gamma })= \mathbb{R }Z_{(-\beta ,\omega )}(-v^{\vee })\), we have an isomorphism
$$\begin{aligned} \Phi _{X \rightarrow X_1}^{\mathbf{F}^{\vee }[1]}: M_{(-\beta ,\omega )}(-v^{\vee }) \rightarrow M_{\widetilde{\omega }}^{-\alpha _1}(w^{\vee }). \end{aligned}$$
(3.10)
By the Grothendieck–Serre duality, we have
$$\begin{aligned} \Phi _{X \rightarrow X_1}^{\mathbf{F}^{\vee }[1]}(E^{\vee }[1])= \Phi _{X \rightarrow X_1}^{\mathbf{E}}(E^{\vee }[1])= D_{X_1} \circ \Phi _{X \rightarrow X_1}^{\mathbf{E}^{\vee }[1]}(E). \end{aligned}$$
Hence the claim holds.\(\square \)
Let \(\mathfrak{k }_y\) be the structure sheaf of a point \(y \in Y\). Since
$$\begin{aligned} \phi _{(\widetilde{\beta },\widetilde{\omega })}(\Phi (E))- \phi _{(\widetilde{\beta },\widetilde{\omega })}(\mathfrak{k }_y[-1])= \phi _{(\beta ,\omega )}(E)- \phi _{(\beta ,\omega )}(\Phi ^{-1}(\mathfrak{k }_y[-1])) \end{aligned}$$
(3.11)
and \(\phi _{(\widetilde{\beta },\widetilde{\omega })}(\mathfrak{k }_y[-1])=0\), we get the following corollary.

Corollary 3.10

$$\begin{aligned} \phi _{(\widetilde{\beta },\widetilde{\omega })}(\Phi (E))= \phi _{(\beta ,\omega )}(E)- \phi _{(\beta ,\omega )}(\Phi ^{-1}(\mathfrak{k }_y[-1])) \end{aligned}$$
(3.12)
for \(\sigma _{(\beta ,\omega )}\)-semi-stable objects \(E\).

Remark 3.11

By a similar argument, we can prove Theorem 3.8 (1).

Proof of Theorem 3.8 (1)

Although the following proof also works for the twisted case, we write the proof for the untwisted case. Let \(\mathcal{C }\) be a chamber containing \((\beta ,\omega )\). We may assume that \(d_{\beta ,\omega }(v)>0\). We write \(\omega =tH\) with \(H \in \mathrm{NS }(X), t>0\). Replacing \(t\) if necessary, we can take \(\gamma :=\beta +\lambda H \in \mathrm{NS }(X)_\mathbb{Q }\) such that \(\mathbb{R }Z_{(\beta ,\omega )}(e^\gamma )=\mathbb{R }Z_{(\beta ,\omega )}(v)\) and \((\beta ,\omega ) \in \mathcal{C }\). We take a primitive vector \(r_1 e^\gamma \) such that \(d_{\beta ,\omega }(r_1 e^\gamma )>0\). For any Fourier–Mukai transform \(\Phi :\mathbf{D}(X) \rightarrow \mathbf{D}(Y)\), let \(v_0 \in \mathbb{Q }\Phi (e^{\gamma })\) be a primitive and positive Mukai vector. Since \(\mathrm{rk }\Phi (e^{\gamma })= -\langle e^\gamma , \Phi ^{-1}(\varrho _Y) \rangle \) by perturbing \(\lambda \) if necessary, we may assume that \(\mathrm{rk }v_0 \ne 0\). We set
$$\begin{aligned} \Phi (e^{\beta +\sqrt{-1} \omega })= -\langle e^{\beta +\sqrt{-1} \omega }, \Phi ^{-1}(\varrho _{Y}) \rangle e^{\widetilde{\beta }+\sqrt{-1} \widetilde{\omega }}, \end{aligned}$$
(3.13)
which gives a commutative diagram similar to (3.2). Then \(X_1 \cong M_{\widetilde{\omega }}(v_0)\) and the universal object \(\mathbf{G}\) on \(X_1 \times Y\) satisfies \(\Phi _{X_1 \rightarrow Y}^{\mathbf{G}[n]}= \Phi \circ \Phi _{X_1 \rightarrow X}^{\mathbf{E}}, n \in \mathbb{Z }\). Since \(\mathbb{R }Z_{(\beta ,\omega )}(e^\gamma ) =\mathbb{R }Z_{(\beta ,\omega )}(v)\), we have \(\mathbb{R }Z_{(\widetilde{\beta },\widetilde{\omega })}(v_0) =\mathbb{R }Z_{(\widetilde{\beta },\widetilde{\omega })}(\Phi (v))\). Then \(\Phi _{X_1 \rightarrow Y}^{\mathbf{G}[n]}\) induces an isomorphism
$$\begin{aligned} M_{H_1}^{\alpha _1}(w) \rightarrow M_{(\widetilde{\beta },\widetilde{\omega })}(\Phi (v)), \end{aligned}$$
where \(H_1\) is the polarization of \(X_1\) which was denoted by \(\widetilde{\omega }\) in Theorem 3.2. Hence we have an isomorphism
$$\begin{aligned} \Phi :M_{(\beta ,\omega )}(v) \rightarrow M_{(\widetilde{\beta },\widetilde{\omega })}(\Phi (v)). \end{aligned}$$
\(\square \)

Remark 3.12

(3.12) also follows from the commutative diagram (3.2) and the preservation of Bridgeland’s stability: Let \(\mathbf{F}\) be a complex such that \(\Phi =\Phi _{X \rightarrow Y}^{\mathbf{F}^{\vee }[1]}\). We note that
$$\begin{aligned} \phi _{(\widetilde{\beta },\widetilde{\omega })}(\Phi (E)) \equiv \phi _{(\beta ,\omega )}(E)+m \mod 2 \end{aligned}$$
by (3.2), where \(\zeta =\exp (\pi \sqrt{-1}m)\). For a \(\sigma _{(\beta ,\omega )}\)-semi-stable object \(E \in \mathbf{D}(X)\) with \(\phi _{(\beta ,\omega )}(\mathbf{F}_{|X \times \{ y \}})+1> \phi _{(\beta ,\omega )}(E)> \phi _{(\beta ,\omega )}(\mathbf{E}_{|X \times \{ y \}})\), we see that \(H^i(\Phi (E))=0\) for \(i \ne -1,0\). Hence \(\Phi (E) \in \mathfrak{A }_{(\widetilde{\beta },\widetilde{\omega })}[1] \cup \mathfrak{A }_{(\widetilde{\beta },\widetilde{\omega })} \cup \mathfrak{A }_{(\widetilde{\beta },\widetilde{\omega })}[-1]\). In particular, \(\phi _{(\widetilde{\beta },\widetilde{\omega })}(\Phi (E)) \in (-1,2)\). Since \(\Phi (\mathbf{F}_{|X \times \{ y \}})=\mathfrak{k }_{y}[-1], \phi _{(\widetilde{\beta },\widetilde{\omega })} (\Phi (\mathbf{F}_{|X \times \{ y \}}))=0\). Then
$$\begin{aligned} \phi _{(\widetilde{\beta },\widetilde{\omega })}(\Phi (E))&= \phi _{(\widetilde{\beta },\widetilde{\omega })}(\Phi (E)) -\phi _{(\widetilde{\beta },\widetilde{\omega })} (\Phi (\mathbf{F}_{|X \times \{ y \}}))\nonumber \\&\equiv \phi _{(\beta ,\omega )}(E) - \phi _{(\beta ,\omega )}(\mathbf{F}_{|X \times \{ y \}}) \mod 2\mathbb{Z }. \end{aligned}$$
(3.14)
Hence \(\phi _{(\widetilde{\beta },\widetilde{\omega })}(\Phi (E)) \in (2k,2k+1)\). Therefore \(k=0\) and
$$\begin{aligned} \phi _{(\widetilde{\beta },\widetilde{\omega })}(\Phi (E))= \phi _{(\beta ,\omega )}(E)- \phi _{(\beta ,\omega )}(\mathbf{F}_{|X \times \{ y \}}). \end{aligned}$$
(3.15)

4 Numerical solutions and the walls

4.1 Semi-homogeneous presentations and numerical solutions

Let us recall the notion of semi-homogeneous presentations introduced in [14].

Definition 4.1

A semi-homogeneous presentation of \(E\in \mathrm{Coh }(X)\) is an exact sequence
$$\begin{aligned} 0\rightarrow E \rightarrow E_1 \rightarrow E_2 \rightarrow 0 \quad \text{ or }\quad 0\rightarrow E_1 \rightarrow E_2 \rightarrow E \rightarrow 0, \end{aligned}$$
where \(E_i\) (\(i=1,2\)) are semi-homogeneous sheaves satisfying the following condition: if we write \(v(E_i)=\ell _i v_i\) with \(\ell _i\) positive integers and \(v_i\) primitive Mukai vectors, then
$$\begin{aligned} (\ell _1-1)(\ell _2-1)=0,\quad \langle v_1^2 \rangle =\langle v_2^2 \rangle =0,\quad \langle v_1,v_2 \rangle =-1. \end{aligned}$$
We call the first sequence the kernel presentation and the second one the cokernel presentation.

Since semi-homogeneous sheaves on abelian varieties are well-known objects, semi-homogeneous presentations give useful information on \(E\). Moreover the property of having a semi-homogeneous presentation is an open condition, and describes the birational structure of the moduli spaces \(M_H(v)\). The numerical data appeared in Definition 4.1 are useful to find a semi-homogeneous presentation. So we introduce the following definition.

Definition 4.2

For a Mukai vector \(v\), the equation
$$\begin{aligned} v&= \pm (\ell _1 v_1-\ell _2 v_2),\\ \ell _1, \ell _2&\in \mathbb{Z }_{>0},\quad v_1, v_2:\text{ positive } \text{ primitive } \text{ Mukai } \text{ vectors }, \end{aligned}$$
is called the numerical equation of\(v\).
A solution \((v_1,v_2,\ell _1,\ell _2)\) of this equation satisfying
$$\begin{aligned} (\ell _1-1)(\ell _2-1)=0,\quad \langle v_1^2 \rangle =\langle v_2^2 \rangle =0,\quad \langle v_1,v_2 \rangle =-1, \end{aligned}$$
is called a numerical solution of \(v\).

Theorem 4.3

[14, Thm. 3.9] Suppose \(\mathrm{NS }(X)=\mathbb Z H\) and let \(v\) be a Mukai vector with \(\langle v^2 \rangle >0\).
(i)

If \(v\) has at least two numerical solutions, then a general member of \(M_H(v)\) has both kernel presentation and cokernel presentation. Each presentation is unique.

(ii)

If \(v\) has only one numerical solution, then a general member of \(M_H(v)\) has either kernel presentation or cokernel presentation. Such a presentation is unique.

For each numerical solution \((v_1,v_2,\ell _1,\ell _2)\) of \(v\), we constructed moduli spaces of simple two-term complexes. These moduli spaces plays an important role to prove [14, Thm. 3.9]. We fix an ample divisor \(H\) on \(X\).

Theorem 4.4

[14, Thm. 4.9] Let \(v\) be a positive Mukai vector with \(\langle v^2\rangle >0\) and \((v_1,v_2,\ell _1,\ell _2)\) be a numerical solution of \(v\).
(i)

We have the fine moduli scheme \(\mathfrak M ^{-}(v_1,v_2,\ell _1,\ell _2)\) of simple complexes \(V^{\bullet }\) such that \(H^i(V^{\bullet })=0\) (\(i \ne -1,0\)), \(H^{-1}(V^{\bullet }) \in \mathcal{M }_H(\ell _1 v_1)^{ss}\) and \(H^0(V^{\bullet }) \in \mathcal{M }_H(\ell _2 v_2)^{ss}\).

(ii)

We have the fine moduli scheme \(\mathfrak M ^+(v_1,v_2,\ell _1,\ell _2)\) of simple complexes \(V^{\bullet }\) such that \(V^{\bullet } \cong [W^{-1} \rightarrow W^0], W^{-1} \in \mathcal{M }_H(\ell _1 v_1)^{ss}\) and \(W^0 \in \mathcal{M }_H(\ell _2 v_2)^{ss}\).

Remark 4.5

Since \(\mathcal{M }_H(\ell _i v_i)^{ss}\) (\(i=1,2\)) are independent of the choice of \(H, \mathfrak M ^\pm (v_1,v_2, \ell _1,\ell _2)\) are independent of the choice of \(H\).

The relation of \(\mathfrak M ^\pm (v_1,v_2,\ell _1,\ell _2)\) is described as follows:

Proposition 4.6

[14, Prop. 4.11] Let \(v\) be a positive Mukai vector with \(\langle v^2\rangle >0\) and \((v_1,v_2,\ell _1,\ell _2)\) be a numerical solution of \(v\). We set \(\ell :=\langle v^2 \rangle /2\). For \(i=1,2\), we denote \(Y_i := M_H(v_i)\), and let \(\mathbf E _i\) be a universal family such that \(v(\Phi _{X \rightarrow Y_i}^\mathbf{E _i^{\vee }}(v_j))=(1,0,0)\) for \(j \ne i\). For \(V^{\bullet } \in \mathfrak M ^{+}(v_1,v_2,\ell _1,\ell _2)\), we set
$$\begin{aligned} \Psi (V^{\bullet }):= \left\{ \begin{array}{l@{\quad }l} \Phi _{Y_1 \rightarrow X}^{\mathbf{E }_1[2]}\,\mathcal D _{Y_1}\, \Phi _{X \rightarrow Y_1}^{\mathbf{E }_1^{\vee }}(V^{\bullet }) &{} (\ell _1,\ell _2)=(\ell ,1),\\ \Phi _{Y_2 \rightarrow X}^{\mathbf{E }_2[1]}\,\mathcal{D }_{Y_2}\, \Phi _{X \rightarrow Y_2}^{\mathbf{E }_2^{\vee }[1]}(V^{\bullet }) &{} (\ell _1,\ell _2)=(1,\ell ).\end{array}\right. \end{aligned}$$
Then \(\Psi \) induces an isomorphism
$$\begin{aligned} \mathfrak M ^{+}(v_1,v_2,\ell _1,\ell _2) \xrightarrow {\ \sim \ }\mathfrak M ^{-}(v_1,v_2,\ell _1,\ell _2). \end{aligned}$$

4.2 Relation with the walls

The operation \(\Psi \) is first introduced in [18] to construct birational maps of moduli spaces, and it is reformulated as an isomorphism of \(\mathfrak M ^\pm (v_1,v_2,\ell _1,\ell _2)\) in [14]. \(\Psi \) plays a fundamental role in these papers. In [7, sect. 4], we explained its relation with Bridgeland’s stability condition. In particular, we showed that \(\mathfrak M ^\pm (v_1,v_2,\ell _1,\ell _2)\) are moduli schemes of stable objects if \(v_1\) defines a wall. We shall slightly generalize this fact.

Lemma 4.7

There is a stability condition \(\sigma _{(\beta ,\omega )}=(\mathfrak{A }_{(\beta ,\omega )},Z_{(\beta ,\omega )})\) such that \(\mathfrak{M }^\pm (v_1,v_2, \ell _1,\ell _2)\) are the moduli schemes \(M_{(\beta ,\omega ^\pm )}(\pm v)\) of stable objects, where \(\omega ^\pm =t^\pm \omega , t^-<1<t^+, t^+-t^- \ll 1\).

Proof

Replacing \(v\) by \(-v\), we assume that \(v=\ell _2 v_2-\ell _1 v_1\). We set \(v=r e^{\gamma }-\frac{\ell }{r}\varrho _X\) and \(v_i=r_i e^{\gamma +\xi _i}\) (\(i=1,2\)). Then \(r=-(r_1 \ell _1-r_2 \ell _2), r_1 \ell _1 \xi _1=r_2 \ell _2 \xi _2\) and \(\frac{\ell }{r}= (r_1 \ell _1 \frac{(\xi _1^2)}{2}-r_2 \ell _2 \frac{(\xi _2^2)}{2})\). Since \(0>\langle e^{\gamma +\xi _1},e^{\gamma +\xi _2} \rangle =-\frac{((\xi _2-\xi _1)^2)}{2}\), there is an ample divisor \(H\) with \((\xi _1-\xi _2,H) \ne 0\). We may assume that \((\xi _1,H)<(\xi _2,H)\). We take \(\beta (x):=\gamma +x \xi _1+(1-x) \xi _2\) with \(0<x<1\). Then
$$\begin{aligned} v_1&= r_1 e^{\beta (x)+(1-x)(\xi _1-\xi _2)}\\&= r_1 \left( e^{\beta (x)}+(1-x)(\xi _1-\xi _2+(\xi _1-\xi _2,\beta )\varrho _X) +\frac{(1-x)^2((\xi _1-\xi _2)^2)}{2}\varrho _X \right) ,\\ v_2&= r_2 e^{\beta (x)+x(\xi _2-\xi _1)}\\&= r_2 \left( e^{\beta (x)}+x(\xi _2-\xi _1+(\xi _2-\xi _1,\beta )\varrho _X) +\frac{x^2((\xi _1-\xi _2)^2)}{2}\varrho _X \right) . \end{aligned}$$
If \(r_1 \ell _1>r_2 \ell _2\), then \((\xi _2,H)>(\xi _1,H)\) implies that \((\xi _2,H)>(\xi _1,H)>0\). Hence \(d_{\beta (x)}(-v_1)>0\) and \(d_{\beta (x)}(v_2)>0\). We have
$$\begin{aligned} \frac{d_{\beta (x)}(-v_1)a_{\beta (x)}(v_2) -d_{\beta (x)}(v_2)a_{\beta (x)}(-v_1)}{d_{\beta (x)}(-v_1)r_{\beta (x)}(v_2)-d_{\beta (x)}(v_2)r_{\beta (x)}(-v_1)} =(1-x)x \frac{((\xi _1-\xi _2)^2)}{2}>0. \end{aligned}$$
Hence there is a positive number \(t\) such that \(Z_{(\beta (x),tH)}(-v_1) \in \mathbb{R }_{>0}Z_{(\beta (x),tH)}(v_2)\) by Lemma 2.14. Then \((\beta ,\omega ):=(\beta (x),tH)\) belongs to the wall defined by \(-v_1\). By [14, Thm. 4.9] and [7, Prop. 4.1.5], \(\mathfrak{M }^\pm (v_1,v_2,\ell _1,\ell _2)=M_{(\beta ,\omega ^\pm )}(v)\).
If \(r_1 \ell _1<r_2 \ell _2\), then we have \((\xi _1,H)<(\xi _2,H)<0\). Hence \(d_\beta (v_1)>0\) and \(d_\beta (-v_2)>0\). In this case, we also have
$$\begin{aligned} \frac{d_{\beta (x)}(v_1)a_{\beta (x)}(-v_2)- d_{\beta (x)}(-v_2)a_{\beta (x)}(v_1)}{d_{\beta (x)}(v_1)r_{\beta (x)}(-v_2)-d_{\beta (x)}(-v_2)r_{\beta (x)}(v_1)} =(1-x)x \frac{((\xi _1-\xi _2)^2)}{2}>0. \end{aligned}$$
(4.1)
Hence there is a positive number \(t\) such that \(Z_{(\beta (x),tH)}(v_1) \in \mathbb{R }_{>0}Z_{(\beta (x),tH)}(-v_2)\). Then \((\beta ,\omega ):=(\beta (x),tH)\) belongs to the wall defined by \(v_1\). By [14, Thm. 4.9] and [7, Prop. 4.1.5], \(\mathfrak{M }^\pm (v_1,v_2,\ell _1,\ell _2)=M_{(\beta ,\omega ^\pm )}(-v)\). \(\square \)

5 Stability conditions on a restricted parameter space

5.1 The structure of walls

In this section, we shall partially generalize the structure of walls in [5]. We note that
$$\begin{aligned}&\{\sigma _{(\xi ,\omega )} \mid \xi \in \mathrm{NS }(X)_\mathbb{R },\ \omega \in \mathrm{Amp }(X)_\mathbb{R } \}\\&\quad = \bigcup _{(\beta ,H) \in \mathrm{NS }(X)_\mathbb{R } \times \mathrm{Amp }(X)_\mathbb{R }} \{ \sigma _{(\beta +sH ,tH)} \mid s \in \mathbb{R },\ t \in \mathbb{R }_{>0} \}. \end{aligned}$$
In this section, we fix \(\beta \in \mathrm{NS }(X)_\mathbb{R }\) and an ample divisor \(H\) on \(X\) and study the wall for the space of special stability conditions:
$$\begin{aligned} \{ \sigma _{(\beta +sH,tH)}\mid s \in \mathbb{R },\ t \in \mathbb{R }_{>0} \}. \end{aligned}$$

Definition 5.1

Let \(v_1\) be a Mukai vector with \(\langle v_1^2 \rangle \ge 0, \langle (v-v_1)^2 \rangle \ge 0\) and \(\langle v_1,v-v_1 \rangle >0\). If \((\mathrm{rk }v_1,d_\beta (v_1),a_\beta (v_1)) \not \in \mathbb{Q }(\mathrm{rk }v,d_\beta (v),a_\beta (v))\), we define a wall for v by
$$\begin{aligned} W_{v_1}^{\beta ,H}:= \{(s,t) \in \mathbb{R }^2 \mid \mathbb{R }Z_{(\beta +sH,tH)}(v_1)= \mathbb{R }Z_{(\beta +sH,tH)}(v) \}. \end{aligned}$$
A connected component of \(\mathbb{R } \times \mathbb{R }_{>0} {\setminus } \cup _{v_1} W_{v_1}^{\beta ,H}\) is called a chamber for \(v\).

Remark 5.2

By [8, Lem. 5.20], for a general \(\beta \in \mathrm{NS }(X)_\mathbb{Q }, (\mathrm{rk }v_1,d_\beta (v_1),a_\beta (v_1)) \not \in \mathbb{Q }(\mathrm{rk }v,d_\beta (v),a_\beta (v))\) for all \(v_1\).

For the study of walls, we collect elementary facts on a family of circles.

Lemma 5.3

We take \(p \in \mathbb{R }\) and \(q \in \mathbb{R }_{>0}\). For \(a \in \mathbb{R }\), let \(C_a:(x+a)^2+y^2=(a+p)^2-q\) be the circle in \((x,y)\)-plane. We also set \(C_\infty :x=p\). Thus \(C_\infty \) is a line in \((x,y)\)-plane.
  1. (1)

    \(C_a \cap C_{a'}=\emptyset \), if \(a \ne a'\).

     
  2. (2)

    \(C_a \cap C_\infty =\emptyset \) for \(a \in \mathbb{R }\).

     

Proof

(1) If \(C_a \cap C_{a'} \not =\emptyset \), then the intersection satisfies \(x=p\). Then \((p+a)^2+y^2=(a+p)^2-q\), which implies that \(y^2=-q<0\). Therefore the claim holds. The proof of (2) is similar. \(\square \)

Remark 5.4

By the proof, we see that \(C_a\)\((a \in \mathbb{R })\) forms a pencil of conics passing through the imaginary points \(\{(p,\pm \sqrt{-q}) \}\).

Lemma 5.5

Assume that \(C_a:(x+a)^2+y^2=(a+p)^2-q\) with \(q>0\) is non-empty. If \(a+p>0\), then \((p-\sqrt{q},0)\) is contained in \(C_a\). If \(a+p<0\), then \((p+\sqrt{q},0)\) is contained in \(C_a\).

Proof

Assume that \(a+p>0\). Then \(a+p > \sqrt{q}\). Then \((a+p)^2-q-(p-\sqrt{q}+a)^2=2\sqrt{q}(a+p-\sqrt{q})>0\). Hence the claim holds. If \(a+p<0\), then \(a+p < -\sqrt{q}\). Hence \((a+p)^2-q-(p+\sqrt{q}+a)^2=-2\sqrt{q}(a+p+\sqrt{q})>0\). Therefore the claim holds.   \(\square \)

We shall study the structure of walls. We first assume that \(r \ne 0\) and set
$$\begin{aligned} v&:= r+dH+D+b\varrho _X \\&= r e^{\beta }+(d_\beta H+D_\beta )+(d_\beta H+D_\beta ,\beta )\varrho _X +a_\beta \varrho _X \\&= r e^{\beta +sH}+(d_\beta -rs)(H+(H,\beta +sH) \varrho _X) +D_\beta +(D_\beta ,\beta )\varrho _X +\widetilde{a}_\beta \varrho _X\\ \end{aligned}$$
Then
$$\begin{aligned} a_\beta =\frac{d_\beta ^2 (H^2)-(\langle v^2 \rangle -(D^2_\beta ))}{2r},\; \widetilde{a}_\beta = \frac{(d_\beta -rs)^2 (H^2)-(\langle v^2 \rangle -(D_\beta ^2))}{2r}. \end{aligned}$$
(5.1)
In particular, we have
$$\begin{aligned} \widetilde{a}_\beta =a_\beta -s d_\beta (H^2)+\frac{r}{2}s^2 (H^2). \end{aligned}$$
(5.2)
We also set
$$\begin{aligned} v_2&:= r_2 e^{\beta }+(d_2 H+D_2)+(d_2 H+D_2,\beta )\varrho _X +a_2 \varrho _X\\&= r_2 e^{\beta +sH}+(d_2-r_2 s)(H+(H,\beta +sH) \varrho _X) +D_2+(D_2,\beta )\varrho _X +\widetilde{a}_2 \varrho _X. \end{aligned}$$

Proposition 5.6

Assume that \(r \ne 0\) and \(\langle v^2 \rangle >0\).
(1)
Assume that \(rd_2-r_2 d_\beta \ne 0\). Then \(\mathbb{R }Z_{(\beta +sH,tH)}(v)=\mathbb{R }Z_{(\beta +sH,tH)}(v_2)\) holds for \((s,t) \in \mathbb{R }^2\) if and only if
$$\begin{aligned}&t^2+\left( s-\frac{a_2 r-a_\beta r_2}{(H^2)(rd_2-r_2 d_\beta )} \right) ^2\nonumber \\&\quad =\left( \frac{d_\beta }{r}-\frac{a_2 r-a_\beta r_2}{(H^2)(rd_2-r_2 d_\beta )} \right) ^2 -\frac{\langle v^2 \rangle -(D_\beta ^2)}{(H^2) r^2}. \end{aligned}$$
(5.3)
(2)
Assume that \(rd_2-r_2 d_\beta =0\) and \(a_2 r-a_\beta r_2 \ne 0\). Then \(\mathbb{R }Z_{(\beta +sH,tH)}(v)=\mathbb{R }Z_{(\beta +sH,tH)}(v_2)\) holds for \((s,t) \in \mathbb{R }^2\) if and only if
$$\begin{aligned} r s-d_\beta =0. \end{aligned}$$

Proof

We shall compute \(\Sigma _{(\beta +sH,tH)}(v,v_1)\) by Lemma 2.14. (1) We first note that
$$\begin{aligned} a_\beta d_2-a_2 d_\beta =\frac{a_\beta }{r}(rd_2-d_\beta r_2) -\frac{d_\beta }{r}(a_2 r-a_\beta r_2). \end{aligned}$$
(5.4)
Applying (5.2) for \(v\) and \(v_2\), we have
$$\begin{aligned}&(d_2-r_2 s)\widetilde{a}_\beta -(d_\beta -rs)\widetilde{a}_2\nonumber \\&\quad = -s^2 \frac{(H^2)}{2}(rd_2-r_2 d_\beta )+s (a_2 r-a_\beta r_2)+ (a_\beta d_2-a_2 d_\beta ). \end{aligned}$$
(5.5)
By using (5.4) and
$$\begin{aligned} \frac{2}{(H^2)}\frac{a_\beta }{r}- \left( \frac{d_\beta }{r}\right) ^2= -\frac{\langle v^2 \rangle -(D_\beta ^2)}{r^2 (H^2)}, \end{aligned}$$
(5.6)
we see that
$$\begin{aligned}&-s^2 \frac{(H^2)}{2}(rd_2-r_2 d_\beta )+s (a_2 r-a_\beta r_2)+ (a_\beta d_2-a_2 d_\beta )\\&\quad = \frac{(H^2)}{2}\left\{ -s^2 (rd_2-r_2 d_\beta )+ s \frac{2}{(H^2)}(a_2 r-a_\beta r_2)+ \frac{2}{(H^2)} \left( \frac{a_\beta }{r}(rd_2-d_\beta r_2)- \frac{d_\beta }{r}(a_2 r-a_\beta r_2) \right) \right\} \\&\quad = \frac{(H^2)}{2}\left\{ -(rd_2-r_2 d_\beta )\left( s-\frac{(a_2 r-a_\beta r_2)}{(H^2)(rd_2-r_2 d_\beta )} \right) ^2+ (rd_2-r_2 d_\beta ) \left( \frac{(a_2 r-a_\beta r_2)}{(H^2)(rd_2-r_2 d_\beta )} \right) ^2 \right. \\&\qquad \left. +\,\frac{2}{(H^2)}\frac{a_\beta }{r}(rd_2-d_\beta r_2) -2(rd_2-r_2 d_\beta )\frac{d_\beta }{r} \frac{(a_2 r-a_\beta r_2)}{(H^2)(rd_2-r_2 d_\beta )} \right\} \\&\quad = \frac{(H^2)}{2}\left\{ -(rd_2-r_2 d_\beta ) \left( s-\frac{(a_2 r-a_\beta r_2)}{(H^2)(rd_2-r_2 d_\beta )} \right) ^2+ (rd_2-r_2 d_\beta ) \left( \frac{d_\beta }{r}-\frac{(a_2 r-a_\beta r_2)}{(H^2)(rd_2-r_2 d_\beta )} \right) ^2 \right. \\&\qquad \left. +\,(rd_2-r_2 d_\beta )\left( \frac{2}{(H^2)}\frac{a_\beta }{r}- \left( \frac{d_\beta }{r}\right) ^2 \right) \right\} \\&\quad = \frac{(rd_2\!-\!r_2 d_\beta )(H^2)}{2}\left\{ \!-\!\left( s-\frac{(a_2 r-a_\beta r_2)}{(H^2)(rd_2-r_2 d_\beta )} \right) ^2\!+\! \left( \frac{d_\beta }{r}\!-\!\frac{(a_2 r\!-\!a_\beta r_2)}{(H^2)(rd_2-r_2 d_\beta )} \right) ^2 \!-\! \frac{\langle v^2 \rangle -(D_\beta ^2)}{r^2 (H^2)}\right\} . \end{aligned}$$
Hence the claim holds.

(2) If \(rd_2-r_2 d_\beta =0\), then \(a_\beta d_2-a_2 d_\beta =a_\beta d_\beta r_2/r-a_2 d_\beta =(a_\beta r_2-a_2 r)d_\beta /r\). Hence the claim follows from (5.5). \(\square \)

By Lemma 5.5, we get the following corollary.

Corollary 5.7

(1)

If \( \frac{d_\beta }{r}-\frac{a_2 r-a_\beta r_2}{(H^2)(rd_2-r_2 d_\beta )}>0\), then \((s,t)=\left( \frac{d_\beta }{r}- \sqrt{\frac{\langle v^2 \rangle -(D_\beta ^2)}{(H^2) r^2}},0 \right) \) is contained in the circle (5.3).

(2)

If \(\frac{d_\beta }{r}-\frac{a_2 r-a_\beta r_2}{(H^2)(rd_2-r_2 d_\beta )}<0\), then \((s,t)= \left( \frac{d_\beta }{r}+ \sqrt{\tfrac{\langle v^2 \rangle -(D_\beta ^2)}{(H^2) r^2}},0 \right) \) is contained in the circle (5.3).

Remark 5.8

  1. (1)

    If \(rd_2-r_2 d_\beta <0\), then \((s,t)\) is surrounded by the circle in Proposition 5.6 if and only if \(\phi _{(\beta +sH,tH)}(v_2) \mod 2\mathbb{Z }\) satisfies \(\phi _{(\beta +sH,tH)}(v)+1>\phi _{(\beta +sH,tH)}(v_2) >\phi _{(\beta +sH,tH)}(v)\).

     
  2. (2)

    If \(rd_2-r_2 d_\beta >0\), then \((s,t)\) is surrounded by the circle in Proposition 5.6 if and only if \(\phi _{(\beta +sH,tH)}(v_2) \mod 2\mathbb{Z }\) satisfies \(\phi _{(\beta +sH,tH)}(v)>\phi _{(\beta +sH,tH)}(v_2) >\phi _{(\beta +sH,tH)}(v)-1\).

     

We next treat the case where \(r=0\). In this case, \(d_\beta \ne 0\) if \(\langle v^2 \rangle >0\).

Proposition 5.9

Assume that \(r=0\) and \(\langle v^2 \rangle >0\). If \(r_2 \ne 0\), then \(\mathbb{R }Z_{(\beta +sH,tH)}(v)=\mathbb{R }Z_{(\beta +sH,tH)}(v_2)\) holds for \((s,t) \in \mathbb{R }^2\) if and only if
$$\begin{aligned} t^2+\left( s-\frac{ a_\beta }{d_\beta (H^2)} \right) ^2 =\left( \frac{a_\beta }{d_\beta (H^2)}-\frac{d_2}{r_2} \right) ^2 -\frac{\langle v_2^2 \rangle - (D_2^2)}{r_2^2(H^2)}. \end{aligned}$$
(5.7)

Proof

By using (5.5) and \(r=0\), we get
$$\begin{aligned}&(d_2-r_2 s)\widetilde{a}_\beta -(d_\beta -rs)\widetilde{a}_2\\&\quad = -s^2 \frac{(H^2)}{2}(rd_2-r_2 d_\beta )+s (a_2 r-a_\beta r_2)+ (a_\beta d_2-a_2 d_\beta )\\&\quad = \frac{r_2 d_\beta (H^2)}{2} \left( s^2- s \frac{2 a_\beta }{d_\beta (H^2)}+ \frac{2(a_\beta d_2-a_2 d_\beta )}{(H^2) r_2 d_\beta }\right) \\&\quad = \frac{r_2 d_\beta (H^2)}{2} \left\{ \left( s-\frac{ a_\beta }{d_\beta (H^2)} \right) ^2 -\left( \frac{a_\beta }{d_\beta (H^2)}-\frac{d_2}{r_2} \right) ^2 +\frac{\langle v_2^2 \rangle - (D_2^2)}{r_2^2(H^2)} \right\} . \end{aligned}$$
By Lemma 2.14, we get the claim. \(\square \)

Corollary 5.10

If \(\sqrt{\frac{\langle v^2 \rangle -(D_\beta ^2)}{(H^2)}} \in \mathbb{Q }\), then there are finitely many walls for \(v\).

Proof

For a fixed \(s \in \mathbb{Q }\), there are finitely many walls by [8, Lem. 5.20 (4)] or [7, Rem. 3.1.4]. Recall that the crucial point of the argument is the fact that the subgroup
$$\begin{aligned} \{ d_{\beta ,H}(E)=(c_1(E)-\mathrm{rk }E\beta ,H) \mid E \in K(X) \} \subset \mathbb{R } \end{aligned}$$
is cyclic.

If \(r=0\), then \((s,t)=(\frac{a_\beta }{d_\beta (H^2)},0)\) is the center of (5.7). Hence every wall intersects with \(s=\frac{a_\beta }{d_\beta (H^2)}\). If \(r \ne 0\), then Corollary 5.7 implies that every wall intersects with \(s=\frac{d_\beta }{r}- \sqrt{\frac{\langle v^2 \rangle -(D_\beta ^2)}{(H^2) r^2}}\) or \(s=\frac{d_\beta }{r}+ \sqrt{\frac{\langle v^2 \rangle -(D_\beta ^2)}{(H^2) r^2}}\). Hence there are finitely many walls for \(v\). \(\square \)

Lemma 5.11

For \(v_2=e^{\beta +\lambda H}\), the condition \(\mathbb{R }Z_{(\beta +sH,tH)}(v)=\mathbb{R }Z_{(\beta +sH,tH)}(v_2)\) for \((s,t) \in \mathbb{R }^2\) is equivalent to the equation of the circle
$$\begin{aligned} C_{v,\lambda }: t^2+(s-\lambda ) \left( s- \frac{1}{(r\lambda -d_\beta )} \left( \lambda d_\beta -\frac{2}{(H^2)}a_\beta \right) \right) =0 \end{aligned}$$
for \(r \lambda -d_\beta \ne 0\) and
$$\begin{aligned} rs-d_\beta =0 \end{aligned}$$
for \(r \lambda -d_\beta = 0\). In particular, the circle \(C_{v,\lambda }\) passes the points \((\lambda ,0)\) and \(\left( \frac{1}{(r\lambda -d_\beta )} (\lambda d_\beta -\frac{2}{(H^2)}a_\beta ),0 \right) \).

Proof

We note that
$$\begin{aligned} v_2&= e^{\beta +\lambda H}=e^\beta +\lambda (H+(H,\beta )\varrho _X)+ \frac{(H^2)}{2}\lambda ^2 \varrho _X \\&= e^{\beta +sH+(\lambda -s) H}= e^{\beta +sH}+(\lambda -s)(H+(H,\beta +sH)\varrho _X)+ \frac{(H^2)}{2}(\lambda -s)^2 \varrho _X. \end{aligned}$$
Then we get
$$\begin{aligned} (d_2-r_2 s)\widetilde{a}_\beta -(d_\beta -rs)\widetilde{a}_2&= (\lambda -s)\widetilde{a}_\beta -(d_\beta -rs)\frac{(H^2)}{2}(\lambda -s)^2 \nonumber \\&= (\lambda -s)(\widetilde{a}_\beta -(\lambda -s)(d_\beta -rs)\frac{(H^2)}{2})\nonumber \\ \!&= \! (\lambda \!-\!s) \left( (r\lambda \!-\!d_\beta )\frac{(H^2)}{2}s\!-\! \left( \lambda d_\beta \frac{(H^2)}{2}\!-\!a_\beta \right) \right) .\quad \end{aligned}$$
(5.8)
Assume that \(r \lambda -d_\beta \ne 0\). Then the condition is given by the circle
$$\begin{aligned} t^2+(s-\lambda ) \left( s- \frac{1}{(r\lambda -d_\beta )}\left( \lambda d_\beta -\frac{2}{(H^2)}a_\beta \right) \right) =0. \end{aligned}$$
In particular, the circle passes the points \((\lambda ,0)\) and \(\left( \frac{1}{(r\lambda -d_\beta )} (\lambda d_\beta -\frac{2}{(H^2)}a_\beta ),0 \right) \).
Assume that \(r \lambda -d_\beta =0\). Then by (5.8), we get
$$\begin{aligned} 0=(\lambda -s)\left( \lambda d_\beta \frac{(H^2)}{2}-a_\beta \right) . \end{aligned}$$
If \(a_\beta =\lambda d_\beta \frac{(H^2)}{2}=r\lambda ^2 \frac{(H^2)}{2}\), then we see that \(v=r e^{\beta +\lambda H}+(D_\beta +(D_\beta ,\beta +\lambda H)\varrho _X)\). Hence \(\langle v^2 \rangle =(D_\beta ^2) \le 0\), which is a contradiction. Therefore we get
$$\begin{aligned} s=\lambda =\frac{d_\beta }{r}. \end{aligned}$$
\(\square \)

Remark 5.12

Assume that \(r \ne 0\). Then
$$\begin{aligned} \lambda \ne \frac{1}{r\lambda -d_\beta } \left( \lambda d_\beta -\frac{2}{(H^2)}a_\beta \right) \Longleftrightarrow \lambda \ne \frac{d_\beta }{r}\pm \sqrt{\frac{\langle v^2 \rangle -(D_\beta ^2)}{(H^2)r^2}}. \end{aligned}$$
In particular, if \(\sqrt{\frac{\langle v^2 \rangle -(D_\beta ^2)}{(H^2)r^2}} \not \in \mathbb{Q }\), then \(C_{v,\lambda }\) is a circle.

Corollary 5.13

Assume that \(v=r e^\beta +a_\beta \varrho _X+d_\beta (H+(H,\beta )\varrho _X)\). For a numerical solution
$$\begin{aligned} (r_1 e^{\beta +\lambda _1 H},r_2 e^{\beta +\lambda _2 H},\ell _1,\ell _2), \end{aligned}$$
we have \(C_{v,\lambda _1}=C_{v,\lambda _2}\) and the equation is given by
$$\begin{aligned} t^2+(s-\lambda _1)(s-\lambda _2)=0. \end{aligned}$$
(5.9)

Proof

Since \(v=\pm (\ell _1 r_1 e^{\beta +\lambda _1 H}- \ell _2 r_2 e^{\beta +\lambda _2 H})\), we have \(C_{v,\lambda _1}=C_{v,\lambda _2}\). Since \((\lambda _i,0) \in C_{v,\lambda _i}\) for \(i=1,2\) and \(\lambda _1 \ne \lambda _2\), we get the claim.\(\square \)

5.2 Relation of stability conditions

All walls except \(rs=d_\beta \) are disjoint to the line \(rs=d_\beta \). By Corollary 5.7, there are at most two unbounded chambers.

Proposition 5.14

Let \(v\) be a positive and primitive Mukai vector such that \(\langle v^2 \rangle >0\). Assume that \((s,t) \in \mathbb{R }^2\) belongs to an unbounded chamber. Then
$$\begin{aligned} M_{(\beta +sH,tH)}(v) \cong \left\{ \begin{array}{ll} M_H^\beta (v) &{} d_{\beta +sH}(v) >0,\\ M_H^{-\beta }(v^{\vee }) &{} d_{\beta +sH}(v) \le 0. \end{array}\right. \end{aligned}$$

Proof

We may assume that \(d_{\beta +sH}(v) \ne 0\). (1) Assume that \(d_{\beta +sH}(v)>0\). If \(\mathrm{rk }v >0\), then Proposition 2.9 and [3, Prop. 14.2] imply \(M_{(\beta +sH,tH)}(v)=M_H^\beta (v)\). If \(\mathrm{rk }v=0\), then we use [7, Cor. 2.2.9].

(2) Assume that \(d_{\beta +sH}(v)< 0\). By Theorem 3.8 (2), we have an isomorphism \(M_{(\beta +sH,tH)}(v) \rightarrow M_{(-(\beta +sH),tH)}(v^{\vee })\) via \(F \mapsto F^{\vee }\). Since \(d_{-\beta -sH}(v^{\vee })>0\), we have \(M_{(-(\beta +sH),tH)}(v^{\vee })=M_H^{-\beta }(v^{\vee })\). Therefore the claim holds.\(\square \)

Lemma 5.15

Let \(\mathcal{C }_0\) and \(\mathcal{C }_1\) be two chambers such that \(\mathcal{C }_0\) is surrounded by \(\mathcal{C }_1\). We take \((s,t_0) \in \mathcal{C }_0\) and \((s,t_1) \in \mathcal{C }_1\). Let \(t_x:=(1-x) t_0+ x t_1\) (\(0 \le x \le 1\)) be a segment connecting \(t_0\) and \(t_1\). If \(E\) is \(\sigma _{(\beta +s H,t_i H)}\)-semi-stable for \(i=0,1\), then \(E\) is \(\sigma _{(\beta +sH,t_x H)}\)-semi-stable for all \(x\).

Proof

Assume that \(E\) is not \(\sigma _{(\beta +sH,t_x H)}\)-semi-stable for some \(x \in (0,1)\). Then there is a subobject \(E_1\) of \(E\) in \(\mathfrak{A }_{(\beta +sH,t_x H)}\) such that \(\phi _{(\beta +sH,t_x H)}(E_1)> \phi _{(\beta +sH,t_x H)}(E)\). Since \(E\) is \(\sigma _{(\beta +s H,t_i H)}\)-semi-stable for \(i=0,1\), \(\phi _{(\beta +sH,t_i H)}(E_1) \le \phi _{(\beta +sH,t_i H)}(E)\). Then there are two numbers \(x_1, x_2\) such that \(0<x_1,x_2<1\) and \(\phi _{(\beta +sH,t_{x_i} H)}(E_1)= \phi _{(\beta +sH,t_{x_i} H)}(E)\). Since \(t_{x_i}\) is uniquely determined by \(v(E_1)\), this does not occur. Therefore \(E\) is \(\sigma _{(\beta +sH,t_x H)}\)-semi-stable for all \(x\).\(\square \)

Definition 5.16

Let \(W\) be a wall for \(v\) in \((s,t)\)-plane. Let \((\beta ,\omega )\) be a point of \(W\) and \((\beta ',\omega ')\) be a point in an adjacent chamber. Then we define the codimension of the wall\(W\) by
$$\begin{aligned} \mathrm{codim }W:=\min _{v=\sum _i v_i } \left\{ \sum _{i<j}\langle v_i,v_j \rangle -\left( \sum _i (\dim \mathcal{M }_H^{\beta '}(v_i)^{ss}- \langle v_i^2 \rangle ) \right) +1 \right\} , \end{aligned}$$
(5.10)
where \(v=\sum \nolimits _i v_i\) are decompositions of \(v\) such that \(\phi _{(\beta ,\omega )}(v)=\phi _{(\beta ,\omega )}(v_i)\) and \(\phi _{(\beta ',\omega ')}(v_i)>\phi _{(\beta ',\omega ')}(v_j), i<j\).

Lemma 5.17

If \(W\) is a codimension 0 wall, then \(W\) is defined by \(v_1\) such that
$$\begin{aligned} v=n v_1+v_2,\; \langle v_1,v_2 \rangle =1,\; \langle v_1^2 \rangle =\langle v_2^2 \rangle =0. \end{aligned}$$
(5.11)

Proof

  1. (1)

    We first assume that \(W\) is a circle, that is, \(W\) is not defined by \(d_\beta -rs =0\). Then the claim is a consequence of [7, Lem. 4.3.4, Rem. 4.3.3].

     
  2. (2)

    We next assume that \(W\) is defined by \(d_\beta -rs=0\). By an auto-equivalence \(\Phi :\mathbf{D}(X) \rightarrow \mathbf{D}(X)\) such that \(\Phi (e^{\beta +d_\beta H/r}) \not \in \mathbb{Q }\varrho _X, \Phi (W)\) becomes a circle. Since \(\Phi \) preserves Bridgeland’s stability conditions, the claim follows from (1). \(\square \)

     

If \(v_1,v_2\) in (5.11) satisfy \(v_1>0,v_2<0\) or \(v_1<0\) and \(v_2>0\), then \((v_1,-v_2,n,1)\) or \((-v_1,v_2,n,1)\) gives a numerical solution of \(v\). Conversely for a numerical solution \((v_1,v_2,\ell _1,\ell _2)\), Lemma 4.7 implies that for a suitable \(\beta \) and \(\omega =tH, v_1\) defines a codimension 0 wall.

Proposition 5.18

Assume that \(\mathrm{NS }(X)=\mathbb{Z }H\). We fix \(\beta \). Then there is a bijective correspondence between a codimension 0 wall and a numerical solution.

Proof

Assume that \(\mathrm{NS }(X)=\mathbb{Z }H\). For a numerical solution \((v_1,v_2,\ell _1,\ell _2), \mathrm{the}\, \mathrm{condition} \beta , c_1(v),c_1(v_1), c_1(v_2) \in \mathbb{Q }H\) implies that \(C_{v,\lambda _1}\) in Corollary 5.13 gives a wall in the \((s,t)\)-plane. Combining Lemma 5.17, we get the claim.\(\square \)

Proposition 5.19

(cf. [7, Prop. 4.3.5]) If \((\beta _1,\omega _1)\) and \((\beta _2,\omega _2)\) are not separated by any codimension 0 wall, then \(M_{(\beta _1,\omega _1)}(v) \cap M_{(\beta _2,\omega _2)}(v) \ne \emptyset \). In particular, \(M_{(\beta _1,\omega _1)}(v)\) and \(M_{(\beta _2,\omega _2)}(v)\) are birationally equivalent.

5.3 Semi-homogeneous presentations and stability.

For a semi-homogeneous presentation, Lemma 4.7 implies that we can relate a \(\sigma _{(\beta ,\omega )}\)-semi-stability. We shall study Gieseker semi-stability of coherent sheaves with two semi-homogeneous presentations.

Lemma 5.20

Let \(v\) be a primitive Mukai vector with \(r:=\mathrm{rk }v>0\). Assume that \(\mathcal{M }_H^\beta (v)^{ss}\) consists of \(\beta \)-stable sheaves. If a simple sheaf \(E\) with \(v(E)=v\) has two semi-homogeneous presentations and \(d_\beta (v)-r s=0\) is not a codimension 0 wall, then \(E\) is a \(\mu \)-stable vector bundle.

Proof

Since \(rs-d_\beta (v)=0\) is not a codimension 0 wall, two semi-homogeneous presentations define two circles \(C_1\) and \(C_2\) which are separated by the line \(rs-d_\beta (v)=0\).

There is a Fourier–Mukai transform \(\Phi :=\Phi _{Y \rightarrow X}^{\mathbf{E}}\) and a complex \(F\) such that \(E=\Phi (F)\) and \(F\) is semi-stable with respect to two chambers \(\mathcal{C }_0, \mathcal{C }_1\) such that \(\mathcal{C }_0\) is surrounded by \(\mathcal{C }_1\) and \(\mathcal{C }_i\) are adjacent to \( \Phi ^{-1} (C_i)\). Let \(\mathcal{C }\) be an unbounded chamber between \(\Phi (\mathcal{C }_1)\) and \(\Phi (\mathcal{C }_2)\). Then \(F\) is semi-stable with respect to \(\Phi ^{-1}(\mathcal{C })\) by Lemma 5.15. Hence \(E\) is semi-stable with respect to all unbounded chambers.

We take an element \(\omega \in \mathrm{Amp }(X)_\mathbb{Q }\). Then \(E\) is \(\sigma _{(\beta +sH,\omega )}\)-semi-stable if \(1 \gg d_\beta (v)-rs>0\). Hence \(E\) is \(\beta \)-twisted semi-stable. Since \(E\) is also \(\sigma _{(\beta +sH,\omega )}\)-semi-stable if \(1 \gg -(d_\beta (v)-rs)>0, E^{\vee }\) is \((-\beta )\)-twisted semi-stable. Hence \(E\) is locally free. Let \(E_1\) be a locally free subsheaf of \(E\). Then there is a generically surjective homomorphism \(E^{\vee } \rightarrow E_1^{\vee }\). Since
$$\begin{aligned} \frac{\chi (E_1(-\beta ))}{\mathrm{rk }E_1}= \frac{\chi (E_1^{\vee }(\beta ))}{\mathrm{rk }E_1^{\vee }}, \end{aligned}$$
\(\frac{d_\beta (E_1)}{\mathrm{rk }E_1}= \frac{d_\beta (E)}{\mathrm{rk }E}\) implies \(\frac{\chi (E_1(-\beta ))}{\mathrm{rk }E_1}= \frac{\chi (E(-\beta ))}{\mathrm{rk }E}\). Thus \(E\) is properly \(\beta \)-twisted semi-stable, which is a contradiction. Therefore \(E\) is \(\mu \)-stable. \(\square \)

Lemma 5.21

Let \(v\) be a primitive Mukai vector with \(r:=\mathrm{rk }v>0\). Assume that \(d_\beta (v)-r s=0\) defines a wall.
  1. (1)
    \(d_\beta (v)-r s=0\) is a codimension 0 wall if and only if
    1. (a)

      \(v=r e^{\xi }-a \varrho _X, \xi \in \mathrm{NS }(X), a \in \mathbb{Z }\), \((r-1)(a-1)=0\) or

       
    2. (b)

      \(v=v_1+\ell v_2, v_i=r_i e^{\frac{\xi _i}{r_i}}, r_1,r_2>0\), \(((r_2 \xi _1-r_1 \xi _2)^2)=-r_1 r_2, (r_2 \xi _1-r_1 \xi _2,H)=0\).

       
     
  2. (2)
    Assume that \(v\) satisfies (a). We take \((s,t)\) such that \(1 \gg d_\beta (v)-rs>0\) and let \(E\) be a \(\sigma _{(\beta +sH,tH)}\)-semi-stable object with \(v(E)=v\). Then
    1. (a1)

      \(r=1\) and \(E=I_Z(\xi )\) or

       
    2. (a2)

      \(a=1\) and \(E=\ker (\oplus _{i=1}^r L_i \rightarrow \mathfrak{k }_x)\), where \(L_i\) are line bundles with \(c_1(L_i)=\xi \).

       
     
  3. (3)

    If \(\mathrm{NS }(X)=\mathbb{Z }H\), then (b) does not occur.

     

Proof

(1), (3) By Lemma 5.17, we have \(v=v_1+\ell v_2, \langle v_i^2 \rangle =0\) and \(\langle v_1,v_2 \rangle =1\), where \(\ell =\langle v^2 \rangle /2\). Assume that \(\mathrm{rk }v_1 \mathrm{rk }v_2 \ne 0\). Then we can set \(v_i:=r_i e^{\frac{\xi _i}{r_i}}\)\((i=1,2)\) with \(((r_2 \xi _1-r_1 \xi _2)^2)=-r_1 r_2\). Since the wall is \(d_\beta -rs=0\), we have \((r_2 \xi _1-r_1 \xi _2,H)=0\). By the Hodge index theorem, \(r_1 r_2>0\). Thus \(r_1, r_2>0\). If \(\mathrm{rk }v_1=0\) or \(\mathrm{rk }v_2=0\), then \(r_2 d_\beta (v_1)-r_1 d_\beta (v_2)=0\) and \(\langle v_1,v_2\rangle =1\) implies that \(\{v_1, v_2 \}=\{(0,0,1),(-1,-\eta ,-(\eta ^2)/2) \}\). Therefore (1) holds. If \(\mathrm{NS }(X)=\mathbb{Z }H\) and \((r_2 \xi _1-r_1 \xi _2,H)=0\), then we have \(r_2 \xi _1-r_1 \xi _2=0\). Hence \(r_1 r_2=0\). Thus (b) does not occur. (2) By the choice of \((s,t)\), \(\sigma _{(\beta +sH,tH)}\)-semi-stability implies \(\beta \)-twisted semi-stability. Thus for \(E \in M_{(\beta +sH,tH)}(v)\), we have a semi-homogeneous presentation
$$\begin{aligned} 0 \rightarrow E \rightarrow E_0 \rightarrow E_1 \rightarrow 0, \end{aligned}$$
where \((v(E_0),v(E_1))=(e^\xi ,\ell \varrho _X)\) or \((v(E_0),v(E_1))=(\ell e^\xi , \varrho _X)\) with \(\langle v^2 \rangle =2\ell \). Hence the claim holds. \(\square \)

Proposition 5.22

Let \(X\) be an abelian surface with \(\mathrm{NS }(X)=\mathbb{Z }H\). Let \(E\) be a simple sheaf on \(X\). If \(E\) has two semi-homogeneous presentations, then \(E\) is Gieseker semi-stable.

Proof

If \(d_\beta (v)-rs=0\) is a codimension 0 wall, then Lemma 5.21 implies the claim. If \(d_\beta (v)-rs=0\) is not a codimension 0 wall, then the claim follows from Lemma 5.20.\(\square \)

5.4 Relation with Fourier–Mukai transforms

We shall study Fourier–Mukai transforms on our space of stability conditions. Let \(r_1 e^\gamma \) be a primitive and isotropic Mukai vector. We set \(X_1:=M_{(\beta +sH,tH)}(r_1 e^{\gamma })\). Let \(\mathbf{E}\) be the universal object on \(X \times X_1\) as a complex of twisted sheaves. Assume that \(\gamma =\beta +\lambda H, \lambda \in \mathbb{Q }\). We consider the Fourier–Mukai transform \(\Phi :=\Phi _{X \rightarrow X_1}^{\mathbf{E}^{\vee }[1]}: \mathbf{D}(X) \rightarrow \mathbf{D}^{\alpha _1}(X_1)\). We set \((\widetilde{\beta +sH},\widetilde{tH})= (\gamma '+s' \widehat{H},t' \widehat{H})\). Then
$$\begin{aligned} s'&= \frac{1}{|r_1|} \frac{2(\lambda -s)}{((\lambda -s)^2+t^2) (H^2)},\nonumber \\ t'&= \frac{1}{|r_1|} \frac{2 t}{((\lambda -s)^2+t^2) (H^2)}. \end{aligned}$$
(5.12)
Since \(((s-\lambda )^2+t^2)({s'}^2+{t'}^2)= \left( \frac{2}{|r_1|(H^2)} \right) ^2\), the image of \((s-\lambda )^2+t^2=\frac{2}{|r_1|(H^2)}\) is \({s'}^2+{t'}^2=\frac{2}{|r_1|(H^2)}\).
If \(\lambda r \ne d_\beta \), then Lemma 5.11 implies that the condition \(\mathbb{R }Z_{(\beta +sH,tH)}(v)= \mathbb{R }Z_{(\beta +sH,tH)}(e^{\beta +\lambda H})\) defines a circle
$$\begin{aligned} C_{v,\lambda }:\; t^2= (\lambda -s) \left( \frac{a_\beta -d_\beta \lambda \frac{(H^2)}{2}}{\lambda r-d_\beta } \frac{2}{(H^2)}+s \right) . \end{aligned}$$
(5.13)
We have
$$\begin{aligned} \frac{a_\beta -d_\beta \lambda \frac{(H^2)}{2}}{\lambda r-d_\beta } \frac{2}{(H^2)}+\lambda&= \frac{(\lambda r-d_\beta )^2 (H^2)- (\langle v^2 \rangle -(D_\beta ^2))}{r(H^2)(\lambda r-d_\beta )} =\frac{2a_\gamma }{-d_\gamma (H^2)}\nonumber \\&\quad = \frac{2\langle e^{\beta +\lambda H},v \rangle }{d_\gamma (H^2)}. \end{aligned}$$
(5.14)
Thus \(C_{v,\lambda }\)is
$$\begin{aligned} \left( s-\left( \lambda +\frac{a_\gamma }{d_\gamma (H^2)} \right) \right) ^2+t^2 =\left( \frac{a_\gamma }{d_\gamma (H^2)} \right) ^2. \end{aligned}$$

Lemma 5.23

The image of
$$\begin{aligned} t^2 \le (\lambda -s) \left( \frac{a_\beta -d_\beta \lambda \frac{(H^2)}{2}}{\lambda r-d_\beta } \frac{2}{(H^2)}+s \right) \end{aligned}$$
(5.15)
by \(\Phi \) is
$$\begin{aligned} \left\{ (s',t') \left| -\tfrac{|r_1| a_\gamma }{d_\gamma }s' \ge 1 \right. \right\} . \end{aligned}$$

Proof

By (5.14) and (5.12), \((s,t)\) satisfies (5.15) if and only if
$$\begin{aligned} 0&\ge (s-\lambda )^2+t^2-(s-\lambda )\frac{2a_\gamma }{d_\gamma (H^2)} \nonumber \\&= \left( (s-\lambda )^2+t^2 \right) \left( 1+s' |r_1| \frac{a_\gamma }{d_\gamma } \right) . \end{aligned}$$
(5.16)
Hence the claim holds. \(\square \)

By Lemma 5.23, we have the following.

Proposition 5.24

Let \(\mathcal{C }^\pm \) be the adjacent chamber of \(C_{v,\lambda }\) such that \(\mathcal{C }^-\) is surrounded by \(\mathcal{C }^+\).
(1)

If   \(\frac{a_\gamma }{d_\gamma }<0\), then \(\Phi (\mathcal{C }^+)\) (resp. \(\Phi (\mathcal{C }^-)\)) is the unbounded chamber satisfying \(s' < -\frac{d_\gamma }{|r_1| a_\gamma }\) (resp. \(s' > -\frac{d_\gamma }{|r_1| a_\gamma }\)).

(2)

If   \(\frac{a_\gamma }{d_\gamma }>0\), then \(\Phi (\mathcal{C }^-)\) (resp. \(\Phi (\mathcal{C }^+)\)) is the unbounded chamber satisfying \(s' < -\frac{d_\gamma }{|r_1| a_\gamma }\) (resp. \(s' > -\frac{d_\gamma }{|r_1| a_\gamma }\)).

We set \(w:=\Phi _{X \rightarrow X_1}^{\mathbf{E}^{\vee }[1]}(v)\). Proposition 5.14 implies \(M_{(\gamma '+s'\widehat{H},t' \widehat{H}) }^{\alpha _1}(w)\) is isomorphic to the moduli scheme of semi-stable sheaves. Then Theorem 3.8 implies a generalization of [8, Thm. 3.3.3] for abelian surfaces.

For the preservation of Gieseker’s semi-stability, we also have the following, which is a generalization of [17].

Proposition 5.25

Let \(v\) be a positive Mukai vector and assume that there are walls for \(v\). Let \(W^{\max }\) be the wall in the region \(rs<d_\beta \) such that \(W^{\max }\) surround all walls in \(rs<d_\beta \), that is, the boundaries of the unbounded chamber are \(rs=d_\beta \) and \(W^{\max }\). We set
$$\begin{aligned} W^{\max } \cap \{(s,0) \mid s \in \mathbb{R } \} =\{(\lambda _1,0), (\lambda _2,0) \},\quad \lambda _1<\lambda _2. \end{aligned}$$
Let \(\Phi :\mathbf{D}(X) \rightarrow \mathbf{D}(X_1)\) be the Fourier–Mukai transform as above. If   \(\lambda \le \lambda _1\) or \(\lambda _2 \le \lambda <\frac{d_\beta }{r}\), then \(\Phi \) or \(\Phi \circ \mathcal{D }_X\) preserves the Gieseker’s semi-stability.

In particular, if \(\lambda \) is sufficiently small, we can apply this proposition, which is nothing but the main result of [17].

Remark 5.26

Assume that \(r:=\mathrm{rk }v>0\). We set
$$\begin{aligned} D:=\min \{((\mathrm{rk }w) c_1(v)-(\mathrm{rk }v) c_1(w),H)>0 \mid w \in H^*(X,\mathbb{Z })_{\mathrm{alg }} \} \end{aligned}$$
and assume that
$$\begin{aligned} D=\min \{(C,H)>0 \mid C \in \mathrm{NS }(X) \}. \end{aligned}$$
We take \(w_0 \in H^*(X,\mathbb{Z })_{\mathrm{alg }}\) such that
$$\begin{aligned} ((\mathrm{rk }w_0) d_\beta (v)-(\mathrm{rk }v) d_\beta (w_0))(H^2)= ((\mathrm{rk }w_0) c_1(v)-(\mathrm{rk }v) c_1(w_0),H)=D. \end{aligned}$$
Replacing \(w_0\) by \(w_0+k v (k \in \mathbb{Z })\), we may assume that \(\mathrm{rk }v \ge \mathrm{rk }w_0>0\). If \(D<\mathrm{rk }w_0 \sqrt{\langle v^2 \rangle (H^2)}\), there is no wall for \(d_\beta (w_0)/\mathrm{rk }w_0 \le s< d_\beta (v)/\mathrm{rk }v\).

6 The chamber structure for an abelian surface \(X\) with \(\mathrm{NS }(X)=\mathbb{Z }H\)

From now on, we assume that \(\mathrm{NS }(X)=\mathbb{Z }H\). Let \(v\) be a primitive Mukai vector with a numerical solution. We shall study the walls and chambers for \(v\). By our assumption, there is an isometry of Mukai lattice sending \(v\) to \(1-\ell \varrho _X\). So we may assume that \(v=1-\ell \varrho _X\). Since a generic classification of stable objects (it induces the birational classification) is most fundamental, we are mainly interested in codimension 0 walls.

6.1 Cohomological Fourier–Mukai transforms

Let \(H_X\) be the ample generator of \(\mathrm{NS }(X)\). We shall describe the action of Fourier–Mukai transforms on the cohomology lattices in [14].

Two smooth projective varieties \(Y_1\) and \(Y_2\) are said to be Fourier–Mukai partners if there is an equivalence \(\mathbf D (Y_1)\simeq \mathbf D (Y_2)\). We denote by \(\mathrm{FM }(X)\) the set of Fourier–Mukai partners of \(X\). The set of equivalences between \(\mathbf D (X)\) and \(\mathbf D (Y)\) is denoted by \(\mathrm{Eq }(\mathbf D (X),\mathbf D (Y))\). For \(Y,Z\in \mathrm{FM }(X)\), we set
$$\begin{aligned}&\mathrm{Eq }_0(\mathbf D (Y),\mathbf D (Z)) := \{\Phi _{Y \rightarrow Z}^\mathbf{E [2k]} \in \mathrm{Eq }(\mathbf D (Y),\mathbf D (Z)) \mid \mathbf E \in \mathrm{Coh }(Y \times Z),\, k \in \mathbb{Z } \},\\&\mathcal E (Z) \!:=\! \bigcup _{Y\in \mathrm{FM }(Z)}\mathrm{Eq }_0(\mathbf D (Y),\mathbf D (Z)), \qquad \mathcal{E } \!:=\! \bigcup _{Z\in \mathrm{FM }(X)}\mathcal{E }(Z) \!=\!\bigcup _{Y,Z\in \mathrm{FM }(X)}\mathrm{Eq }_0(\mathbf D (Y),\mathbf D (Z)). \end{aligned}$$
Note that \(\mathcal{E }\) is a groupoid with respect to the composition of the equivalences. For \(Y \in \mathrm{FM }(X)\), we have \((H_Y^2)=(H_X^2)\). We set \(n:= (H^2_X)/2\).
In [14, sect. 6.4], we constructed an isomorphism of lattices
$$\begin{aligned} \iota _X: (H^*(X,\mathbb Z )_{\mathrm{alg }},\langle \cdot ,\cdot \rangle ) \xrightarrow {\ \sim \ }(\mathrm{Sym }_2(\mathbb Z , n),B), \quad (r,dH_X,a) \mapsto \left( \begin{array}{l@{\quad }l} r &{} d\sqrt{n} \\ d\sqrt{n} &{} a \end{array}\right) , \end{aligned}$$
where \(\mathrm{Sym }_2(\mathbb Z , n)\) is given by
$$\begin{aligned} \mathrm{Sym }_2(\mathbb Z ,n):=\left\{ \left( \begin{array}{l@{\quad }l} x &{} y \sqrt{n}\\ y\sqrt{n}&{}z\\ \end{array}\right) \,\Bigg |\, x,y,z\in \mathbb Z \right\} , \end{aligned}$$
and the bilinear form \(B\) on \(\mathrm{Sym }_2(\mathbb Z ,n)\) is given by
$$\begin{aligned} B(X_1,X_2) := 2ny_1 y_2-(x_1 z_2+z_1 x_2) \end{aligned}$$
for \(X_i = \left( \begin{array}{l@{\quad }l}x_i &{} y_i \sqrt{n} \\ y_i \sqrt{n} &{}z_i \end{array}\right) \in \mathrm{Sym }_2(\mathbb Z ,n)\) (\(i=1,2\)).
Each \(\Phi _{X \rightarrow Y}\) gives an isometry
$$\begin{aligned} \iota _{Y} \circ \Phi ^H_{X \rightarrow Y} \circ \iota _{X}^{-1} \in \mathrm{O }(\mathrm{Sym }_2(\mathbb Z , n)), \end{aligned}$$
(6.1)
where \(\mathrm{O }(\mathrm{Sym }_2(\mathbb Z , n))\) is the isometry group of the lattice \((\mathrm{Sym }_2(\mathbb Z , n),B)\). Thus we have a map
$$\begin{aligned} \eta : \mathcal{E } \rightarrow \mathrm{O }(\mathrm{Sym }_2(\mathbb Z , n)) \end{aligned}$$
which preserves the structures of multiplications.

Definition 6.1

We set
$$\begin{aligned} \widehat{G}&:= \left\{ \left( \begin{array}{l@{\quad }l} a \sqrt{r} &{} b \sqrt{s}\\ c \sqrt{s} &{} d \sqrt{r} \end{array}\right) \Bigg |\, \begin{aligned} a,b,c,d,r,s \in \mathbb Z ,\, r,s>0\\ rs=n, \, adr-bcs = \pm 1 \end{aligned} \right\} ,\\ G&:= \widehat{G}\cap \mathrm{SL }(2,\mathbb R ). \end{aligned}$$
We have a right action \(\cdot \) of \(\widehat{G}\) on the lattice \((\mathrm{Sym }_2(\mathbb Z , n),B)\):
$$\begin{aligned} \left( \begin{array}{l@{\quad }l} r &{}d \sqrt{n}\\ d \sqrt{n}&{}a \end{array}\right) \cdot g := {}^t g \left( \begin{array}{l@{\quad }l} r &{}d \sqrt{n}\\ d \sqrt{n}&{}a \end{array}\right) g,\; g \in \widehat{G}. \end{aligned}$$
(6.2)
Thus we have an anti-homomorphism:
$$\begin{aligned} \alpha :\widehat{G}/\{\pm 1\} \rightarrow \mathrm{O }(\mathrm{Sym }_2(n,\mathbb{Z })). \end{aligned}$$

Theorem 6.2

[14, Thm. 6.16, Prop. 6.19] Let \(\Phi \in \mathrm{Eq }_0(\mathbf D (Y),\mathbf D (X))\) be an equivalence.
(1)
\(v_1:=v(\Phi (\mathcal{O }_Y))\) and \(v_2:=\Phi (\varrho _Y)\) are positive isotropic Mukai vectors with \(\langle v_1,v_2 \rangle =-1\) and we can write
$$\begin{aligned}&v_1=(p_1^2 r_1,p_1 q_1 H_Y, q_1^2 r_2),\quad v_2=(p_2^2 r_2,p_2 q_2 H_Y, q_2^2 r_1),\nonumber \\&p_1,q_1,p_2,q_2, r_1, r_2 \in \mathbb{Z },\;\; p_1,r_1,r_2 >0,\nonumber \\&r_1 r_2=n,\;\; p_1 q_2 r_1-p_2 q_1 r_2=1. \end{aligned}$$
(6.3)
(2)
We set
$$\begin{aligned} \theta (\Phi ):=\pm \left( \begin{array}{l@{\quad }l} p_1 \sqrt{r_1} &{} q_1 \sqrt{r_2}\\ p_2 \sqrt{r_2} &{} q_2 \sqrt{r_1} \end{array}\right) \in G/\{\pm 1\}. \end{aligned}$$
(6.4)
Then \(\theta (\Phi )\) is uniquely determined by \(\Phi \) and we have a map
$$\begin{aligned} \theta :\mathcal{E } \rightarrow G/\{\pm 1\}. \end{aligned}$$
(6.5)
(3)
The action of \(\theta (\Phi )\) on \(\mathrm{Sym }_2(n,\mathbb{Z })\) is the action of \(\Phi \) on \(\mathrm{Sym }_2(n,\mathbb{Z })\):
$$\begin{aligned} \iota _X \circ \Phi (v) =\iota _Y(v)\cdot \theta (\Phi ). \end{aligned}$$
(6.6)
Thus we have the following commutative diagram:
https://static-content.springer.com/image/art%3A10.1007%2Fs00209-013-1214-1/MediaObjects/209_2013_1214_Equ49_HTML.gif
(6.7)

From now on, we identify the Mukai lattice \(H^*(X,\mathbb{Z })_{\mathrm{alg }}\) with \(\mathrm{Sym }_2(n,\mathbb{Z })\) via \(\iota _X\). Then for \(g \in \widehat{G}\) and \(v \in H^*(X,\mathbb{Z })_{\mathrm{alg }}, v \cdot g\) means \(\iota _X(v \cdot g)=\iota _X(v) \cdot g\).

We also need to treat the composition of a Fourier–Mukai transform and the dualizing functor \(\mathcal{D }_X\). For a Fourier–Mukai transform \(\Phi _{X \rightarrow Y}^\mathbf{E } \in \mathrm{Eq }(\mathbf D (X),\mathbf D (Y))\), we set
$$\begin{aligned} \theta (\Phi _{X \rightarrow Y}^\mathbf{E }\mathcal{D }_X) := \left( \begin{array}{l@{\quad }l} 1 &{} 0\\ 0 &{} -1 \end{array}\right) \theta (\Phi _{X \rightarrow Y}^\mathbf{E }) \in \widehat{G}/\{\pm 1\}. \end{aligned}$$
Then the action of \(\theta (\Phi _{X \rightarrow Y}^\mathbf{E }\mathcal{D }_X)\) on \(\mathrm{Sym }_2(\mathbb{Z },n)\) is the same as the action of \(\Phi _{X \rightarrow Y}^\mathbf{E }\mathcal{D }_X\).

Lemma 6.3

[14, Lemma 6.18] If \(\theta (\Phi _{X \rightarrow Y}^\mathbf{E })= \left( \begin{array}{l@{\quad }l} a &{} b\\ c &{} d \end{array}\right) \), then
$$\begin{aligned} \theta (\Phi _{Y \rightarrow X}^\mathbf{E })= \pm \left( \begin{array}{l@{\quad }l} d &{} b\\ c &{} a \end{array}\right) ,\quad \theta (\Phi _{Y \rightarrow X}^\mathbf{E ^{\vee }[2]})= \pm \left( \begin{array}{c@{\quad }c} d &{} -b\\ -c &{} a \end{array}\right) ,\quad \theta (\Phi _{X \rightarrow Y}^\mathbf{E ^{\vee }[2]})= \pm \left( \begin{array}{c@{\quad }c} a &{} -b\\ -c &{} d \end{array}\right) . \end{aligned}$$

6.2 The arithmetic group \(G\) and numerical solutions for the ideal sheaf

Let \(\ell \in \mathbb Z _{>0}\). We assume that \(\sqrt{\ell n}\notin \mathbb Z \). Our next task is to describe the numerical solution of the ideal sheaf of 0-dimensional subscheme. First we introduce an arithmetic group \(S_{n,\ell }\).

Definition 6.4

For \((x,y)\in \mathbb R ^2\), set
$$\begin{aligned} P(x,y):= \left( \begin{array}{l@{\quad }l} y&{}\ell x\\ x&{}y \end{array}\right) . \end{aligned}$$
We also set
$$\begin{aligned} S_{n,\ell }:= \left\{ \left( \begin{array}{l@{\quad }l} y&{}\ell x\\ x&{}y \end{array}\right) \,\bigg |\, \begin{aligned} x=a \sqrt{r}, y=b \sqrt{s},\; a,b,r, s \in \mathbb Z \\ r,s>0,\; rs=n,\; y^2-\ell x^2=\pm 1 \end{aligned} \right\} . \end{aligned}$$

Lemma 6.5

(1)

\(S_{n,\ell }\) is a commutative subgroup of \(\mathrm{GL }(2,\mathbb R )\).

(2)
We have a homomorphism
$$\begin{aligned} \begin{array}{c c c c} \phi :&{} S_{n,\ell } &{} \longrightarrow &{} \mathbb R ^\times \\ &{} P(x,y) &{} \mapsto &{} y+x\sqrt{\ell }. \end{array} \end{aligned}$$
(3)
For \(\ell >1, \phi \) is injective. For \(\ell =1\), we have
$$\begin{aligned} \mathrm{Ker }\phi =\left\langle \ \left( \begin{array}{l@{\quad }l} 0&{}1 \\ 1&{}0 \end{array}\right) \right\rangle . \end{aligned}$$
(4)
We set a subgroup \(G_{n,\ell }\) of \(\widehat{G}\) (Definition 6.1) to be
$$\begin{aligned} G_{n,\ell }:= \left\{ g \in \widehat{G} \left| \; {}^{t} g \left( \begin{array}{c@{\quad }c} 1 &{} 0 \\ 0 &{} -\ell \end{array}\right) g =\pm \left( \begin{array}{c@{\quad }c} 1 &{} 0 \\ 0 &{} -\ell \end{array}\right) \right. \right\} . \end{aligned}$$
Then
$$\begin{aligned} G_{n,\ell } =S_{n,\ell } \rtimes \left\langle \left( \begin{array}{c@{\quad }c} 1 &{} 0 \\ 0 &{} -1 \end{array}\right) \right\rangle . \end{aligned}$$

Proof

The proofs of (1) and (2) are straightforward.

For (3), assume that \(x,y\in \mathbb R \) with \(x^2,y^2,x y/\sqrt{n}\in \mathbb Q \) satisfy \(y+x\sqrt{\ell }=1\). Then \((y^2+\ell x^2)+2(x y/\sqrt{n})\sqrt{\ell n}=(y+x\sqrt{\ell })^2=1\). Our assumptions yield \(y^2+\ell x^2=1\) and \(x y=0\). If \(x=0\), then \(y=\pm 1\). If \(y=0\), then \(\ell =1\) and \(x=1\). Hence the conclusion holds.

(4) follows from direct computations. \(\square \)

Then the Dirichlet unit theorem yields the following corollary.

Corollary 6.6

If \(\ell >1\), then \(S_{n,\ell }\cong \mathbb Z \oplus \mathbb Z /2\mathbb Z \).

Proof

Let \(p_1,\ldots ,p_m\) be the prime divisors of \(\ell n\) and \(\mathfrak o \) be the ring of algebraic integers in \(\mathbb Q (\sqrt{p_1},\ldots ,\sqrt{p_m})\). By Dirichlet unit theorem \(\mathfrak o ^\times \) is a finitely generated abelian group whose torsion subgroup is \(\{\pm 1\}\). Hence \(\phi (S_{n,\ell })\) is a finitely generated abelian group whose torsion subgroup is \(\{\pm 1\}\). For \(A\in S_{n,\ell }\), we have \(\phi (A^2)\in \mathbb Z [\sqrt{\ell n}]\). Since \(\mathbb Z [\sqrt{n\ell }]^\times \cong \mathbb Z \oplus \mathbb Z /2\mathbb Z \), we get \(S_{n,\ell }\cong \mathbb Z \oplus \mathbb Z /2\mathbb Z \).\(\square \)

Remark 6.7

If \(n=1\) and \(\ell >1\), then \(S_{1,\ell }\) is the group of units of \(\mathbb Z [\sqrt{\ell }]\). Moreover if \(\ell \) is square free and \(\ell \equiv 2,3 \pmod {4}\), then since \(\mathbb Z [\sqrt{\ell }]\) is the ring of the integers of \(\mathbb Q [\sqrt{\ell }]\), a generator of \(S_{1,\ell }\) becomes a fundamental unit.

Lemma 6.8

For two positive isotropic Mukai vectors \(w_0,w_1\) on the fixed abelian surface \(X\), the condition
$$\begin{aligned} (1,0,-\ell )=\pm (\ell w_0-w_1),\quad \langle w_0,w_1\rangle =-1 \end{aligned}$$
is equivalent to
$$\begin{aligned} w_0=(p^2,-\tfrac{p q}{\sqrt{n}}H,q^2),\ w_1=(q^2,-\tfrac{\ell p q}{\sqrt{n}}H,\ell ^2 p^2),\quad P(p,q)\in S_{n,\ell }. \end{aligned}$$

Proof

Assume that there are isotropic Mukai vectors with the first condition, Since \(w_0\) is isotropic and \(\mathrm{rk }w_0>0\), we can write \(w_0=(p^2,-\tfrac{p q}{\sqrt{n}}H,q^2)\), where \(p^2, \tfrac{p q}{\sqrt{n}}, q^2 \in \mathbb{Z }\). Since \(\langle (1,0,-\ell ),w_0 \rangle =\pm 1\), we have \(-q^2+\ell p^2=\pm 1\). Then we see that \(w_1=\ell w_0 \mp (1,0,-\ell ) =(q^2,-\tfrac{\ell p q}{\sqrt{n}}H,\ell ^2 p^2)\). Then we also see that \(P(p,q)\in S_{n,\ell }\). The converse is obvious.\(\square \)

Corollary 6.9

Recall the action \(\cdot \) of \(\mathrm{GL }(2,\mathbb R )\) given in (6.2). By the correspondence
$$\begin{aligned} \!G_{n,\!\ell } \!\ni \!g \!\mapsto (w_0,w_1),\quad w_0:=(0,0,1)\cdot g, \ w_1:=(1,0,0)\cdot g, \end{aligned}$$
we have a bijective correspondence:
$$\begin{aligned} \begin{array}{ccc} \!G_{\!n,\ell } \Big / \left\langle \pm \left( \begin{array}{l@{\quad }l} 1 &{} 0\\ 0 &{} -1 \end{array}\right) \right\rangle \cong S_{n,\ell }/\{\pm 1\} &{} \!\longleftrightarrow \! &{} \left\{ (w_0,w_1) \left| \begin{aligned} \!\langle \!w_0,\!w_1\rangle \!=\!-\!1,\, \!\langle w_0^2 \rangle =\langle w_1^2\rangle \!=\!0,\\ \!w_0,\!w_1\!>\!0,\, (1,0,\!-\!\ell )\!=\!\pm (\!\ell \!w_0\!-\!w_1) \end{aligned} \right. \right\} \end{array} \end{aligned}$$

Definition 6.10

Assume that \(\ell >1\). Let
$$\begin{aligned} A_\ell := \left( \begin{array}{l@{\quad }l} q &{} \ell p\\ p &{} q \end{array}\right) ,\;p,q>0 \end{aligned}$$
be the generator of \(S_{n,\ell }/\{\pm 1\}\). We set \(\epsilon := q^2-\ell p^2\in \{\pm 1\}\). For \(m \in \mathbb Z \), we set
$$\begin{aligned} \left( \begin{array}{l@{\quad }l} q &{} \ell p \\ p &{} q \end{array}\right) ^m= \left( \begin{array}{l@{\quad }l} b_m &{} \ell a_m\\ a_m &{} b_m \end{array}\right) . \end{aligned}$$
By the definition we have
$$\begin{aligned} (a_0,b_0)=(0,1),\quad (a_{-m},b_{-m})=\epsilon ^m(-a_m,b_m),\, m\in \mathbb Z _{>0} \end{aligned}$$
and
$$\begin{aligned} S_{n,\ell }= \left\{ \left. \pm \left( \begin{array}{l@{\quad }l} b_m &{} \ell a_m\\ a_m &{} b_m \end{array}\right) \right| m \in \mathbb Z \right\} . \end{aligned}$$
(6.8)
Next we consider the right action of \(\mathrm{GL }(2,\mathbb R )\) on \(\mathbb R ^2\)
$$\begin{aligned} (x,y) \mapsto (x,y) X, \quad X \in \mathrm{GL }(2,\mathbb R ). \end{aligned}$$
(6.9)
Then the quadratic map
$$\begin{aligned} \begin{array}{c c c} \mathbb R ^2 &{} \rightarrow &{} \mathrm{Sym }_2(\mathbb R )\\ (x,y) &{} \mapsto &{} \left( \begin{array}{l} x \\ y \end{array}\right) \left( \begin{array}{l@{\quad }l} x&y \end{array}\right) \end{array} \end{aligned}$$
is \(\mathrm{GL }(2,\mathbb R )\)-equivariant. Using this action, we have the next descriptions of the topological invariants of fine moduli spaces \(M_H(v)\) of dimension 2.
$$\begin{aligned}&\left\{ v \left| \begin{aligned} v \in H^*(X,\mathbb Z )_{\mathrm{alg }},\, \langle v^2\rangle =0,\, v>0,\nonumber \\ \langle w,v\rangle =-1,\,\exists w \in H^*(X,\mathbb Z )_{\mathrm{alg }} \end{aligned} \right. \right\} \nonumber \\&\quad \overset{\varphi _1}{\longleftrightarrow } \left\{ (a\sqrt{r}, b\sqrt{s}) \!\in \! \mathbb R ^2 \left| \begin{aligned} a,b\in \mathbb Z ,\, r,s\in \mathbb Z _{>0},\nonumber \\ r s\!=\!n,\, \gcd (a r,b s)\!=\!1 \end{aligned} \right. \right\} /\{\pm 1\} = \left\{ (0,1) X \mid X \in G \right\} /\{\pm 1\}\nonumber \\&\quad \overset{\varphi _2}{\longleftrightarrow } \left\{ \left. \dfrac{b \sqrt{n}}{a r} \in \mathbb{P }^1(\mathbb R ) =\mathbb R \cup \{ \infty \} \right| r s=n,\; \gcd (a r,b s)=1 \right\} , \end{aligned}$$
(6.10)
where we used the correspondences
$$\begin{aligned} v=(a^2 r, a b H,b^2 s) \overset{\varphi _1}{\longleftrightarrow } \pm (a\sqrt{r},b\sqrt{s}) \overset{\varphi _2}{\longleftrightarrow } \dfrac{\mu (v)}{2\sqrt{n}}=\dfrac{b \sqrt{n}}{a r}. \end{aligned}$$
Here we used the slope for the Mukai vector defined by \(\mu (v):=(H,c_1(v))/\mathrm{rk }v\). These correspondences are \(\widehat{G}\)-equivariant under the action (6.9). Lemma 6.8, (6.8) and (6.10) imply the following one to one correspondence:
$$\begin{aligned} \begin{array}{c c c} \left\{ \{ \!v_1,\!v_2 \} \left| \begin{aligned} &{}\text { \!There is a numerical solution}\\ &{}(\!v_1,\!v_2,\!\ell _1,\!\ell _2)\text { of }(\!1,0,\!-\!\ell ) \end{aligned} \right. \right\} &{} \!\!\longleftrightarrow \! &{} \left\{ \left. \left\{ \frac{\!b_m}{\!a_m}, \frac{\!\ell a_m}{b_m} \right\} \subset \mathbb{P }^1(\mathbb R ) \right| m \in \mathbb Z \right\} \\ \{ \!v_1,\!v_2 \} &{} \!\!\longleftrightarrow \! &{} \left\{ \frac{\mu (v_1)}{2\sqrt{n}},\frac{\!\mu (v_2)}{2\sqrt{n}} \right\} , \end{array}\qquad \quad \end{aligned}$$
(6.11)
where \((\ell _i,\ell _j)=(\ell ,1)\) if and only if \(\left( \frac{\mu (v_i)}{2\sqrt{n}},\frac{\mu (v_j)}{2\sqrt{n}}\right) = \left( \frac{b_m}{a_m},\frac{\ell a_m}{b_m}\right) \).

Definition 6.11

For \(m \in \mathbb{Z }\), we set
$$\begin{aligned} u_m&:= a_m^2 e^{\frac{b_m}{a_m \sqrt{n}}H}= \left( a_m^2,\tfrac{a_m b_m}{\sqrt{n}}H,b_m^2\right) ,\nonumber \\ u_m'&:= b_m^2 e^{\frac{\ell a_m}{b_m \sqrt{n}}H}= \left( b_m^2,\tfrac{\ell a_m b_m}{\sqrt{n}}H,\ell ^2 a_m^2\right) . \end{aligned}$$
(6.12)

6.3 Codimension 0 walls and the action of \(G_{n,\ell }\)

Definition 6.12

\(C_0\) is the wall associated to the numerical solution \((1,\varrho _X,1,\ell )\). For \(m \ne 0\), let \(C_m\) be the wall associated to the numerical solution \((u_m,u_m',\ell ,1)\).

Proposition 6.13

(1)
\(C_0\) is the \(t\)-axis and \(C_m\)\((m \ne 0)\) is the circle defined by
$$\begin{aligned} \left( s-\frac{1}{\sqrt{n}}\frac{b_m}{a_m} \right) \left( s-\frac{1}{\sqrt{n}}\frac{\ell a_m}{b_m}\right) +t^2=0. \end{aligned}$$
(2)

\(\{C_m \mid m \in \mathbb{Z }\}\) is the set of codimension 0 walls.

Proof

By Corollary 5.13 and Definition 6.11, (1) follows. By Lemma 5.18, numerical equations correspond to codimension 0 walls. Hence the claim follows from the correspondence (6.11).\(\square \)

Definition 6.14

  1. (1)
    For \(C_m\) (\(m \in \mathbb{Z }\)), we define adjacent chambers \(C_m^\pm \) as follows:
    • \(C_m^-\) is surrounded by \(C_m^+\) for \(m<0\).

    • \(C_0^- \subset \{(s,t) \mid s<0 \}\) and \(C_0^+ \subset \{(s,t) \mid s>0 \}\).

    • \(C_m^+\) is surrounded by \(C_m^-\) for \(m>0\).

     
  2. (2)

    Let \(M_{C_m^\pm }(v)\) be the moduli scheme \(M_{(sH,tH)}(v)\) of stable objects for \((s,t) \in C_m^\pm \).

     
Then we have
$$\begin{aligned} M_{C_m^\pm }(v) = \left\{ \begin{array}{l@{\quad }l} \mathfrak M ^{\pm }(u_m,u_m',\ell ,1), &{} \frac{b_m}{a_m}<\frac{\ell a_m}{b_m}\\ \mathfrak M ^{\pm }(u_m',u_m,1,\ell ), &{} \frac{b_m}{a_m}>\frac{\ell a_m}{b_m} \end{array}\right. \end{aligned}$$
for \(m<0\) and
$$\begin{aligned} M_{C_m^\pm }(v) = \left\{ \begin{array}{l@{\quad }l} \mathfrak M ^{\mp }(u_m,u_m',\ell ,1)[-1], &{} \frac{b_m}{a_m}<\frac{\ell a_m}{b_m}\\ \mathfrak M ^{\mp }(u_m',u_m,1,\ell )[-1], &{} \frac{b_m}{a_m}>\frac{\ell a_m}{b_m} \end{array}\right. \end{aligned}$$
for \(m \ge 0\). In particular, we have
$$\begin{aligned} M_{C_0^-}(v)=\mathfrak{M }^+(1,\varrho _X,1,\ell )[-1]= M_H(1,0,-\ell ). \end{aligned}$$
(6.13)
We note that \(\Psi \) in Proposition 4.6 satisfies \(\Psi ^{-1}=\Psi \). By Proposition 4.6, we have the following isomorphisms.
$$\begin{aligned} \Psi _m:M_{C_m^\pm }(v) \rightarrow M_{C_m^\mp }(v). \end{aligned}$$
Let \(\Phi _{X_1 \rightarrow X}^{\mathbf{E}_m}\) be a Fourier–Mukai transform such that \(\Phi _{X_1 \rightarrow X}^{\mathbf{E}_m}(\varrho _{X_1})=u_m\) and \(\Phi _{X_1 \rightarrow X}^{\mathbf{E}_m}(1)=u_m'\), where \(X_1=M_H(u_m)\). Then we have
$$\begin{aligned} \theta (\Phi _{X_1 \rightarrow X}^{\mathbf{E}_m})= \left( \begin{array}{l@{\quad }l} b_m &{} \ell a_m\\ \epsilon ^m a_m &{} \epsilon ^m b_m \end{array}\right) = \left( \begin{array}{c@{\quad }c} 1 &{} 0\\ 0 &{} -1 \end{array}\right) ^{\frac{-1+\epsilon ^m}{2}} \left( \begin{array}{l@{\quad }l} q &{} \ell p\\ p &{} q \end{array}\right) ^m. \end{aligned}$$
(6.14)
Thus we get
$$\begin{aligned} \theta (\Psi _m)= \left( \begin{array}{l@{\quad }l} q &{} \ell p\\ p &{} q \end{array}\right) ^{-m} \left( \begin{array}{c@{\quad }c} 1 &{} 0\\ 0 &{} -1 \end{array}\right) \left( \begin{array}{l@{\quad }l} q &{} \ell p\\ p &{} q \end{array}\right) ^m. \end{aligned}$$
(6.15)
We also have
$$\begin{aligned} \left( \begin{array}{l@{\quad }l} q &{} \ell p\\ p &{} q \end{array}\right) ^{m+k} \theta (\Psi _m)&= \left( \begin{array}{l@{\quad }l} q &{} \ell p\\ p &{} q \end{array}\right) ^{m+k} \left( \begin{array}{l@{\quad }l} q &{} \ell p\\ p &{} q \end{array}\right) ^{-m} \left( \begin{array}{c@{\quad }c} 1 &{} 0\\ 0 &{} -1 \end{array}\right) \left( \begin{array}{l@{\quad }l} q &{} \ell p\\ p &{} q \end{array}\right) ^m\nonumber \\&= \left( \begin{array}{c@{\quad }c} 1 &{} 0\\ 0 &{} -1 \end{array}\right) \left( \begin{array}{l@{\quad }l} q &{} \ell p\\ p &{} q \end{array}\right) ^{m-k}. \end{aligned}$$
(6.16)
Hence \((\Psi _m(u_{m+k}),\Psi (u_{m+k}'))=(u_{m-k},u_{m-k}')\). Thus we get the following proposition.

Proposition 6.15

\(\Psi _m(C_{m+k})=C_{m-k}\) and \(\Psi _m(C_{m+k}^\pm )=C_{m-k}^\mp \). In particular, \(\Psi _m\) induces an isomorphism
$$\begin{aligned} M_{C_{m+k}^\pm }(v) \rightarrow M_{C_{m-k}^\mp }(v). \end{aligned}$$

Remark 6.16

$$\begin{aligned} \theta (\Phi _{X_1 \rightarrow X}^{\mathbf{E}_m} \circ \Phi _{X \rightarrow X_1}^{\mathbf{E}_m})= \pm A_\ell ^{2m}. \end{aligned}$$
Hence \(M_H(u_{2m}) \cong X\).

Proposition 6.17

(1)

There are finitely many walls between \(C_0\) and \(C_{-1}\).

(2)

By the action of \(G_{n,\ell }\), every wall is transformed to a wall between \(C_0\) and \(C_{-1}\).

Proof

  1. (1)

    We set \(\lambda _0:=-\frac{q}{p \sqrt{n}}\). Then for \(\beta =\lambda _0 H\), there is finitely many walls. Since every wall between \(C_0\) and \(C_{-1}\) intersects with the line \(s=\lambda _0\), the claim holds.

     
  2. (2)

    Let \(C\) be a wall between \(C_m\) and \(C_{m-1}\). Since \(\Psi _0(C)\) is a wall between \(C_{-m}\) and \(C_{-m+1}\), we may assume that \(m < 0\). By Proposition 6.15, we see that \(\Psi _m(C)\) is a wall between \(C_{m+1}\) and \(C_m\). Therefore \(\Psi _{-1}\circ \cdots \circ \Psi _{m-1}\circ \Psi _m(C)\) is a wall between \(C_0\) and \(C_{-1}\).

     
\(\square \)

Remark 6.18

$$\begin{aligned} \theta (\Psi _{-1}\circ \cdots \circ \Psi _{m-1}\circ \Psi _m) = {\left\{ \begin{array}{ll} \theta (\Psi _{-1})A_\ell ^{-m-1},&{} 2 \not |m,\\ A_\ell ^{-m},&{} 2 |m. \end{array}\right. } \end{aligned}$$

In Proposition 6.15, we did not specify the correspondence of complexes. Since \(\phi _{(sH,tH)}(v) \mod 2\mathbb{Z }\) is well-defined, the correspondence is determined up to shift \([2k]\) (\(k \in \mathbb{Z }\)). We next fix the ambiguity of this shift. We note that \(\theta (\Psi _m)=\theta ([1] \circ \mathcal{D }_X \circ \Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]})\), where \([1]\) is the shift functor. Since \(Z_{(sH,tH)}(v) \in \mathbb{C } {\setminus } \mathbb{R }_{<0}\), we take \(\phi _{(sH,tH)}(v) \in (-1,1)\) to consider the moduli space \(M_{(sH,tH)}(v)\) as in Definition 2.3 (cf. Remark 2.5). Then the isomorphisms in Proposition 6.15 are given by the following proposition.

Proposition 6.19

Assume that \(m<0\). \([1] \circ \mathcal{D }_X \circ \Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]}\) induces isomorphisms
$$\begin{aligned} M_{C_{m+k}^\pm }(v)&\rightarrow M_{C_{m-k}^\mp }(v)\\ E&\mapsto (\Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]}(E))^{\vee }[1]. \end{aligned}$$

Proof

Since \(b_{2m}^2-\ell a_{2m}^2=1\), we have \(\frac{b_{2m}}{a_{2m}}<\frac{\ell a_{2m}}{b_{2m}}<0\) for \(m<0\). For \((s,t) \in C_{2m}, \phi _{(sH,tH)}((\mathbf{E}_{2m})_{|X \times \{x \}}) \in (-1,0]\). Hence \(\phi _{(sH,tH)}((\mathbf{E}_{2m})_{|X \times \{x \}}[1])= \phi _{(sH,tH)}(E)\) for \((s,t) \in C_{2m}\) and \(E \in M_{(sH,tH)}(v)\). For \(E \in M_{(sH,tH)}(v)\), we also have
$$\begin{aligned} {\left\{ \begin{array}{ll} \phi _{(sH,tH)}((\mathbf{E}_{2m})_{|X \times \{x \}}[1])> \phi _{(sH,tH)}(E)>\phi _{(sH,tH)}((\mathbf{E}_{2m})_{|X \times \{x \}}),\; &{} (s,t) \text { is outside of } C_{2m},\\ \phi _{(sH,tH)}((\mathbf{E}_{2m})_{|X \times \{x \}}[2])> \phi _{(sH,tH)}(E)>\phi _{(sH,tH)}((\mathbf{E}_{2m})_{|X \times \{x \}}[1]),\; &{} (s,t) \text { is inside of } C_{2m}. \end{array}\right. } \end{aligned}$$
If \((s,t)\) is outside of \(C_{2m}\), then since \(\phi _{(\widetilde{sH},\widetilde{tH})} (\Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]}((\mathbf{E}_{2m})_{|X \times \{x \}})) =0\), we have \(\phi _{(\widetilde{sH},\widetilde{tH})} (\Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]}(E)) \in (0,1)\). Hence
$$\begin{aligned} \phi _{(-\widetilde{sH},\widetilde{tH})} ((\Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]}(E))^{\vee }[1]) \in (0,1). \end{aligned}$$
If \((s,t)\) is inside of \(C_{2m}\), then \(\phi _{(\widetilde{sH},\widetilde{tH})} (\Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]}(E)) \in (1,2)\). Hence
$$\begin{aligned} \phi _{(-\widetilde{sH},\widetilde{tH})} ((\Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]}(E))^{\vee }[1]) \in (-1,0). \end{aligned}$$
Therefore the claim holds. \(\square \)

Remark 6.20

Assume that \(m>0\). Then \([-1] \circ \mathcal{D }_X \circ \Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]}\) induces an isomorphism
$$\begin{aligned} M_{C_{m+k}^\pm }(v)&\rightarrow M_{C_{m-k}^\mp }(v)\\ E&\mapsto (\Phi _{X \rightarrow X}^{\mathbf{E}_{2m}^{\vee }[1]}(E))^{\vee }[-1]. \end{aligned}$$

Proposition 6.21

We set
$$\begin{aligned} M_m:=M_{C_{m-1}^+}(v) \cap M_{C_m^-}(v). \end{aligned}$$
(1)

\(M_m \ne \emptyset \) and \(M_m\) is birationally equivalent to \(M_{C_{m-1}^+}(v)\) and \(M_{C_m^-}(v)\).

(2)
We have a sequence of isomorphisms
$$\begin{aligned} \cdots \overset{\Psi _{-3}}{\rightarrow } M_{-2} \overset{\Psi _{-2}}{\rightarrow } M_{-1} \overset{\Psi _{-1}}{\rightarrow } M_0 \overset{\Psi _{0}}{\rightarrow } M_{1} \overset{\Psi _{1}}{\rightarrow } M_{2} \overset{\Psi _{2}}{\rightarrow } \cdots . \end{aligned}$$

Proof

  1. (1)

    Since there is no codimension 0 wall between \(C_{m-1}\) and \(C_m, M_m \ne \emptyset \). Since \(M_{C_{m-1}^+}(v)\) and \(M_{C_m^-}(v)\) are irreducible, \(M_m\) is birationally equivalent to \(M_{C_{m-1}^+}(v)\) and \(M_{C_m^-}(v)\).

     
  2. (2)
    By Proposition 6.15 or Proposition 6.19 (and Remark 6.20), we have isomorphisms
    $$\begin{aligned} \Psi _m&:&M_{C_{m-1}^+}(v) \rightarrow M_{C_{m+1}^-}(v),\nonumber \\ \Psi _m&:&M_{C_{m}^-}(v) \rightarrow M_{C_m^+}(v). \end{aligned}$$
    (6.17)
    Hence we have an isomorphism
    $$\begin{aligned} \Psi _m:M_{C_{m-1}^+}(v) \cap M_{C_m^-}(v) \rightarrow M_{C_m^+}(v) \cap M_{C_{m+1}^-}(v). \end{aligned}$$
    Thus the claim holds.
     
\(\square \)
We note that \(M_0\) is an open subset of \(M_{C_0^-}(v)=\{I_Z \otimes L \mid I_Z \in \mathrm{Hilb }^{\ell }(X), L \in \mathrm{Pic }^0(X)\}\). Hence we have two semi-homogeneous presentations of \(I_Z \otimes L \in M_0\):
$$\begin{aligned} 0 \rightarrow I_Z \otimes L \rightarrow L \rightarrow \mathcal{O }_Z \rightarrow 0 \end{aligned}$$
(6.18)
and
$$\begin{aligned} 0 \rightarrow E_{-1} \rightarrow E_0 \rightarrow I_Z \otimes L \rightarrow 0. \end{aligned}$$
(6.19)
Starting from these two semi-homogeneous presentations, we have a sequence of complexes \(F_m^\bullet \in M_m\) such that \(F_0^{\bullet }=I_Z \otimes L\) and \(\Psi _m(F_m^\bullet )=F_{m+1}^\bullet \). Then we have exact triangles
$$\begin{aligned}&V_{m-1}^+ \rightarrow F_m^{\bullet } \rightarrow V_{m-1}^- \rightarrow V_{m-1}^+[1]\nonumber \\&W_{m}^- \rightarrow F_m^{\bullet } \rightarrow W_{m}^+ \rightarrow W_{m}^-[1] \end{aligned}$$
(6.20)
such that
  • \((W_0^-,W_0^+)=(\mathcal{O }_Z[-1],L)\), \((V_{-1}^+,V_{-1}^-)=(E_0,E_{-1}[1])\),

  • \(\Psi _m(W_m^\pm )=V_m^\pm , \Psi _m(V_{m-1}^\pm )=W_{m+1}^\pm \),

  • for \((s,t) \in C_{m-1}, V_{m-1}^\pm \) are \(\sigma _{(sH,tH)}\)-semi-stable objects with the same phase and define the wall \(C_{m-1}\),

  • for \((s,t) \in C_{m}, W_{m}^\pm \) are \(\sigma _{(sH,tH)}\)-semi-stable objects with the same phase and define the wall \(C_m\).

6.4 Fourier–Mukai transforms of the families \(F_m^{\bullet }\)

We first assume \(\epsilon =q^2-\ell p^2=-1\). In this case, the algebraic integers \(a_m,b_m\) in Definition 6.10 satisfy the following relations:
$$\begin{aligned} \dfrac{b_{2 k - 1}}{a_{2 k - 1}}&< \dfrac{\ell a_{2 k}}{b_{2 k}} <\dfrac{b_{2 k + 1}}{a_{2 k + 1}}<\sqrt{\ell } <\dfrac{\ell a_{2 k + 1}}{b_{2 k + 1}}<\dfrac{b_{2 k}}{a_{2 k}} <\dfrac{\ell a_{2 k - 1}}{b_{2 k - 1}} \quad (k\in \mathbb Z _{>0}), \\ \lim _{k\rightarrow \infty }\dfrac{b_k}{a_k}&= \lim _{k\rightarrow \infty } \dfrac{\ell a_k}{b_k}= \sqrt{\ell }. \end{aligned}$$
Thus \(\pm \sqrt{\frac{\ell }{n}}\) are the accumulation points of \(\cup _m C_m\).
We regard \((a_m:b_m)\) and \((b_m:\ell a_m)\) as elements of \(\mathbb{P }^1(\mathbb{R })\). Then the inhomogeneous coordinates of these points give a sequence
$$\begin{aligned} -\infty&= \dfrac{b_0}{a_0} <-\dfrac{\ell p}{q}=\dfrac{\ell a_{-1}}{b_{-1}} <\dfrac{b_{-2}}{a_{-2}}<\cdots <-\sqrt{\ell }<\cdots <\dfrac{\ell a_{-2}}{b_{-2}}<\dfrac{b_{-1}}{a_{-1}}=-\dfrac{q}{p}\nonumber \\&< \dfrac{\ell a_0}{b_0}=0<\dfrac{b_1}{a_1}=\dfrac{q}{p} <\dfrac{\ell a_2}{b_2}<\cdots <\sqrt{\ell }<\cdots <\dfrac{b_2}{a_2} <\dfrac{\ell a_1}{b_1}=\dfrac{\ell p}{q}<\dfrac{b_0}{a_0}=\infty ,\nonumber \\ \end{aligned}$$
(6.21)
where we write the inhomogeneous coordinate of \((0:1)\) as \(\infty \) or \(-\infty \).

For a Fourier–Mukai transform \(\Phi _{X \rightarrow X'}^\mathbf{G ^{\vee }}:\mathbf D (X) \rightarrow \mathbf D (X')\), we write \(c_1(\mathbf{G }_{|X \times \{x'\}})/ \mathrm{rk }\mathbf{G }_{|X \times \{x'\}}=(\lambda /\sqrt{n}) H, x_1 \in X_1\). If \(-\ell p / q<\lambda <-q/p\), then \(\Phi _{X \rightarrow X'}^{\mathbf{G ^{\vee }}}(F_0^\bullet )\) is not a sheaf for all \(F_0^\bullet \).

Definition 6.22

We set
$$\begin{aligned} I_1&:= \Bigg [0,\tfrac{b_1}{a_1}\Bigg ) \cup \Bigg [\tfrac{\ell a_1}{b_1},\infty \Bigg ), I_0\!:=\! \Bigg [-\infty ,-\tfrac{\ell a_1}{b_1}\Bigg ) \cup \Bigg [-\tfrac{b_1}{a_1},0\Bigg ),\\ I_{2k}&:= \Bigg [\tfrac{b_{2k-1}}{a_{2k-1}},\tfrac{\ell a_{2k}}{b_{2k}}\Bigg ) \cup \Bigg [\tfrac{b_{2k}}{a_{2k}},\tfrac{\ell a_{2k-1}}{b_{2k-1}}\Bigg ), I_{-2k}\!:=\! \Bigg [-\tfrac{b_{2k}}{a_{2k}},-\tfrac{\ell a_{2k+1}}{b_{2k+1}}\Bigg ) \cup \Bigg [-\tfrac{b_{2k+1}}{a_{2k+1}},-\tfrac{\ell a_{2k}}{b_{2k}}\Bigg ),\\ I_{2k+1}&:= \Bigg [\tfrac{\ell a_{2k}}{b_{2k}},\tfrac{b_{2k+1}}{a_{2k+1}}\Bigg ) \cup \Bigg [\tfrac{\ell a_{2k+1}}{b_{2k+1}},\tfrac{b_{2k}}{a_{2k}}\Bigg ), I_{-2k+1}:= \Bigg [-\tfrac{\ell a_{2k-1}}{b_{2k-1}},-\tfrac{b_{2k}}{a_{2k}}\Bigg ) \cup \Bigg [-\tfrac{\ell a_{2k}}{b_{2k}},\!-\!\tfrac{b_{2k-1}}{a_{2k-1}}\Bigg ). \end{aligned}$$
For \(I=\coprod _i [s_i, t_i)\), we denote \(I^*:=\coprod _i (s_i,t_i]\).

By (6.21), we have decompositions \(\mathbb P ^1(\mathbb{R }) {\setminus }\{\pm \sqrt{\ell }\} = \coprod _{m \in \mathbb{Z }}I_m= \coprod _{m \in \mathbb{Z }}I_m^*\).

Theorem 6.23

(1)

If \(\lambda \in I_m\)\((m \le 0)\), then \(\Phi _{X \rightarrow X'}^{\mathbf{G ^{\vee }}}(F_m^\bullet )\) is a stable sheaf up to shift.

(2)

If \(\lambda \in I_m^*\)\((m \le 0)\), then \(\mathcal{D }_{X'}\Phi _{X \rightarrow X'}^{\mathbf{G ^{\vee }}}(F_m^\bullet )= \Phi _{X \rightarrow X'}^{\mathbf{G }[2]}(F_m^{\bullet \vee })\) is a stable sheaf up to shift.

Proof

(1) For a small number \(t>0, (\lambda ,t)\) belongs to the interior of the annulus bounded by \(C_{-m-1}\) and \(C_{-m}\). By Lemma 5.15, \(F_m^{\bullet }\) is \(\sigma _{(\lambda H,tH)}\)-semi-stable. By Proposition , \(\Phi _{X \rightarrow X'}^{\mathbf{G}^{\vee }[n]}(F_m^{\bullet })\) is a stable sheaf, where \(n=1\) for \(\lambda > -\sqrt{\ell }\) and \(n=2\) for \(\lambda <-\sqrt{\ell }\). The proof of (2) is similar. \(\square \)

By this theorem, we have semi-homogeneous presentations for a general member of \(M_H(w), w=\Phi _{X \rightarrow X'}^{\mathbf{G}^{\vee }}(v)\).

Remark 6.24

The claim also follows from [14, Lem. 4.4].

We next assume that \(\epsilon =q^2-\ell p^2=1\). Then the algebraic integers \(a_m,b_m\) in Definition 6.10 satisfy
$$\begin{aligned} 0<\dfrac{b_m}{a_m} -\sqrt{\ell }<\dfrac{b_{m-1}}{a_{m-1}} -\sqrt{\ell } \end{aligned}$$
for \(m\in \mathbb Z _{>0}\). We also have the following sequence of inequalities:
$$\begin{aligned} -\infty&= \dfrac{b_0}{a_0} <-\dfrac{q}{p}=\dfrac{b_{-1}}{a_{-1}} <\dfrac{b_{-2}}{a_{-2}}<\cdots <-\sqrt{\ell }<\cdots <\dfrac{\ell a_{-2}}{b_{-2}}<\dfrac{\ell a_{-1}}{b_{-1}}=-\dfrac{\ell p}{q}\\&< \dfrac{\ell a_0}{b_1}=0 <\dfrac{\ell a_1}{b_1}=\dfrac{\ell p}{q} <\dfrac{\ell a_2}{b_2} <\cdots <\sqrt{\ell } <\cdots <\dfrac{b_2}{a_2}<\dfrac{b_1}{a_1}=\dfrac{q}{p} <\dfrac{b_0}{a_0}=\infty . \end{aligned}$$

Definition 6.25

We set
$$\begin{aligned} I_1&:= \Bigg [0,\tfrac{\ell a_1}{b_1}\Bigg ) \cup \Bigg [\tfrac{b_1}{a_1},\infty \Bigg ), I_0 :=\Bigg [-\infty ,-\tfrac{b_1}{a_1}\Bigg ) \cup \Bigg [-\tfrac{\ell a_1}{b_1},0\Bigg ),\\ I_{m+1}&:= \Bigg [\tfrac{\ell a_m}{b_m},\tfrac{\ell a_{m+1}}{b_{m+1}}\Bigg ) \cup \Bigg [\tfrac{b_{m+1}}{a_{m+1}},\tfrac{b_{m} }{a_{m} }\Bigg ), I_{-m}:=\Bigg [\!-\!\tfrac{b_m}{a_m},\!-\!\tfrac{b_{m+1}}{a_{m+1}}\Bigg ) \cup \Bigg [\!-\!\tfrac{\ell a_{m+1}}{b_{m+1}},-\tfrac{\ell a_{m}}{b_{m}}\Bigg )\,\, m \ge 1. \end{aligned}$$

Then we have \(\mathbb P ^1(\mathbb{R }) {\setminus } \{\pm \sqrt{\ell }\}= \coprod _{m \in \mathbb Z } I_m =\coprod _{m \in \mathbb Z } I_m^*\).

Theorem 6.26

For a Fourier–Mukai transform \(\Phi _{X \rightarrow X'}^{\mathbf{G ^{\vee }}}:\mathbf D (X) \rightarrow \mathbf D (X')\), we write \(c_1(\mathbf{G }_{|X \times \{x'\}})/ \mathrm{rk }\mathbf{G }_{|X \times \{x'\}}=(\lambda /\sqrt{n}) H, x_1 \in X_1\).
(1)

If \(\lambda \in I_m\)\((m \le 0)\), then \(\Phi _{X \rightarrow X'}^\mathbf{G ^{\vee }}(F_m^\bullet )\) is a stable sheaf up to shift.

(2)

If \(\lambda \in I_m^*\)\((m \le 0)\), then \(\mathcal{D }_{X'}\, \Phi _{X \rightarrow X'}^\mathbf{G ^{\vee }}(F_m^\bullet )= \Phi _{X \rightarrow X'}^{\mathbf{G }[2]}\, \mathcal{D }_{X}(F_m^\bullet )\) is a stable sheaf.

The proof is based on the calculation in this subsection and is similar to that of Theorem 6.23. We omit the detail.

7 Examples

Let \(X\) be a principally polarized abelian surface with \(\mathrm{NS }(X)=\mathbb{Z }H\). We shall study walls for \(v=(1,0,-\ell )\). We note that \(n:=\frac{(H^2)}{2}=1\).

By using Corollary 5.7, it is easy to see that \(s=0\) is the unique wall for \(\ell =1\). So we assume that \(\ell \ge 2\). We use the notations in Sect. 6. (1) Assume that \(\ell =2\). In this case, \(S_{1,2}/\{\pm 1 \}\) is generated by
$$\begin{aligned} A_2:= \left( \begin{array}{l@{\quad }l} 1 &{} 2\\ 1 &{} 1 \end{array}\right) . \end{aligned}$$
Hence we have a numerical solution \(v=2(1,-H,1)-(1,-2H,4)\). We have \(u_{-1}=(1,-H,1)\) and \(C_{-1}=W_{u_{-1}}\) is the circle in the \((s,t)\)-plane
$$\begin{aligned} \left( s+\frac{3}{2} \right) ^2+t^2=\frac{1}{2^2}. \end{aligned}$$
For \(s=-1\), we get \(c_1(v e^{-sH})=H\). Thus there is no wall intersecting with \(s=-1\). Then we see that there is no wall between \(C_{-1}\) and \(C_0:s=0\) (see Fig. 1). Hence we get the following result.
https://static-content.springer.com/image/art%3A10.1007%2Fs00209-013-1214-1/MediaObjects/209_2013_1214_Fig1_HTML.gif
Fig. 1

Walls for \(v=1-2\varrho _X\)

Lemma 7.1

The set of walls for \(v=(1,0,-2)\) is given by \(\{C_n \mid n \in \mathbb{Z }\}\).

Proposition 7.2

Let \(v\) be a positive and primitive Mukai vector with \(\langle v^2 \rangle =4\). Then \(M_H(v)\) is isomorphic to \(\mathrm{Hilb }^{2}(X) \times X\).

Proof

There is a Fourier–Mukai transform \(\Phi _{X \rightarrow X}^{\mathbf{E}^{\vee }}:\mathbf{D}(X) \rightarrow \mathbf{D}(X)\) such that \(\Phi _{X \rightarrow X}^{\mathbf{E}^{\vee }}((1,0, -2))=v\). Then we have an isomorphism \(M_{(\beta ,\omega )}(1,0,-2) \rightarrow M_H(v)\), where \(\beta =c_1(\mathbf{E}_{|\{ x\} \times X})/\mathrm{rk }\mathbf{E}_{|\{ x\} \times X}\) and \((\omega ^2) \ll 1\). By Theorem 3.8 and Proposition  6.17, \(M_{(\beta ,\omega )}(1,0,-2) \cong \mathrm{Hilb }^{2}(X) \times X\), which implies the claim. \(\square \)

(2) Assume that \(\ell =3\). In this case, \(S_{1,3}/\{\pm 1 \}\) is generated by
$$\begin{aligned} A_3:= \left( \begin{array}{l@{\quad }l} 2 &{} 3\\ 1 &{} 2 \end{array}\right) . \end{aligned}$$
Hence we have a numerical solution \(v=(4,-6H,9)-3(1,-2H,4)\). We have \(u_{-1}=(1,-2H,4)\) and \(W_{u_{-1}}\) is the circle in \((s,t)\)-plane defined by
$$\begin{aligned} \left( s+\frac{7}{4} \right) ^2+t^2=\frac{1}{4^2}. \end{aligned}$$
We set \(w_{-1}:=(1,-H,1)\). Then \(v=w_{-1}+(0,H,-4)\) and \(w_{-1}\) defines a wall \(W_{w_{-1}}\) while the defining equation is

Lemma 7.3

\(W_{w_{-1}}\) is the unique wall between \(C_0\) and \(C_{-1}=W_{u_{-1}}\) (see Fig. 2).
https://static-content.springer.com/image/art%3A10.1007%2Fs00209-013-1214-1/MediaObjects/209_2013_1214_Fig2_HTML.gif
Fig. 2

Walls for \(v=1-3\varrho _X\)

Proof

Since \(W_{u_{-1}}\) passes the point \((s,t)=(-2,0)\), it is sufficient to classify walls for \(s=-2\). Assume that \(w e^{2H}=(r,dH,a)\) defines a wall for \(s=0\). Since \(v e^{2H}=(1,2H,1)\), we have \(d=1\). We set \(w':=(v-w)\) and write \(w' e^{2H}=(r',H,a')\). By the definition of walls, we have \(\langle w^2 \rangle \ge 0, \langle {w'}^2 \rangle \ge 0\) and \(\langle w,w' \rangle >0\). Thus \(ra \le 1, r'a' \le 1\) and \(1-r a'-r' a>0\). Since \(r+r'=1\), we may assume that \(r>0\) and \(r' \le 0\). Since \(a+a'=1\), we have \(0<1-r a'-r' a=(2a-1)r+(1-a)\). If \(a \le 0\), then \((2a-1)r+(1-a) \le (2a-1)+(1-a)=a \le 0\). Hence \(a \ge 1\). Since \(ra \le 1\), we have \(r=a=1\). Therefore \(w e^{2H}=(1,H,1)\), which implies that \(w=(1,-H,1)\). \(\square \)

We have \(u_0=(0,0,1), u_{-1}=(1,-2H,4), u_{-2}=(4^2,-28H,7^2)\) and so on. We define \(w_n \in H^*(X,\mathbb{Z })_{\mathrm{alg }}\) by
$$\begin{aligned} w_n:=(a_n^2,a_n b_n H,b_n^2),\; (a_n,b_n)=(1,-1)A_3^{n-1}. \end{aligned}$$
Thus \(w_{-1}=(1,-H,1), w_{-2}=(9,-15H,5^2)\) and so on. By Propositions 2.11 and 6.17, we get the following.

Lemma 7.4

The set of walls for \(v=(1,0,-3)\) is given by \(\{W_{u_n},W_{w_n} \mid n \in \mathbb{Z }\}\).

Therefore all moduli spaces are isomorphic to \(\mathrm{Hilb }^{3}(X) \times X\).

Proposition 7.5

Let \(v\) be a positive and primitive Mukai vector with \(\langle v^2 \rangle =6\). Then \(M_H(v)\) is isomorphic to \(\mathrm{Hilb }^{3}(X) \times X\).

Proof

There is a Fourier–Mukai transform \(\Phi _{X \rightarrow X}^{\mathbf{E}^{\vee }}:\mathbf{D}(X) \rightarrow \mathbf{D}(X)\) such that \(\Phi _{X \rightarrow X}^{\mathbf{E}^{\vee }} ((1,0,-3))=v\). Then we have an isomorphism \(M_{(\beta ,\omega )}(1,0,-3) \rightarrow M_H(v)\), where \(\beta =c_1(\mathbf{E}_{|\{ x\} \times X})/\mathrm{rk }\mathbf{E}_{|\{ x\} \times X}\) and \((\omega ^2) \ll 1\). By Theorem 3.8 and Proposition 6.17, \(M_{(\beta ,\omega )}(1,0,-3) \cong \mathrm{Hilb }^{3}(X) \times X\), which implies the claim. \(\square \)

(3) Assume that \(\ell =4\). By Corollary 5.7, all walls in \(s<0\) intersect with the line \(s=-\sqrt{4}=-2\). Assume that \(v_1\) defines a wall for \(v\). Since \(v e^{2H}=(1,2H,0), u_1 e^{2H}=(r,H,a)\). Hence \(v_1=(r,(1-2r)H,a-4+4r)\) and \(v-v_1=(1-r,(2r-1)H,-a-4r)\). Replacing \(v_1\) by \(v-v_1\) if necessary, we may assume that \(r> 0\). We have \(2n:=\langle v_1^2 \rangle \ge 0, \langle (v-v_1)^2 \rangle \ge 0\) and \(\langle v_1,v-v_1 \rangle >0\). Hence \(n=1-ra, 4-2n>a \ge -n\). Then \(n=0,1,2,3\), which implies that \(ra=1,0,-1,-2\). We also have \(\left| \frac{a-4+4r+4r}{2(1-2r)}\right| >2\) by (5.3), which implies that \(a(8(2r-1)+a)>0\). In particular, \(a \ne 0\). Then \(ra=1,0,-1,-2\) and \(r> 0\) implies that \(r=1,2\). If \(r=1\), then \(a(a+8)>0\) and \(ra=1,0,-1,-2\) imply that \(a=1\). Thus \(v_1=(1,-H,1)\). If \(r=2\), then \(a=-1\) and \(a(a+24)>0\), which is impossible. Therefore \(v_1=(1,-H,1)\). Thus \(W_{v_1}\) is the unique wall which is defined by
$$\begin{aligned} \left( s+\frac{5}{2} \right) ^2+t^2=\frac{3^2}{2^2}. \end{aligned}$$

Proposition 7.6

Let \(v\) be a positive and primitive Mukai vector with \(\langle v^2 \rangle =8\). Then \(M_H(v)\) is isomorphic to \(\mathrm{Hilb }^{4}(X) \times X\) or \(M_H(0,2H,-1)\).

Proof

We first prove that https://static-content.springer.com/image/art%3A10.1007%2Fs00209-013-1214-1/MediaObjects/209_2013_1214_IEq1653_HTML.gif. We note that the Hilbert–Chow morphism of \(\mathrm{Hilb }^{4}(X)\) induces a divisorial contraction of \(M_H(1,0,-4)\). We note that \(M_H(3,H,-1)\) has a morphism to the Uhlenbeck compactification of the moduli of stable vector bundles, which contracts a \(\mathbb{P }^2\)-bundle over \(M_H(3,H,0)\). Let \(\mathbf{P}\) be the Poincaré line bundle on \(X \times X\), where we identify \(\mathrm{Pic }^0(X)\) with \(X\). Then we have an isomorphism \(M_H(1,0,-4) \cong M_H(1,H,-3) \cong M_H(3,H,-1)\) by sending \(E\) to \(\Phi _{X \rightarrow X}^{\mathbf{P}[1]}(E(H))\) [16, Prop. 3.5]. Hence \(M_H(1,0,-4)\) has another contraction. There is no other contraction by a similar argument in [16, Example 7.2]. On the other hand, \(M_H(0,2H,-1)\) has a Lagrangian fibration. Therefore https://static-content.springer.com/image/art%3A10.1007%2Fs00209-013-1214-1/MediaObjects/209_2013_1214_IEq1668_HTML.gif.

For \(s=-2\) and \(v=(1,0,-4)\), we have two moduli spaces \(M_{(-2H,t_1 H)}(1,0,-4)\) and \(M_{(-2H,t_2 H)}(1,0,-4)\), where \(t_1 >\sqrt{2}\) and \(t_2<\sqrt{2}\). For \(E \in M_{(-2H,t_2 H)}(1,0,-4), \Phi _{X \rightarrow X}^{\mathbf{P}[1]}(E(2H)) \in M_H(0,2H,-1)\) and we have an isomorphism \(M_{(-2H,t_2 H)}(1,0,-4) \cong M_H(0,2H,-1)\).

By [14, sect. 7.3], every quadratic form \(r x^2+2d xy+a y^2\) is equivalent to \(x^2-4y^2\). Since \((1,0,-4)^{\vee }=(1,0,-4)\), there is an auto-equivalence \(\Phi \) such that \(\Phi (v)=(1,0,-4)\). Then \(\Phi (M_H(v)) \cong M_{(sH,tH)}(1,0,-4)\) for a suitable \((s,t)\). Therefore the claim holds. \(\square \)

(4) Assume that \(\ell =5\). In this case, \(S_{1,5}/\{ \pm 1 \}\) is generated by
$$\begin{aligned} A_5:= \left( \begin{array}{l@{\quad }l} 2 &{} 5\\ 1 &{} 2 \end{array}\right) . \end{aligned}$$
Hence we have a numerical solution \(v=5(1,-2H,4)-(4,-10H,25)\). Then we have \(u_{-1}=(1,-2H,4)\) and \(W_{u_{-1}}\) is the circle defined by
$$\begin{aligned} \left( s+\frac{9}{4} \right) ^2+t^2=\frac{1}{4^2}. \end{aligned}$$
(5) Assume that \(\ell =6\). In this case, \(S_{1,6}/\{ \pm 1 \}\) is generated by
$$\begin{aligned} A_6 := \left( \begin{array}{l@{\quad }l} 5 &{} 12\\ 2 &{} 5 \end{array}\right) . \end{aligned}$$
Hence we have a numerical solution \(v=(5^2,-60H,12^2)-6(4,-10H,5^2)\). Then we have \(u_{-1}=(5^2,-10H,5^2)\) and \(W_{u_{-1}}\) is the circle defined by
$$\begin{aligned} \left( s+\frac{49}{20} \right) ^2+t^2=\frac{1}{20^2}. \end{aligned}$$

Remark 7.7

For the classification of moduli spaces \(M_H(v)\) with \(\langle v^2 \rangle /2=1\), it is not sufficient to treat the case where \(v\) is represented by \((1,0,-1)\) modulo the action of \(\mathrm{GL }(2,\mathbb{Z })\). Actually there is another orbit represented by \(v=(0,H,0)\). Both cases are treated by Mukai [11, Thm. 5.4]. In particular, we have \(M_H(v) \cong \mathrm{Pic }^0(X) \times X\).

Acknowledgments

We would like to thank the referee for valuable suggestions.

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