The Picard index of a surface with torus action

Abstract

We consider normal rational projective surfaces with torus action and provide a formula for their Picard index, that means the index of the Picard group inside the divisor class group. As an application, we classify the log del Pezzo surfaces with torus action of Picard number one up to Picard index $10\,000$.

1 Introduction

Consider a normal variety $X$ defined over an algebraically closed field $\mathbb {K}$ of characteristic zero. If $X$ is normal, then the Picard group $\textrm{Pic}(X)$ embeds into the divisor class group $\textrm{Cl}(X)$ as the subgroup consisting of the Cartier divisor classes, and the Picard index $[\textrm{Cl}(X):\textrm{Pic}(X)]$ measures the amount of non-invertible reflexive rank one sheaves on $X$. For rational normal projective surfaces admitting a (non-trivial) action of the multiplicative group $\mathbb {K}^*$, we provide the following formula, involving the torsion part $\textrm{Cl}(X)^{\textrm{tors}}$ and the local class groups $\textrm{Cl}(X,x)$, hosting the Weil divisors modulo those being principal near $x \in X$.

Theorem 1.1

The Picard index of a normal rational projective $\mathbb {K}^*$-surface $X$ is given by

$$\begin{aligned}{}[\textrm{Cl}(X): \textrm{Pic}(X)]\ =\ \frac{1}{|\textrm{Cl}(X)^{\textrm{tors}}|} \prod _{x \in X} |\textrm{Cl}(X,x)|. \end{aligned}$$

Note that rationality forces our $\mathbb {K}^*$-surface $X$ to be $\mathbb {Q}$-factorial and $\textrm{Cl}(X)$ to be finitely generated, see for instance [2, Thm. 5.4.1.5]. Moreover, by normality of $X$, there are only finitely many singular points and these are the only possible contributors of non-trivial local class groups. Thus, all terms in our formula are indeed finite.

Beyond the $\mathbb {K}^*$-surfaces, the formula trivially holds for all smooth projective surfaces with a finitely generated and torsion free divisor class group. As soon as we allow torsion, the r.h.s. is no longer integral in the smooth case and thus the formula fails. Concrete examples are the Enriques surfaces, having divisor class group $\mathbb {Z}^{10} \times \mathbb {Z}/2\mathbb {Z}$. A singular counterexample without $\mathbb {K}^*$-action is provided by the $D_8$-singular log del Pezzo surface of Picard number one: it is $\mathbb {Q}$-factorial with divisor class group $\mathbb {Z}\times \mathbb {Z}/ 2 \mathbb {Z}$ and doesn’t satisfy the formula, see Example 7.6.

Our motivation to consider the Picard index arises from the study of log del Pezzo surfaces. Recall that these are normal projective surfaces with an ample anticanonical divisor and at most finite quotient singularities. The log del Pezzo surfaces form an infinite class, which can be filtered into finite subclasses by further conditions on the singularities. Common conditions are bounding the Gorenstein index or the log terminality; for the state of the art we refer to [1, 5, 17] in the general case and to [7, 8, 10] in the case of log del Pezzo surfaces with $\mathbb {K}^*$-action. The idea of filtering by the Picard index has appeared in [12], where not-necessarily log terminal Fano varieties with divisor class group $\mathbb {Z}$ and a torus action of complexity one have been considered. Here, we use Theorem 1.1 to derive in Picard number one suitable bounds on toric and non-toric log del Pezzo $\mathbb {K}^*$-surfaces and then present a classification algorithm. Explicit results are obtained up to Picard index $1\,000\,000$ in the toric case and up to Picard index $10\,000$ in the non-toric case. The defining matrices for the surfaces are available under [9], the toric ones up to Picard index $10\,000$ and the non-toric ones up to Picard index $2\,500$.

Theorem 1.2

There are $1\,415\,486$ families of log del Pezzo $\mathbb {K}^*$-surfaces of Picard number one and Picard index at most $10\,000$. Of those, $68\,053$ are toric and $1\,347\,433$ are non-toric. The number of families for given Picard index develops as follows:

Let us give an idea of the proof of Theorem 1.1. We use the approach of [11, 13], encoding rational projective $\mathbb {K}^*$ -surfaces $X$ via integral $n \times r$ matrices  $P$, see also [2, Sections 3.4, 5.4]. The columns $v_1, \ldots , v_r$ of $P$ are the primitive ray generators of the fan $\Sigma$ of a toric variety $Z$ hosting our $X$ as a closed $\mathbb {K}^*$-invariant subvariety. As we will mention in Proposition 3.8, a basic feature of this encoding is that $X$ inherits divisor class group, Picard group and local class groups from its ambient toric variety:

$$\begin{aligned} \textrm{Cl}(X) = \textrm{Cl}(Z), \qquad \textrm{Pic}(X)\ =\ \textrm{Pic}(Z), \qquad \textrm{Cl}(X,x)\ =\ \textrm{Cl}(Z,x). \end{aligned}$$

This allows us to work entirely in the setting of toric varieties and to provide in Proposition 2.7 a first formula of the Picard index $\iota _{\textrm{Pic}}(X)$, which is in fact, independently from the embedded $X$, valid for any $\mathbb {Q}$-factorial toric variety $Z$:

$$\begin{aligned} \iota _{\textrm{Pic}}(X) = \iota _{\textrm{Pic}}(Z) \ = \ \frac{1}{\vert \textrm{coker}({\hat{P}}^*) \vert } \prod _{\sigma \in S} \vert \textrm{Cl}(Z,z_\sigma ) \vert . \end{aligned}$$

Here, $S \subseteq \Sigma$ contains the cones $\sigma$ defining a closed torus orbit $\mathbb {T}^n \cdot z_{\sigma } \subseteq Z$. The lattice homomorphism ${\hat{P}}$ shows up when we describe the Picard group inside the divisor class group as the intersection over the kernels of the projections onto the local class groups of the points $z_\sigma \in Z$. In terms of the defining data, ${\hat{P}}$ is given as follows. Every cone $\sigma \in \Sigma$ defines lattices

$$\begin{aligned} {\textstyle N_{\sigma } \,= \ \textrm{lin}_{\mathbb {Q}}(\sigma ) \cap \mathbb {Z}^n, \qquad \qquad F_{\sigma } \,= \ \underset{v_i\in \sigma }{\bigoplus }\ \mathbb {Z}\cdot f_{\sigma ,i}. } \end{aligned}$$

Moreover, we have the homomorphisms $P_{\sigma } :F_{\sigma } \rightarrow N_{\sigma }$, sending the basis vector $f_{\sigma ,i} \in F_\sigma$ to $v_i \in N_\sigma$. These fit together to a homomorphism

$$\begin{aligned} {\textstyle \underset{\sigma \in S}{\bigoplus }\ P_{\sigma } :\underset{\sigma \in S}{\bigoplus }\ F_{\sigma } \ \rightarrow \ \underset{\sigma \in S}{\bigoplus }\ N_{\sigma }. } \end{aligned}$$

The desired lattice homomorphism ${\hat{P}} :\ker (\beta ) \rightarrow \ker (\alpha )$ is the (well defined) restriction of $\oplus P_{\sigma }$ to the kernels of

$$\begin{aligned} {\textstyle \beta :\underset{\sigma \in S}{\bigoplus } F_{\sigma } \rightarrow \mathbb {Z}^r, \quad f_{\sigma , i} \mapsto e_i, \qquad \qquad \alpha :\underset{\sigma \in S}{\bigoplus }\ N_{\sigma } \rightarrow \mathbb {Z}^n, \quad (v_{\sigma })_{\sigma \in S} \mapsto \underset{\sigma \in S}{\sum }\ v_{\sigma }. } \end{aligned}$$

At this point the serious and technical part of the proof of Theorem 1.1 begins, namely to verify the identity

$$\begin{aligned} \vert \textrm{coker}({\hat{P}}^*) \vert \ = \ \vert \textrm{Cl}(Z)^{\textrm{tors}} \vert . \end{aligned}$$

From toric geometry, we know $\textrm{Cl}(Z)^{\textrm{tors}} = \textrm{coker}(P)$. In order to compare the cokernel orders, we give an explicit matrix representation of ${\hat{P}}$ and work with maximal minors on both sides, see Proposition 7.1. This part of the proof heavily depends on the specific shape of the defining matrices $P$ for $\mathbb {K}^*$-surfaces. Once this is done, Theorem 1.1 follows from the observation that the points $x \in X$ with non-trivial local class group correspond to points $z_{\sigma }$ with $\sigma \in S$.

This article is structured as follows. In Sect. 2, we begin the study of the Picard index in a purely toric setting. In Sect. 3, we recall the combinatorial theory of $\mathbb {K}^*$-surfaces in terms of the matrices $P$ mentioned before. The technical part of the proof of Theorem 1.1 starts in Sect. 4, where we study maximal minors of $P$. In Sect. 5, we give an explicit matrix representation of ${\hat{P}}$ and in Sect. 6, we study its maximal minors. In Sect. 7, we establish the equality of the cokernel orders of $P$ and ${\hat{P}}^*$, completing the proof of Theorem 1.1. Finally, in Sect. 8, we present the classification algorithm for log del Pezzo $\mathbb {K}^*$-surfaces of Picard number one with given Picard index, proving Theorem 1.2.

The author would like to thank Prof. Jürgen Hausen for his valuable feedback and advice. Moreover, the author thanks the referees for their very helpful comments and suggestions for improving the presentation of the article.

2 The Picard group of a toric variety

In this section, we develop our approach to the Picard group of toric varieties, which yields in Corollary 2.4 a criterion for torsion-freeness and in Proposition 2.7 a first formula involving the Picard index and local class groups. The reader is assumed to be familiar with the basics of toric geometry [4, 6].

Construction 2.1

Let $Z = Z_{\Sigma }$ be a toric variety coming from a fan $\Sigma$ in the lattice $N:= \mathbb {Z}^r$. We assume $\Sigma$ to be non-degenerate, i.e. its primitive ray generators $v_1, \dots , v_n$ span $\mathbb {Q}^r$ as a convex cone. We allow $\Sigma$ to be non-complete. With $F:= \mathbb {Z}^n$, we have the generator map

$$\begin{aligned} P :F \rightarrow N, \qquad e_i \mapsto v_i. \end{aligned}$$

To any maximal cone $\sigma = \textrm{cone}(v_{i_1}, \dots , v_{i_{n_\sigma }}) \in \Sigma _{\max }$, we associate the lattices

$$\begin{aligned} N_\sigma \,=\ \textrm{lin}_{\mathbb {Q}}(\sigma ) \cap N, \qquad \qquad F_\sigma \,=\ \mathbb {Z}^{n_\sigma }. \end{aligned}$$

We define the local generator map associated to $\sigma$ by

$$\begin{aligned} P_\sigma :F_\sigma \rightarrow N_\sigma , \qquad e_j \mapsto v_{i_j}. \end{aligned}$$

With the inclusion $\alpha _\sigma :N_\sigma \hookrightarrow N$ and the map $\beta _\sigma :F_\sigma \rightarrow F$ sending $e_j$ to $e_{i_j}$, we obtain a commutative diagram

Consider the dual lattices

$$\begin{aligned} M:= N^*, \qquad E:= F^*,\qquad M_\sigma := N_\sigma ^*,\qquad E_\sigma := F_\sigma ^*. \end{aligned}$$

Setting $K:= M/\textrm{im}(P^*)$ and $K_\sigma := M_\sigma / \textrm{im}(P_\sigma ^*)$, we obtain a map $\pi _\sigma :K \rightarrow K_\sigma$ fitting into the commutative diagram with exact rows

By standard toric geometry, we have isomorphisms $K \cong \textrm{Cl}(Z)$ and $K_\sigma \cong \textrm{Cl}(U_\sigma )$, where $U_\sigma$ is the affine toric chart associated to $\sigma$. Moreover, the map $\pi _\sigma$ corresponds to the restriction of divisor classes $[D] \mapsto [D|_{U_\sigma }]$. In particular, its kernel consists of those divisor classes that are principal on $U_\sigma$.

Construction 2.2

In the setting of Construction 2.1, we define lattices $\textbf{N}$ and $\textbf{F}$ and a lattice homomorphism $\textbf{P}:\textbf{N}\rightarrow \textbf{F}$ by

$$\begin{aligned} \textbf{N}:= \bigoplus _{\sigma \in \Sigma _{\max }} N_\sigma , \qquad \textbf{F}:= \bigoplus _{\sigma \in \Sigma _{\max }} F_\sigma , \qquad \textbf{P}:= \bigoplus _{\sigma \in \Sigma _{\max }} P_\sigma . \end{aligned}$$

Furthermore, we define lattice homomorphisms

$$\begin{aligned} \alpha :\textbf{N}\rightarrow N,&\qquad N_\sigma \ni v \mapsto \alpha _\sigma (v),\\ \beta :\textbf{F}\rightarrow F,&\qquad F_\sigma \ni w \mapsto \beta _\sigma (w). \end{aligned}$$

Let $\gamma :{\hat{N}} \rightarrow \textbf{N}$ be a kernel of $\alpha$ and $\delta :{\hat{F}} \rightarrow \textbf{F}$ be a kernel of $\beta$. We obtain an induced map ${\hat{P}} :{\hat{F}} \rightarrow {\hat{N}}$ making the following diagram commute:

Now consider the dual lattices $\textbf{M}:= \textbf{N}^*$ and $\textbf{E}:= \textbf{F}^*$ as well as the abelian group $\textbf{K}:= \bigoplus _{\sigma \in \Sigma _{\max }} K_\sigma$. We define the map

$$\begin{aligned} \pi :K \rightarrow \textbf{K}, \quad w \mapsto (\pi _\sigma (w))_{\sigma \in \Sigma _{\max }}. \end{aligned}$$

Setting ${\hat{M}}:= \textbf{M}/ \textrm{im}(\alpha ^*)$ and ${\hat{E}}:= \textbf{E}/ \textrm{im}(\beta ^*)$ as well as ${\hat{K}}:= \textbf{K}/ \textrm{im}(\pi )$, we obtain a map ${\hat{P}}' :{\hat{M}} \rightarrow {\hat{E}}$ fitting into the following commutative diagram with exact rows:

Proposition 2.3

In Construction 2.2, the map $\beta$ is surjective and there is an exact sequence

Moreover, if $\alpha$ is surjective, ${\hat{M}}$ is torsion-free and ${\hat{P}}' = {\hat{P}}^*$.

Proof

Every primitive generator of a ray of $\Sigma$ is a generator of some maximal cone. This implies that $\beta$ is surjective, hence $\beta ^*$ is injective. As a subgroup of $K$, the Picard group $\textrm{Pic}(Z)$ consists of the Cartier divisor classes, i.e. those that are principal on all affine toric charts $U_{\sigma }$ for $\sigma \in \Sigma _{\max }$. This means

$$\begin{aligned} \textrm{Pic}(Z) = \bigcap _{\sigma \in \Sigma _{\max }} \ker (\pi _\sigma ) = \ker (\pi ). \end{aligned}$$

Applying the snake lemma to the lower diagram of Construction 2.2, gives the exact sequence of the Proposition. The last statement is clear. $\square$

Corollary 2.4

Assume that in Construction 2.2, the map $\alpha$ is surjective. Then the Picard group $\textrm{Pic}(Z)$ is torsion-free.

Remark 2.5

Corollary 2.4 generalizes the well-known fact that if $Z$ has a toric fixed point, its Picard group is torsion-free. Indeed, having a toric fixed point means having a cone $\sigma \in \Sigma$ of maximal dimension. This implies $N_\sigma = N$, hence $\alpha$ is surjective.

Definition 2.6

The Picard index of a normal $\mathbb {Q}$-factorial variety $X$ is defined as

$$\begin{aligned} \iota _{\textrm{Pic}}(X)\,=\ [\textrm{Cl}(X): \textrm{Pic}(Z)]. \end{aligned}$$

Proposition 2.7

Let $Z = Z_{\Sigma }$ be a toric variety with a non-degenerate simplicial fan $\Sigma$. In the notation of Construction 2.2, we have

$$\begin{aligned} \iota _{\textrm{Pic}}(Z)\ =\ \frac{1}{|{\hat{K}}|} \prod _{\sigma \in \Sigma _{\max }} |K_\sigma |. \end{aligned}$$

Proof

Recall that $\textrm{Pic}(Z) = \ker (\pi )$ and ${\hat{K}} = \textbf{K}/ \textrm{im}(\pi )$. Since $\Sigma$ is simplicial, each $K_\sigma$ is finite, hence so is $\textbf{K}$. We obtain

$$\begin{aligned} \iota _{\textrm{Pic}}(Z) = [K: \ker (\pi )] = |\textrm{im}(\pi )| = \frac{|\textbf{K}|}{|{\hat{K}}|} = \frac{1}{|{\hat{K}}|} \prod _{\sigma \in \Sigma _{\max }} |K_\sigma |. \end{aligned}$$

$\square$

3 Background on $\mathbb {K}^{*}$ -surfaces

We recall the construction of rational $\mathbb {K}^*$-surfaces X in terms of defining matrices P from [11, 13, 15] and provide the necessary facts around it and fix our notation for the subsequent sections; see also [2, Sections 3.4, 5.4]. As a basic feature of this approach, X comes embedded into a specific toric variety Z which determines a significant part of the geometry of X. In particular and most relevant for us, $X \subseteq Z$ directly inherits divisor class group, Picard group, local class groups and hence also the Picard index; see Proposition 3.8. This will allow us to prove Theorem 1.1 entirely in terms of toric geometry.

We start by recalling some aspects of the geometry of $\mathbb {K}^*$-surfaces and their fixed points, the major part of which has been developed in [18,19,20]. A $\mathbb {K}^*$-surface is an irreducible, normal surface $X$ coming with an effective morphical action $\mathbb {K}^* \times X \rightarrow X$. Let $X$ be a projective $\mathbb {K}^*$-surface. For each point $x \in X$, the orbit map $t \mapsto t \cdot x$ extends to a morphism $\varphi _x :\mathbb {P}_1 \rightarrow X$. This allows one to define

$$\begin{aligned} x_0\,=\ \varphi _x(0), \qquad \qquad x_{\infty }\,=\ \varphi _x(\infty ). \end{aligned}$$

The points $x_0$ and $x_{\infty }$ are fixed points for the $\mathbb {K}^*$-action and they lie in the closure of the orbit $\mathbb {K}^* \cdot x$. There are three types of fixed points: A fixed point is called parabolic (hyperbolic, elliptic), if it lies in the closure of precisely one (precisely two, infinitely many) non-trivial $\mathbb {K}^*$-orbits. Hyperbolic and elliptic fixed points are isolated, hence their number is finite. Parabolic fixed points form a closed smooth curve with at most two connected components. Every projective $\mathbb {K}^*$-surface has a source and a sink, i.e. two irreducible components of the fixed point set $F^+, F^- \subseteq X$ such that there exist non-empty $\mathbb {K}^*$-invariant open subsets $U^+, U^- \subseteq X$ with

$$\begin{aligned} x_0 \in F^+ \text { for all } x \in U^+, \qquad \qquad x_{\infty } \in F^- \text { for all } x \in U^-. \end{aligned}$$

The source either consists of a single elliptic fixed point or it is a smooth curve of parabolic fixed points. The same holds for the sink.

Before giving the general construction of $\mathbb {K}^*$-surfaces in terms of integral matrices, we start with an example, which will reappear as a running example throughout the article.

Example 3.1

Consider the integral matrix

$$\begin{aligned} P = \begin{bmatrix} v_{01}&v_{02}&v_{11}&v_{21} \end{bmatrix} = \begin{bmatrix} -1 &{} -1 &{} 8 &{} 0 \\ -1 &{} -1 &{} 0 &{} 4 \\ -1 &{} -2 &{} 7 &{} 3 \end{bmatrix}. \end{aligned}$$

Let $Z$ be the toric variety coming from the lattice fan $\Sigma$ with maximal cones

$$\begin{aligned} \sigma ^+:= \textrm{cone}(v_{01},v_{11},v_{21}), \quad \sigma ^-:= \textrm{cone}(v_{02},v_{11},v_{21}), \quad \tau _{01}:= \textrm{cone}(v_{01},v_{02}). \end{aligned}$$

We write $U_1,U_2,U_3$ for the coordinate functions of the acting torus $\mathbb {T}^3$ of $Z$ and set $h:= 1 + U_1 + U_2$. Taking the closure of the zero locus defines an irreducible normal surface

$$\begin{aligned} X:= \overline{V(h)} \subseteq Z, \end{aligned}$$

with a $\mathbb {K}^*$-action given on $X \cap \mathbb {T}^3$ by $t \cdot z = (z_1,z_2,tz_3)$. There are two elliptic fixed points $x^{\pm } = z_{\sigma ^{\pm }}$ and one hyperbolic fixed point $\{ x_{01} \} = X \cap \mathbb {T}^3 \cdot z_{\tau _{01}}$. As we will see in Proposition 3.8, divisor class group and Picard group can be computed via toric geometry:

$$\begin{aligned} \textrm{Cl}(X)\ =\ \mathbb {Z}\times \mathbb {Z}/4\mathbb {Z}, \qquad \textrm{Pic}(X)\ =\ \mathbb {Z}\cdot (15,\overline{1}) \subseteq \textrm{Cl}(X). \end{aligned}$$

In particular, we have $\iota _{\textrm{Pic}}(X) = 60$.

We come to the general construction of $\mathbb {K}^*$-surfaces that will be our working environment for the rest of this article. In a first step, we produce our defining matrices $P$.

Construction 3.2

Fix positive integers $r, n_0, \dots , n_r$. We start with integral vectors $l_i = (l_{i1}, \dots , l_{in_i}) \in \mathbb {Z}_{\ge 1}^{n_i}$ and $d_i = (d_{i1}, \dots , d_{in_i}) \in \mathbb {Z}^{n_i}$ such that

$$\begin{aligned} \gcd (l_{ij}, d_{ij}) = 1, \qquad \frac{d_{i1}}{l_{i1}}> \dots > \frac{d_{in_i}}{d_{in_i}}, \qquad i = 0, \dots , r. \end{aligned}$$

The building blocks for our defining matrices are

$$\begin{aligned} L\,=\ \begin{bmatrix} -l_0 &{} l_1 &{} \dots &{} 0 \\ \vdots &{} &{} \ddots &{} \vdots \\ -l_0 &{} 0 &{} \dots &{} l_r \end{bmatrix}, \qquad d\,=\ \begin{bmatrix} d_0&d_1&\dots&d_r \end{bmatrix}. \end{aligned}$$

According to the possible constellations of source and sink, we introduce four types of integral matrices:

$$\begin{aligned} \begin{array}{crclcrcl} \text {(ee)} &{} P &{} = &{} \begin{bmatrix} L \\ d \end{bmatrix}, \qquad \qquad &{} \text {(pe)} &{} P &{} = &{} \begin{bmatrix} L &{} 0 \\ d &{} 1 \end{bmatrix}, \\ \\ \text {(ep)} &{} P &{} = &{} \begin{bmatrix} L &{} 0 \\ d &{} -1 \end{bmatrix}, \qquad \qquad &{} \text {(pp)} &{} P &{} = &{} \begin{bmatrix} L &{} 0 &{} 0 \\ d &{} 1 &{} -1 \end{bmatrix}. \end{array} \end{aligned}$$

With the canonical basis vectors $e_1, \dots , e_{r+1}$ of $\mathbb {Z}^{r+1}$ and $e_0:= -(e_1 + \dots + e_r)$, the columns of $P$ are

$$\begin{aligned} v_{ij}\,=\ l_{ij} e_i + d_{ij} e_{r+1}, \qquad v^+\,=\ e_{r+1}, \qquad v^-\,=\ -e_{r+1}, \end{aligned}$$

where $i = 0, \dots , r$ and $j = 1, \dots , n_i$. We call $P$ a defining matrix, if its columns generate $\mathbb {Q}^{r+1}$ as a convex cone.

Next, we will construct the fan for the ambient toric varieties for our $\mathbb {K}^*$-surfaces.

Construction 3.3

Let $P$ be a defining matrix. Setting $v_{i0}:= v^+$ and $v_{in_i+1}:= v^-$ for all $i$, we define the cones

$$\begin{aligned} \sigma ^+:= \textrm{cone}(v_{01}, \dots , v_{r1}), \qquad \sigma ^-:= \textrm{cone}(v_{0n_0}, \dots , v_{rn_r}), \end{aligned}$$

$$\begin{aligned} \tau _{ij}:= \textrm{cone}(v_{ij}, v_{ij+1}), \quad \text {for } i = 0, \dots , r \text { and } j = 0, \dots , n_i. \end{aligned}$$

According to the type of $P$, we define $\Sigma$ to be the fan with the following maximal cones:

$$\begin{aligned} \begin{array}{cccccc} \text {(ee)} &{} \{ \sigma ^+ \} &{} \cup &{} \{ \tau _{i1}, \dots , \tau _{in_i-1}\;\ i = 0, \dots , r \} &{} \cup &{} \{ \sigma ^- \}, \\ \text {(pe)} &{} \{ \tau _{00}, \dots , \tau _{r0} \} &{} \cup &{} \{ \tau _{i1}, \dots , \tau _{in_i-1}\;\ i = 0, \dots , r \} &{} \cup &{} \{ \sigma ^- \}, \\ \text {(ep)} &{} \{ \sigma ^+ \} &{} \cup &{} \{ \tau _{i1}, \dots , \tau _{in_i-1}\;\ i = 0, \dots , r \} &{} \cup &{} \{ \tau _{0n_0}, \dots , \tau _{rn_r} \}, \\ \text {(pp)} &{} \{ \tau _{00}, \dots , \tau _{r0} \} &{} \cup &{} \{ \tau _{i1}, \dots , \tau _{in_i-1}\;\ i = 0, \dots , r \} &{} \cup &{} \{ \tau _{0n_0}, \dots , \tau _{rn_r} \}. \\ \end{array} \end{aligned}$$

Note that $\Sigma$ is a non-degenerate simplicial lattice fan in $\mathbb {Z}^{r+1}$. However, it is in general not complete.

Construction 3.4

Let $P$ be a defining matrix. Consider the toric variety $Z = Z_{\Sigma }$, where $\Sigma$ is as in Construction 3.3. Let $U_1, \dots , U_{r+1}$ be the coordinate functions on the acting torus $\mathbb {T}^{r+1}$ of $Z$. Fix pairwise different $1 = \lambda _2, \dots , \lambda _r \in \mathbb {K}^*$ and set

$$\begin{aligned} h_i:= \lambda _i + U_1 + U_i, \qquad \qquad i = 2, \dots , r. \end{aligned}$$

Passing to the closure of the common set of zeroes of $h_2, \dots , h_r$, we obtain an irreducible rational normal projective surface

$$\begin{aligned} X(P):= \overline{V(h_2, \dots , h_r)} \subseteq Z. \end{aligned}$$

Since the $h_i$ do not depend on the last coordinate $U_{r+1}$ of $\mathbb {T}^{r+1}$, we get an effective $\mathbb {K}^*$-action on $X(P)$ as a subtorus of $\mathbb {T}^{r+1}$ by

$$\begin{aligned} t \cdot x:= (1, \dots , 1, t) \cdot x. \end{aligned}$$

Example 3.5

The $\mathbb {K}^*$-surface from Example 3.1 arises from Constructions 3.2 to 3.4 by setting $(n_0,n_1,n_2) = (2,1,1)$ as well as $(l_0,l_1,l_2) = ((1,1),(8),(4))$ and $(d_0,d_1,d_2) = ((-1,-2), (7), (3))$ and choosing the type (ee).

Remark 3.6

Consider a $\mathbb {K}^*$-surface $X = X(P)$. Let $Z = Z_{\Sigma }$ be the ambient toric variety with acting torus $T = \mathbb {T}^{r+1}$. The fixed points of $X$ are given as follows. For every $\tau _{ij} \in \Sigma _{\max }$, the associated toric orbit $T \cdot z_{\tau _{ij}}$ intersects $X$ in a fixed point

$$\begin{aligned} \{x_{ij}\}\ =\ X\ \cap \ T \cdot z_{\tau _{ij}}. \end{aligned}$$

If $1 \le j \le n_i-1$, the fixed point $x_{ij}$ is hyperbolic and all hyperbolic fixed points arise this way. For $j \in \{0, n_i\}$, the fixed point $x_{ij}$ is parabolic. According to the type of $P$, we have the following.

• (ee) There are two elliptic fixed points $x^+ = z_{\sigma ^+}$ and $x^- = z_{\sigma ^-}$ and no parabolic fixed points.

• (pe) There is one elliptic fixed point $x^- = z_{\sigma ^-}$. There are parabolic fixed points $x_{i0} \in F^+$ and all parabolic fixed points in $F^+ \backslash \{x_{00}, \dots , x_{r0}\}$ are smooth.

• (ep) There is one elliptic fixed point $x^+ = z_{\sigma ^+}$. There are parabolic fixed points $x_{in_i} \in F^-$ and all parabolic fixed points in $F^- \backslash \{x_{0n_0}, \dots , x_{rn_r}\}$ are smooth.

• (pp) There are no elliptic fixed points. There are parabolic fixed points $x_{i0} \in F^+$ and $x_{in_i} \in F^-$ and all parabolic fixed points in $F^+ \backslash \{x_{00}, \dots , x_{r0}\}$ and $F^- \backslash \{x_{0n_0}, \dots , x_{rn_r} \}$ are smooth.

Theorem 3.7

(See [2, Thm. 5.4.1.5]) Every rational projective $\mathbb {K}^*$-surface is isomorphic to a $\mathbb {K}^*$-surface $X(P)$ arising from Construction 3.4.

Proposition 3.8

Let $X = X(P) \subseteq Z$ arise from Construction 3.4. Then we have

• (i) $\textrm{Cl}(X) \cong \textrm{Cl}(Z)$,

• (ii) $\textrm{Pic}(X) \cong \textrm{Pic}(Z)$,

• (iii) $\textrm{Cl}(X, x) \cong \textrm{Cl}(Z,x)$ for all $x \in X$,

• (iv) $\iota _{\textrm{Pic}}(X) = \iota _{\textrm{Pic}}(Z)$.

Proof

By [2, Prop. 5.4.1.8], the embedding $X \subseteq Z$ is the canonical toric embedding in the sense of [2, Sec. 3.2.5]. Now use [2, 3.2.5.4,3.3.1.7]. $\square$

4 Maximal minors of P

As a first step towards the equality of the cokernel orders of P and ${\hat{P}}^*$, we study in this section the set M(P) of maximal minors of our defining matrices P. We figure out relations among the maximal minors of P and, based on that, show that $\gcd (M(P))$ equals $\gcd (M'(P))$ for a proper, more accessible subset $M'(P) \subset M(P)$; see Proposition 4.11. First, we introduce a suitable lattice basis for $\mathbb {Z}^n$, reflecting the block structure of P.

Construction 4.1

Let $P$ be a defining matrix. Set $n:= n_0 + \dots + n_r$. Then $P$ is an integral $(r+1) \times (n+m)$-matrix, where $m \in \{0,1,2\}$. Write $e_1, \dots , e_{r+1}$ for the canonical basis vectors of $\mathbb {Z}^{r+1}$ and set $u:= e_{r+1}$. Then we define

$$\begin{aligned} N \,= \ \mathbb {Z}^{r+1} \ = \ \mathbb {Z}e_1 \oplus \dots \oplus \mathbb {Z}e_r \oplus \mathbb {Z}u. \end{aligned}$$

For each $i = 0, \dots , r$, we introduce a set of symbols to be used as a lattice basis:

Setting

$$\begin{aligned} {\mathcal {F}}:= {\mathcal {F}}_0 \cup \dots \cup {\mathcal {F}}_r \cup {\mathcal {F}}', \qquad \qquad F \,= \ F_0 \oplus \dots \oplus F_r \oplus F', \end{aligned}$$

we obtain an isomorphism $F \cong \mathbb {Z}^{n+m}$. Set $e_0:= -(e_1 + \dots + e_r)$. Then we can view $P$ as a lattice map given by

$$\begin{aligned} P :F \rightarrow N, \quad {\left\{ \begin{array}{ll} f_{ij} \mapsto v_{ij}:= l_{ij} e_i + d_{ij} u &{} \\ f^+ \mapsto u, &{} \text { if } f^+ \in {\mathcal {F}}' \\ f^- \mapsto -u, &{} \text { if } f^- \in {\mathcal {F}}'. \end{array}\right. } \end{aligned}$$

Definition 4.2

Let a subset $A \subseteq \mathcal F$ with $|A| = r+1$ be given. Then we have a sublattice $F_A:= \bigoplus _{f \in A} \mathbb {Z}\cdot f \subseteq F$ and an induced map $P_A :F_A \rightarrow N$ as in the commutative diagram

We call $|\det (P_A)| \in \mathbb {Z}_{\ge 0}$ the maximal minor of $P$ associated to $A$. The set of maximal minors of $P$ is

$$\begin{aligned} M(P)\,=\ \{ |\det (P_A)|\;\ A \subseteq \mathcal F,\ |A| = r+1 \}. \end{aligned}$$

Example 4.3

With $P$ as in Example 3.1, we have $\mathcal F = \{f_{01}, f_{02}, f_{11}, f_{21} \}$. Setting $A_{ij}:= \mathcal F \backslash \{f_{ij}\}$, the maximal minors of $P$ are

$$\begin{aligned} \vert \det (P_{A_{01}}) \vert&= \vert d_{02}l_{11}l_{21} + l_{02}d_{11}l_{21}+l_{02}l_{11}d_{21} \vert = 12, \\ \vert \det (P_{A_{02}}) \vert&= \vert d_{01}l_{11}l_{21} + l_{01}d_{11}l_{21}+l_{01}l_{11}d_{21}\vert = 20, \\ \vert \det (P_{A_{11}}) \vert&= \vert l_{21}(d_{01}l_{02}-l_{01}d_{02}) \vert = 4, \\ \vert \det (P_{A_{21}}) \vert&= \vert l_{11}(d_{01}l_{02}-l_{01}d_{02}) \vert = 8. \end{aligned}$$

Hence, we have $M(P) = \{12,20,4,8\}$.

The following lemma gives a vanishing criterion for maximal minors of $P$. In particular, it will allow us to describe the nonzero maximal minors of $P$ explicitly.

Lemma 4.4

Let $A \subseteq \mathcal {F}$ with $|A| = r + 1$. Assume that there exist $0 \le i_0 < i_1 \le r$ such that $A \cap \mathcal {F}_{i_0} = A \cap \mathcal {F}_{i_1} = \emptyset$. Then $\det (P_A) = 0$.

Proof

Consider the dual bases $\{f^*_{ij}\}$ and $\{e_1^*, \dots , e_r^*, u^*\}$ of $F^*$ and $N^*$ respectively. Then we have

$$\begin{aligned} P^*(e_i^*) = \left( \sum _{j=1}^{n_i} l_{ij} f^*_{ij} \right) - \left( \sum _{j=1}^{n_0} l_{0j} f^*_{0j} \right) . \end{aligned}$$

If $i_0 = 0$, this implies $P^*_A(e_{i_1}^*) = 0$. If $i_0$ and $i_1$ are both nonzero, we have

$$\begin{aligned} P^*_A(e_{i_0}^*) = - \left( \sum _{j=1}^{n_0} l_{0j} f^*_{0j} \right) = P^*_A(e_{i_1}^*). \end{aligned}$$

Thus in both cases, $\det (P_A) = \det (P_A^*) = 0$. $\square$

Definition 4.5

• (i) Let numbers $1 \le j_i \le n_i$ for all $i = 0, \dots , r$ be given. We define

  $$\begin{aligned} \mu (j_0, \dots , j_r)\,=\ \sum _{i_0=0}^{r} d_{i_0 j_{i_0}} \prod _{i \ne i_0} l_{ij_i}, \qquad \qquad {\hat{\mu }}\,=\ \mu (n_0, \dots , n_r). \end{aligned}$$

• (ii) Let $0 \le i \le r$ and $0 \le j, j' \le n_i$ be given. We define

  $$\begin{aligned} \nu (i, j, j')\,=\ l_{ij} d_{ij'} - l_{ij'} d_{ij}, \qquad \qquad {\hat{\nu }}(i, j)\,=\ \nu (i, j, n_i). \end{aligned}$$

Lemma 4.6

Let $A \subseteq \mathcal {F}$ with $|A| = r + 1$ such that $\det (P_A) \ne 0$. Assume that $A \cap \mathcal F' = \emptyset$ holds. Then either (i) or (ii) must hold.

• (i) We have $|A \cap \mathcal {F}_i| = 1$ for all $i = 0, \dots , r$ and $|\det (P_A)| = |\mu (j_0, \dots , j_r)|$ for some numbers $1 \le j_i \le n_i$.

• (ii) There exist $0 \le i_0, i_1 \le r$ with

  $$\begin{aligned} |A \cap \mathcal {F}_i| = {\left\{ \begin{array}{ll} 2, &{} i = i_0 \\ 0, &{} i = i_1 \\ 1, &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

  and we have $|\det (P_A)| = |\nu (i_0, j_{i_0}, j_{i_0}')| \prod _{i \ne i_0,i_1} l_{ij_i}$ for some numbers $1 \le j_i \le n_i$ as well as $1 \le j_{i_0}' \le n_{i_0}$.

Proof

Lemma 4.4 implies that there is at most one $i = 0, \dots , r$ with $A \cap \mathcal F_i = \emptyset$. Since $A \cap \mathcal F' = \emptyset$ and $|A| = r+1$, the conditions on $|A \cap \mathcal F_i|$ follow. The expressions for $\det (P_A)$ then come from cofactor expansion. $\square$

Example 4.7

Applying Lemma 4.6 to Example 4.3, we see that the minors associated to $A_{01}$ and $A_{02}$ satisfy (i), while those associated to $A_{11}$ and $A_{21}$ satisfy (ii).

Lemma 4.8

Let $A \subseteq \mathcal F$ with $|A| = r+1$ such that $\det (P_A) \ne 0$. Assume that $A \cap \mathcal F' \ne \emptyset$ holds. Then there exists an $i_1 = 0, \dots , r$ such that

$$\begin{aligned} |A \cap \mathcal F_i| = {\left\{ \begin{array}{ll} 0, &{} i = i_1 \\ 1, &{} \text {otherwise} \end{array}\right. } \end{aligned}$$

and we have $|\det (P_A)| = \prod _{i \ne i_1} l_{ij_i}$.

Proof

Note that $|A \cap \mathcal F'| = 2$ cannot hold, since this would imply $\det (P_A) = 0$. Hence $|A \cap \mathcal F'| = 1$. Lemma 4.4 forces the condition on $|A \cap \mathcal F_i|$. Cofactor expansion gives the expression for $\det (P_A)$. $\square$

Lemma 4.9

• (i) Let $i = 0, \dots , r$ and $1 \le j,j' \le n_i$. There exist integers $\alpha , \beta \in \mathbb {Z}$ such that

  $$\begin{aligned} \nu (i,j,j')\ =\ \alpha {\hat{\nu }}(i,j) - \beta {\hat{\nu }}(i,j'). \end{aligned}$$

• (ii) Let $i = 0, \dots , r$ and $j_i = 1, \dots , n_i$ for all $i$. There exist integers $\beta _i$ such that

  $$\begin{aligned} \mu (j_0, \dots , j_r)&= \beta _{i_0}\mu (j_0, \dots , j_{i_0-1}, n_{i_0}, j_{i_0+1}, \dots , j_r) \\&\quad + \sum _{i_1 \ne i_0} \beta _{i_1} {\hat{\nu }}(i_0, j_{i_0}) \prod _{i \notin \{i_0, i_1\}} l_{i j_i}. \end{aligned}$$

Proof

We show (i). Since $\gcd (l_{in_i}, d_{in_i}) = 1$, we find integers $x, y \in \mathbb {Z}$ such that $x l_{in_i} + y d_{in_i} = 1$. Set $\alpha := x l_{ij} + y d_{ij}$ and $\beta := x l_{ij'} + y d_{ij'}$. Then the rows of $3 \times 3$-matrix

$$\begin{aligned} \begin{bmatrix} l_{ij} &{} l_{ij'} &{} l_{in_i} \\ d_{ij} &{} d_{ij'} &{} d_{in_i} \\ \alpha &{} \beta &{} 1 \end{bmatrix} \end{aligned}$$

are linearly dependent, hence its determinant vanishes. Cofactor expansion by the last row gives the desired equality. For (ii), we consider exemplarily the case $i_0 = 0$. Since $\gcd (l_{0n_0}, d_{0n_0}) = 1$, we find $x, y \in \mathbb {Z}$ such that $-x l_{0n_0} + y d_{0n_0} = 1$. Now set

$$\begin{aligned} \beta _0:= -x l_{0 j_0} + y d_{0 j_0}, \qquad \qquad \beta _1:= x l_{1 j_1} + y d_{1 j_1}, \end{aligned}$$

as well as $\beta _i:= yd_{ij_i}$ for $i = 2, \dots , r$. Then the rows of the $(r+2) \times (r+2)$ matrix

$$\begin{aligned} \begin{bmatrix} -l_{0j_0} &{} -l_{0n_0} &{} l_{1 j_1} &{} 0 &{} \dots &{} 0\\ -l_{0j_0} &{} -l_{0n_0} &{} 0 &{} l_{2 j_2} &{} \dots &{} 0\\ \vdots &{} \vdots &{} \vdots &{} &{} \ddots &{} \vdots \\ -l_{0j_0} &{} -l_{0n_0} &{} 0 &{} \dots &{} \dots &{} l_{r j_r}\\ d_{0j_0} &{} d_{0n_0} &{} d_{1j_1} &{} d_{2j_2} &{} \dots &{} d_{r j_r}\\ \beta _0 &{} 1 &{} \beta _1 &{} \beta _2 &{} \dots &{} \beta _r \end{bmatrix} \end{aligned}$$

are linearly dependent, hence its determinant vanishes. Cofactor expansion by the last row and adjusting the signs of the $\beta _i$ gives the desired equality. $\square$

Definition 4.10

According to the type of $P$, we define the set

$$\begin{aligned} M'(P):= {\left\{ \begin{array}{ll} \displaystyle \{ |{\hat{\mu }}| \} \cup \left\{ |{\hat{\nu }}(i_0, j_{i_0})| \prod _{i \ne i_0, i_1} l_{i j_i}\;\ \begin{aligned} &{} 0 \le i_0, i_1 \le r,\text { with } i_0 \ne i_1, \\ &{} 1 \le j_i \le n_i \text { for all } i \ne i_1 \end{aligned} \right\} , \quad \text {(ee)} \\ \displaystyle \left\{ \prod _{i \ne i_1} l_{ij_i}\;\ 0 \le i_1 \le r,\ 1 \le j_i \le n_i \text { for all } i \ne i_1 \right\} , \quad \text {(pe), (ep), (pp).} \end{array}\right. } \end{aligned}$$

Proposition 4.11

We have $\gcd (M(P)) = \gcd (M'(P))$.

Proof

Consider first the case (ee). Then $\mathcal F' = \emptyset$, hence Lemma 4.6 implies $M'(P) \subseteq M(P)$. This shows that $\gcd (M(P))$ divides $\gcd (M'(P))$. On the other hand, by repeated application of Lemma 4.9, we see that every maximal minor of $P$ can be written as a $\mathbb {Z}$-linear combination of elements of $M'(P)$. This implies that $\gcd (M'(P))$ divides $\gcd (M(P))$. For cases (pe), (ep) and (pp), Lemma 4.8 implies $M'(P) \subseteq M(P)$, hence $\gcd (M(P))$ divides $\gcd (M'(P))$. On the other hand, Lemma 4.6 implies that every maximal minor of $P$ is a $\mathbb {Z}$-linear combination of elements of $M'(P)$, hence $\gcd (M'(P))$ divides $\gcd (M(P))$. $\square$

Example 4.12

In Example 4.3, we can apply Lemma 4.9 (ii) to express $|\det (P_{A_{02}})|$ as a $\mathbb {Z}$-linear combination of the other maximal minors. Hence we have $M'(P) = M(P) \backslash \{|\det (P_{A_{02}})|\} = \{12,8,4\}$.

5 The construction of ${\hat{P}}$

The aim of this section is to provide an explicit matrix representation of the map ${\hat{P}}$ from Construction 2.2, which will be used to describe its maximal minors in Sect. 6.

Construction 5.1

Let $\Sigma$ be the fan of an ambient toric variety of a $\mathbb {K}^*$-surface, as defined in Construction 3.3. Consider the lattices $F_{\sigma }:= \mathbb {Z}^{n_{\sigma }}$ and $N_{\sigma }:= \textrm{lin}_{N_{\mathbb {Q}}}(\sigma ) \cap N$ from Construction 2.1. We have $n_{\sigma ^+} = n_{\sigma ^-} = r+1$ and $n_{\tau _{ij}} = 2$. Moreover, we have $N_{\sigma ^+} = N_{\sigma ^-} = N = \mathbb {Z}^{r+1}$ and $N_{\tau _{ij}} = \mathbb {Z}e_i + \mathbb {Z}u \subseteq N$. We will work with the identifications

$$\begin{aligned} \begin{array}{lclclcl} F_{\sigma ^+} &{} \cong &{} \mathbb {Z}f_{01}^+ \oplus \dots \oplus \mathbb {Z}f_{r1}^+, &{} \qquad \qquad &{} N_{\sigma ^+} &{} \cong &{} \mathbb {Z}e_1^+ \oplus \dots \oplus \mathbb {Z}e_r^+ \oplus \mathbb {Z}u^+, \\ F_{\tau _{ij}} &{} \cong &{} \mathbb {Z}f_{ij}^- \oplus \mathbb {Z}f_{ij+1}^+, &{} \qquad \qquad &{} N_{\tau _{ij}} &{} \cong &{} \mathbb {Z}e_{ij} \oplus \mathbb {Z}u_{ij}, \\ F_{\sigma ^-} &{} \cong &{} \mathbb {Z}f_{0n_0}^- \oplus \dots \oplus \mathbb {Z}f_{rn_r}^-, &{} \qquad \qquad &{} N_{\sigma ^-} &{} \cong &{} \mathbb {Z}e_1^- \oplus \dots \oplus \mathbb {Z}e_r^- \oplus \mathbb {Z}u^-. \\ \end{array} \end{aligned}$$

Then according to the type of $P$, a lattice basis of $\textbf{N}$ is given by

$$\begin{aligned} \begin{array}{cccccc} \text {(ee)} &{} \{e_1^+, \dots , e_r^+, u^+\} &{} \cup &{} S &{} \cup &{} \{ e_1^-, \dots , e_r^-, u^- \}, \\ \text {(pe)} &{} \{ e_{i0}, u_{i0}\;\ i = 0, \dots , r \} &{} \cup &{} S &{} \cup &{} \{ e_1^-, \dots , e_r^-, u^- \}, \\ \text {(ep)} &{} \{e_1^+, \dots , e_r^+, u^+\} &{} \cup &{} S &{} \cup &{} \{ e_{in_i}, u_{in_i}\;\ i = 0, \dots , r \}, \\ \text {(pp)} &{} \{ e_{i0}, u_{i0}\;\ i = 0, \dots , r \} &{} \cup &{} S &{} \cup &{} \{ e_{in_i}, u_{in_i}\;\ i = 0, \dots , r \}, \end{array} \end{aligned}$$

where $S:= \{ e_{ij}, u_{ij}\;\ i = 0, \dots , r,\ j = 1, \dots , n_i-1 \}$. A lattice basis of $\textbf{F}$ is given by

$$\begin{aligned} \begin{array}{cccc} \text {(ee)} &{} \{ f_{ij}^-\;\ i = 0, \dots , r,\ j = 1, \dots , n_i \} &{} \cup &{} \{ f_{ij}^+\;\ i = 0, \dots , r,\ j = 1, \dots , n_i \}, \\ \text {(pe)} &{} \{ f_{ij}^-\;\ i = 0, \dots , r,\ j = 0, \dots , n_i \} &{} \cup &{} \{ f_{ij}^+\;\ i = 0, \dots , r,\ j = 1, \dots , n_i \}, \\ \text {(ep)} &{} \{ f_{ij}^-\;\ i = 0, \dots , r,\ j = 1, \dots , n_i \} &{} \cup &{} \{ f_{ij}^+\;\ i = 0, \dots , r,\ j = 1, \dots , n_i+1 \}, \\ \text {(pp)} &{} \{ f_{ij}^-\;\ i = 0, \dots , r,\ j = 0, \dots , n_i \} &{} \cup &{} \{ f_{ij}^+\;\ i = 0, \dots , r,\ j = 1, \dots , n_i+1 \}. \end{array} \end{aligned}$$

In particular, we have $\textrm{rank}(\textbf{F}) = \textrm{rank}(\textbf{N}) = 2n+m(r+1)$. With respect to these bases, the maps $\alpha$ and $\beta$ from Construction 2.2 are

$$\begin{aligned} \begin{array}{ll} \alpha :\textbf{N}\rightarrow N, &{} e_{ij}, e_i^+, e_i^- \mapsto e_i, \qquad u_{ij}, u^+, u^- \mapsto u,\\ \beta :\textbf{F}\rightarrow F, &{} f_{ij}^+, f_{ij}^- \mapsto f_{ij}. \end{array} \end{aligned}$$

Setting $e^{+}_0:= -(e^{+}_1 + \dots + e^{+}_r)$ and $e^{-}_0:= -(e^{-}_1 + \dots + e^{-}_r)$, the maps $P_{\sigma ^{+}}, P_{\sigma ^-}$ and $P_{\tau _{ij}}$ are then given as

$$\begin{aligned} \begin{array}{ll} P_{\sigma ^+} :F_{\sigma ^+} \rightarrow N_{\sigma ^-}, &{} f^+_{i1} \mapsto l_{i1} e^+_i + d_{i1} u^+,\\ P_{\sigma ^-} :F_{\sigma ^-} \rightarrow N_{\sigma ^-}, &{} f^-_{in_i} \mapsto l_{in_i} e^-_i + d_{in_i} u^-,\\ P_{\tau _{ij}} :F_{\tau _{ij}} \rightarrow N_{\tau _{ij}}, &{} f^-_{ij} \mapsto l_{ij} e_{ij} + d_{ij} u_{ij}\\ &{}f^+_{ij+1} \mapsto l_{ij+1} e_{ij} + d_{ij+1} u_{ij}. \end{array} \end{aligned}$$

Remark 5.2

Clearly, the map $\alpha$ in Construction 5.1 is surjective in all cases. Hence Corollary 2.4 implies that the Picard group of a projective rational $\mathbb {K}^*$-surface is torsion-free.

Example 5.3

With $P$ as in Example 3.1, we have $\{e_1^+,e_2^+,u^+,e_{01},u_{01},e_1^-,e_2^-,u^-\}$ as a basis for $\textbf{N}$ and $\{f_{01}^+, f_{11}^+, f_{21}^+, f_{01}^-, f_{02}^+, f_{02}^-, f_{11}^-, f_{21}^- \}$ as a basis for $\textbf{F}$. The matrix representations of $\alpha$ and $\beta$ are

$$\begin{aligned} \alpha \ =\ \begin{bmatrix} 1 &{} 0 &{} 0 &{} -1 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} -1 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \end{bmatrix}, \qquad \beta \ =\ \begin{bmatrix} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \end{bmatrix}. \end{aligned}$$

Construction 5.4

We continue in the setting of Construction 5.1. We will give explicit descriptions of the maps $\gamma , \delta$ and ${\hat{P}}$ from Construction 2.2. According to the type of $P$, let us set for all $i = 0, \dots , r$

$$\begin{aligned} n_i':= {\left\{ \begin{array}{ll} n_i - 1, &{} \text {(ee)}, \\ n_i, &{} \text {(pe), (ep), (pp)}, \end{array}\right. } \end{aligned}$$

$$\begin{aligned} \hat{\mathcal {N}}_i:= \{ {\hat{e}}_{ij}, {\hat{u}}_{ij}\;\ j = 1, \dots , n_i' \}, \qquad \qquad \hat{\mathcal {F}}_i:= \{{\hat{f}}_{ij}\;\ j = 1, \dots n_i \}. \end{aligned}$$

Now define the sets of symbols

Let ${\hat{N}}$ and ${\hat{F}}$ be the free lattices over $\hat{\mathcal {N}}$ and $\hat{\mathcal {F}}$ respectively. In particular, we have $\textrm{rank}({\hat{N}}) = 2n + (m-1)(r+1)$ and $\textrm{rank}({\hat{F}}) = n + mr$. According to the type of $P$, we define a map $\gamma :{\hat{N}} \rightarrow \textbf{N}$ as follows:

$$\begin{aligned} \begin{array}{ll} (\textrm{ee}) &{} \begin{array}{lcl} {\hat{e}}_{ij} &{} \mapsto &{} {\left\{ \begin{array}{ll} e_i^+ - e_{i1}, &{} j = 1 \\ e_{ij-1} - e_{ij}, &{} 2 \le j \le n_i-1 \end{array}\right. } \\ {\hat{u}}_{ij} &{} \mapsto &{} {\left\{ \begin{array}{ll} u_i^+ - u_{i1}, &{} j = 1 \\ u_{ij-1} - u_{ij}, &{} 2 \le j \le n_i-1 \end{array}\right. } \\ {\tilde{e}}_i &{} \mapsto &{} e_i^+ - e_i^- \\ {\tilde{u}} &{} \mapsto &{} u^+ - u^-, \end{array} \\ (\textrm{pe}) &{} \begin{array}{lcl} {\hat{e}}_{ij} &{} \mapsto &{} {\left\{ \begin{array}{ll} e_{ij-1} - e_{ij}, &{} 1 \le j \le n_i-1 \\ e_{in_i-1} - e_i^-, &{} j = n_i \end{array}\right. } \\ {\hat{u}}_{ij} &{} \mapsto &{} {\left\{ \begin{array}{ll} u_{ij-1} - u_{ij}, &{} 1 \le j \le n_i-1 \\ u_{in_i-1} - u^-, &{} j = n_i \end{array}\right. } \end{array} \\ (\textrm{ep}) &{} \begin{array}{lcl} {\hat{e}}_{ij} &{} \mapsto &{} {\left\{ \begin{array}{ll} e_i^+ - e_{i1}, &{} j = 1, \\ e_{ij-1} - e_{ij}, &{} 2 \le j \le n_i \\ \end{array}\right. } \\ {\hat{u}}_{ij} &{} \mapsto &{} {\left\{ \begin{array}{ll} u^+ - u_{i1}, &{} j = 1, \\ u_{ij-1} - u_{ij}, &{} 2 \le j \le n_i \\ \end{array}\right. } \end{array} \\ (\textrm{pp}) &{} \begin{array}{lcl} {\hat{e}}_{ij} &{} \mapsto &{} e_{ij-1} - e_{ij}, \\ {\hat{u}}_{ij} &{} \mapsto &{} u_{ij-1} - u_{ij}, \\ {\tilde{u}}_i &{} \mapsto &{} u_{in_i} - u_{i-1,n_{i-1}} \\ {\tilde{e}} &{} \mapsto &{} (e_{00} + e_{10} + \dots + e_{r0}). \end{array} \end{array} \end{aligned}$$

Then $\gamma$ is a kernel of $\alpha :\textbf{N}\rightarrow N$. Next, we define a map $\delta :{\hat{F}} \rightarrow \textbf{F}$ as follows:

$$\begin{aligned} \begin{array}{ll} (\textrm{ee})&\begin{array}{lcl} {\hat{f}}_{ij} &{} \mapsto &{} f_{ij}^+ - f_{ij}^- \end{array} \\ (\textrm{pe}) &{} \begin{array}{lcl} {\hat{f}}_{ij} &{} \mapsto &{} f_{ij}^+ - f_{ij}^- \\ {\hat{f}}_i^- &{} \mapsto &{} f_{i-1,0}^- - f_{i,0}^- \end{array} \\ (\textrm{ep}) &{} \begin{array}{lcl} {\hat{f}}_{ij} &{} \mapsto &{} f_{ij}^+ - f_{ij}^-\\ {\hat{f}}_i^+ &{} \mapsto &{} f_{i-1,n_{i-1}+1}^+ - f_{i,n_i+1}^+ \end{array} \\ (\textrm{pp}) &{} \begin{array}{lcl} {\hat{f}}_{ij} &{} \mapsto &{} f_{ij}^+ - f_{ij}^-\\ {\hat{f}}_i^- &{} \mapsto &{} f_{i-1,0}^- - f_{i,0}^-\\ {\hat{f}}_i^+ &{} \mapsto &{} f_{i-1,n_{i-1}+1}^+ - f_{i,n_i+1}^+. \end{array} \\ \end{array} \end{aligned}$$

Then $\delta$ is a kernel of $\beta :\textbf{F}\rightarrow F$. If $P$ is of type (ee), set ${\tilde{e}}_0:= -({\tilde{e}}_1 + \dots + {\tilde{e}}_r)$. Then the induced map ${\hat{P}} :{\hat{F}} \rightarrow {\hat{N}}$ is given as follows.

Proof

In all cases, we can verify that $\alpha \circ \gamma = 0$ and $\beta \circ \delta = 0$. Moreover, $\gamma$ and $\delta$ are both injective and we have

$$\begin{aligned} \textrm{rank}({\hat{N}})&= 2n - (m-1)(r+1) = \textrm{rank}(\textbf{N}) - \textrm{rank}(N), \\ \textrm{rank}({\hat{F}})&= n + mr = \textrm{rank}(\textbf{F}) - \textrm{rank}(F). \end{aligned}$$

This shows that $\gamma$ and $\delta$ are kernels of $\alpha$ and $\beta$ respectively. Furthermore, direct computations verify that $\gamma \circ {\hat{P}} = \textbf{P}\circ \delta$ holds in all cases. This shows that ${\hat{P}}$ is the induced map between the kernels. $\square$

Example 5.5

Continuing Example 5.3, we get $\hat{\mathcal {N}} = \{{\hat{e}}_{01}, {\hat{u}}_{01}, {\tilde{e}}_1, {\tilde{e}}_2, {\tilde{u}} \}$ and $\hat{\mathcal {F}} = \{{\hat{f}}_{01}, {\hat{f}}_{02}, {\hat{f}}_{11}, {\hat{f}}_{21} \}$. We obtain the matrix representations

$$\begin{aligned} {\tiny \gamma = \begin{bmatrix} -1 &{} 0 &{} 1 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 1 \\ -1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} -1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \end{bmatrix}, \quad \delta = \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 \\ -1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} -1 &{} 0 &{} 0 \\ 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} -1 \end{bmatrix},\quad {\hat{P}}= \begin{bmatrix} l_{01} &{} -l_{02} &{} 0 &{} 0 \\ d_{01} &{} -d_{02} &{} 0 &{} 0 \\ 0 &{} -l_{02} &{} l_{11} &{} 0 \\ 0 &{} -l_{02} &{} 0 &{} l_{21} \\ 0 &{} d_{02} &{} d_{11} &{} d_{21} \end{bmatrix}. } \end{aligned}$$

Remark 5.6

We give the general matrix representation of ${\hat{P}}$ from Construction 5.4 for the case (ee). Let ${\hat{F}}_i$ and ${\hat{N}}_i$ be the free lattices over $\hat{\mathcal {F}}_i$ and $\hat{\mathcal {N}}_i$ respectively. Let ${\tilde{N}}$ be the free lattice over $\tilde{\mathcal {N}}$. We define the lattice maps

$$\begin{aligned} \begin{array}{lclcl} {\hat{P}}_i :{\hat{F}}_i \rightarrow {\hat{N}}_i, &{} \qquad &{} {\hat{f}}_{ij} &{} \mapsto &{} {\left\{ \begin{array}{ll} l_{ij} {\hat{e}}_{ij} + d_{ij} {\hat{u}}_{ij}, &{} 1 \le j \le n_i-1 \\ \displaystyle -l_{in_i} \left( \sum _{k=1}^{n_{i}-1} {\hat{e}}_{ik} \right) - d_{in_i} \left( \sum _{k=1}^{n_i-1} {\hat{u}}_{ik} \right) , &{} j = n_i, \end{array}\right. } \\ {\tilde{P}}_i :{\hat{F}}_i \rightarrow {\tilde{N}}, &{} \qquad &{} {\hat{f}}_{ij} &{} \mapsto &{} {\left\{ \begin{array}{ll} 0, &{} 1 \le j \le n_i-1 \\ l_{in_i} {\tilde{e}}_i + d_{in_i} {\tilde{u}}, &{} j = n_i. \end{array}\right. }. \end{array} \end{aligned}$$

Then we have ${\hat{P}}({\hat{f}}_{ij}) = {\hat{P}}_i({\hat{f}}_{ij}) + \tilde{P}_i({\hat{f}}_{ij})$. We obtain the matrix representations:

$$\begin{aligned} \begin{array}{lclclcl} {\tilde{P}}_0 &{} = &{} \begin{bmatrix} 0 &{} \dots &{} 0 &{} -l_{0n_0} \\ 0 &{} \dots &{} 0 &{} -l_{0n_0} \\ \vdots &{} &{} \vdots &{} \vdots \\ 0 &{} \dots &{} 0 &{} -l_{0n_0} \\ 0 &{} \dots &{} 0 &{} d_{0n_0} \end{bmatrix}, &{} \qquad &{} {\tilde{P}}_i &{} = &{} \begin{bmatrix} 0 &{} \dots &{} 0 &{} 0 \\ \vdots &{} &{} \vdots &{} \vdots \\ 0 &{} \dots &{} 0 &{} l_{in_i} \\ \vdots &{} &{} \vdots &{} \vdots \\ 0 &{} \dots &{} 0 &{} 0 \\ 0 &{} \dots &{} 0 &{} d_{in_i} \end{bmatrix} \quad (i \ge 1), \\ \\ {\hat{P}}_i &{} = &{} \begin{bmatrix} l_{i1} &{} 0 &{} \dots &{} 0 &{} -l_{in_i} \\ d_{i1} &{} 0 &{} \dots &{} 0 &{} -d_{in_i} \\ 0 &{} l_{i2} &{} &{} 0 &{} -l_{in_i} \\ 0 &{} d_{i2} &{} &{} 0 &{} -d_{in_i} \\ \vdots &{} &{} \ddots &{} &{} \vdots \\ 0 &{} 0 &{} \dots &{} l_{in_i-1} &{} -l_{in_i} \\ 0 &{} 0 &{} \dots &{} d_{in_i-1} &{} -d_{in_i} \end{bmatrix}, &{} \qquad &{} {\hat{P}} &{} = &{} \begin{bmatrix} {\hat{P}}_0 &{} 0 &{} \dots &{} 0 \\ 0 &{} {\hat{P}}_1 &{} \dots &{} 0 \\ \vdots &{} &{} \ddots &{} \\ 0 &{} 0 &{} &{} {\hat{P}}_r \\ {\tilde{P}}_0 &{} {\tilde{P}}_1 &{} \dots &{} {\tilde{P}}_r \end{bmatrix}. \end{array} \end{aligned}$$

6 Maximal minors of ${\hat{P}}$

In this section, we consider the set $M({\hat{P}})$ of maximal minors arising from the matrix representation of ${\hat{P}}$ given in the preceding section. A series of reduction steps turns $M({\hat{P}})$ into a smaller set having the same greatest common divisor, which finally enables us to compare with $M'(P)$. Whereas the cases (pe), (ep) and (pp) are are the simpler ones; see Proposition 6.7, the case (ee) takes up the second half of the section, see Proposition 6.17.

Definition 6.1

Let a subset $A \subseteq \hat{\mathcal {N}}$ with $|A| = |\hat{\mathcal {F}}|$ be given. Then we have a sublattice ${\hat{N}}_A:= \bigoplus _{x \in A} \mathbb {Z}\cdot x \subseteq {\hat{N}}$ and an induced map ${\hat{P}}_A :{\hat{F}} \rightarrow {\hat{N}}_A$ as in the commutative diagram

We call $|\det ({\hat{P}}_A)| \in \mathbb {Z}$ the maximal minor of ${\hat{P}}$ associated to $A$. The set of all maximal minors of ${\hat{P}}$ is defined as

$$\begin{aligned} M({\hat{P}}):= \{ |\det ({\hat{P}}_A)| \;\ A \subseteq \hat{\mathcal {N}},\ |A| = |\hat{\mathcal {F}}| \}. \end{aligned}$$

Construction 6.2

Let $A \subseteq \hat{\mathcal {N}}$ with $|A| = |\hat{\mathcal {F}}|$. We define

$$\begin{aligned} \begin{aligned} \hat{\mathcal {N}}_A^{\text {sing}}\ :=\ {}&\{{\hat{e}}_{ij}\ ;\ {\hat{e}}_{ij} \in A \text{ and } {\hat{u}}_{ij} \notin A \}\ \cup \ \{{\hat{u}}_{ij}\ ;\ {\hat{e}}_{ij} \notin A \text{ and } {\hat{u}}_{ij} \in A \}\ \subseteq \ A, \\ \hat{\mathcal {F}}_A^{\text {sing}}\ :=\ {}&\{{\hat{f}}_{ij}\ ;\ {\hat{e}}_{ij} \in A \text{ and } {\hat{u}}_{ij} \notin A \}\ \cup \ \{{\hat{f}}_{ij}\ ;\ {\hat{e}}_{ij} \notin A \text{ and } {\hat{u}}_{ij} \in A \}\ \subseteq \ \hat{\mathcal {F}}, \end{aligned} \end{aligned}$$

$$\begin{aligned} \hat{\mathcal {N}}_A^{\textrm{red}}\,=\ A\ \backslash \ \hat{\mathcal {N}}_A^{\textrm{sing}}, \qquad \qquad \hat{\mathcal {F}}_A^{\textrm{red}}\,=\ \hat{\mathcal {F}}\ \backslash \ \hat{\mathcal {F}}_A^{\textrm{sing}}. \end{aligned}$$

Note that we have $|\hat{\mathcal {N}}_A^{\textrm{red}}| = |\hat{\mathcal {F}}_A^{\textrm{red}}|$. Let ${\hat{N}}_A^{\textrm{red}}$ and ${\hat{F}}_A^{\textrm{red}}$ be the free lattices over $\hat{\mathcal {N}}_A^{\textrm{red}}$ and $\hat{\mathcal {F}}_A^{\textrm{red}}$ respectively. We obtain an induced map ${\hat{P}}_A^{\textrm{red}}$ as in the commutative diagram

We call $|\det ({\hat{P}}_A^{\textrm{red}})|$ the reduced minor of ${\hat{P}}$ associated to $A$. We define the set of reduced minors of ${\hat{P}}$ as

$$\begin{aligned} M^{\textrm{red}}({\hat{P}}):= \{ |\det ({\hat{P}}_A^{\textrm{red}})| \;\ A \subseteq \hat{\mathcal {N}},\ |A| = |\hat{\mathcal {F}}|\}. \end{aligned}$$

Example 6.3

The matrix ${\hat{P}}$ from Example 5.5 has five maximal minors and four reduced minors:

$$\begin{aligned} M({\hat{P}})&= \{d_{01}|{\hat{\mu }}|,\quad l_{01}|{\hat{\mu }}|,\quad d_{11}l_{21}|{\hat{\nu }}(0,1)|,\quad l_{11}d_{21}|{\hat{\nu }}(0,1)|,\quad l_{11}l_{21}|{\hat{\nu }}(0,1)| \} \\ M^{\textrm{red}}({\hat{P}})&= \{|{\hat{\mu }}|,\quad d_{11}l_{21}|{\hat{\nu }}(0,1)|,\quad l_{11}d_{21}|{\hat{\nu }}(0,1)|,\quad l_{11}l_{21}|{\hat{\nu }}(0,1)| \}. \end{aligned}$$

Proposition 6.4

Let $A \subseteq \hat{\mathcal {N}}$ with $|A| = |\hat{\mathcal {F}}|$. Then we have

$$\begin{aligned} \det ({\hat{P}}_A) = \det ({\hat{P}}_A^{\textrm{red}}) \prod _{{\hat{f}}_{ij} \in \hat{\mathcal {F}}_A^{\textrm{sing}}} x_{ij}, \quad \text {where} \quad x_{ij}:= {\left\{ \begin{array}{ll} l_{ij}, &{} {\hat{e}}_{ij} \in A, \\ d_{ij}, &{} {\hat{u}}_{ij} \in A. \end{array}\right. } \end{aligned}$$

Moreover, $\gcd (M({\hat{P}})) = \gcd (M^{\textrm{red}}({\hat{P}}))$ holds.

Proof

If ${\hat{f}}_{ij} \in \hat{\mathcal {F}}_A^{\textrm{sing}}$, we have

$$\begin{aligned} {\hat{P}}_A({\hat{f}}_{ij}) = {\left\{ \begin{array}{ll} l_{ij}{\hat{e}}_{ij}, &{} {\hat{e}}_{ij} \in A, \\ d_{ij}{\hat{u}}_{ij}, &{} {\hat{u}}_{ij} \in A. \end{array}\right. } \end{aligned}$$

In other words, the matrix representation of ${\hat{P}}_A$ has a column with the single entry $x_{ij}$ and zeroes elsewhere. Doing cofactor expansion by all these columns amounts to passing from $\det ({\hat{P}}_A)$ to $\det ({\hat{P}}_A^{\textrm{red}})$. This shows the first claim. The second one then follows from $\gcd (l_{ij}, d_{ij}) = 1$. $\square$

Definition 6.5

Set $\mathcal L:= \{(i,j)\;\ i = 0, \dots , r,\ j = 1, \dots , n_i' \}$. For $A \subseteq \hat{\mathcal N}$ with $|A| = |\hat{\mathcal {F}}|$, we define

$$\begin{aligned} L(A)\,=\ \{(i,j)\;\ {\hat{e}}_{ij} \in A \text { and } {\hat{u}}_{ij} \in A \}\ \subseteq \ \mathcal L. \end{aligned}$$

Lemma 6.6

Let $A \subseteq \hat{\mathcal N}$ with $|A| = |\hat{\mathcal {F}}|$.

• (i) If ${\hat{e}}_{ij} \notin A$ and ${\hat{u}}_{ij} \notin A$ for some $i = 0, \dots , r$ and $j = 1, \dots , n_i'$, we have $\det ({\hat{P}}_A) = 0$.

• (ii) If $(i,j_0), (i,j_1) \in L(A)$ for some $i = 0, \dots , r$ and $1 \le j_0 < j_1 \le n_i'$, we have $\det ({\hat{P}}_A) = 0$.

Proof

For (i), we have ${\hat{P}}_A({\hat{f}}_{ij}) = 0$, hence $\det ({\hat{P}}_A) = 0$. We show (ii). Consider first the case (ee). Then we have $n_i' = n_i - 1$. Set $\hat{\mathcal {N}}_{A,i}:= A \cap \hat{\mathcal {N}}_i$ and ${\hat{N}}_{A,i}:= {\hat{N}}_A \cap {\hat{N}}_i$. By (i), we may assume that ${\hat{e}}_{ij} \in \hat{\mathcal {N}}_{A,i}$ or ${\hat{u}}_{ij} \in \hat{\mathcal {N}}_{A,i}$ holds for all $1 \le j \le n_i-1$. Since $(i,j_0), (i,j_1) \in L(A)$, we thus have $|\hat{\mathcal {N}}_{A,i}| > n_i$. Consider the map ${\hat{P}}_{A,i} :{\hat{N}}_{A,i} \rightarrow {\hat{F}}_i$. In the matrix representation from Remark 5.6, we have

$$\begin{aligned} {\hat{P}}_A = \begin{bmatrix} *&{} 0 &{} 0\\ *&{} \boxed {\begin{array}{c} {\hat{P}}_{A,i} \quad 0\end{array}} &{}0\\ *&{} *&{} *\end{bmatrix}, \end{aligned}$$

where the outlined box is a square $|\hat{\mathcal {N}}_{A,i}| \times |\hat{\mathcal {N}}_{A,i}|$-matrix. Since the determinant of the outlined box vanishes, also $\det ({\hat{P}}_A) = 0$.

Now let $P$ be of type (pe), (ep) or (pp). Then $n_i' = n_i$. We define the set

$$\begin{aligned} \bar{\mathcal {N}}_A\,=\ \{ {\hat{u}}_{ij}\;\ (i,j) \in L(A) \}\ \subseteq \ \hat{\mathcal {N}}_A^{\textrm{red}}. \end{aligned}$$

Writing ${\bar{N}}_A$ for the free lattice over $\bar{\mathcal {N}}_A$ and ${\bar{F}}$ for the free lattice over $\bar{\mathcal {F}}$, we obtain an induced map ${\bar{P}}_A :{\bar{F}} \rightarrow {\bar{N}}_A$ as in the commutative diagram

Note that if $(i,j) \in L(A)$, we have $(\hat{P}_A^{\textrm{red}})^*({\hat{e}}_{ij}^*) = l_{ij} {\hat{f}}_{ij}^*$. That means, ${\hat{P}}_A^{\textrm{red}}$ contains a row with a single entry $l_{ij}$ and zeroes elsewhere. Doing cofactor expansion, we arrive at

$$\begin{aligned} \det ({\hat{P}}_A^{\textrm{red}}) = \det ({\bar{P}}_A) \prod _{(i,j) \in L(A)} l_{ij}. \end{aligned}$$

But since $(i,j_0), (i,j_1) \in L(A)$, we have ${\bar{P}}_A^*(\hat{u}_{ij_0}^*) = {\bar{P}}_A^*({\hat{u}}_{ij_1}^*)$, i.e. ${\bar{P}}_A$ contains two equal rows. Hence we have $\det ({\hat{P}}_A) = \det (\hat{P}_A^{\textrm{red}}) = \det ({\bar{P}}_A) = 0$. $\square$

Proposition 6.7

Let $P$ be of type (pe), (ep) or (pp). Let $A \subseteq \hat{\mathcal {N}}$ with $|A| = |\hat{\mathcal {F}}|$ such that $\det ({\hat{P}}_A) \ne 0$. Then we have $|L(A)| = r$. Furthermore, there exists an $i_1 = 0, \dots , r$ and $j_i = 1, \dots , n_i$ for all $i \ne i_1$ such that

$$\begin{aligned} |\det ({\hat{P}}_A^{\textrm{red}})| = \prod _{i \ne i_1} l_{ij_i}. \end{aligned}$$

In particular, we have $\gcd (M^{\textrm{red}}({\hat{P}})) = \gcd (M'(P))$.

Proof

By Lemma 6.6 (i), we have $\hat{e}_{ij} \in A$ or ${\hat{u}}_{ij} \in A$ for all $i$ and $j$. By Lemma 6.6 (ii), for each $i$ there is at most one $j$ with ${\hat{e}}_{ij} \in A$ and ${\hat{u}}_{ij} \in A$. Writing $\pi _1 :\mathbb {Z}\times \mathbb {Z}\rightarrow \mathbb {Z}$ for the projection onto the first coordinate, this implies that $|\pi _1(L(A))| = |L(A)|$. Together, we obtain

$$\begin{aligned} |A \cap \hat{\mathcal {N}}_i| = {\left\{ \begin{array}{ll} n_i, &{} i \notin \pi _1(L(A)), \\ n_i + 1, &{} i \in \pi _1(L(A)). \end{array}\right. } \end{aligned}$$

Thus, we have $|A| = n + |L(A)| + |A \cap \tilde{\mathcal {N}}|$. Recall that for the cases (pe) and (ep), we have $\tilde{\mathcal {N}} = \emptyset$ and $|A| = |\hat{\mathcal {F}}| = n + r$, hence $|L(A)| = r$. For (pp), we must have ${\tilde{u}}_i \in A$ for all $i = 1, \dots , r$, since otherwise ${\hat{P}}_A({\hat{f}}_i^+) = 0$. Hence $|A \cap \tilde{\mathcal {N}}| = r$. Since in this case, $|A| = n+2r$, we also arrive at $|L(A)| = r$. This implies that we have $L(A) = \{(i,j_i)\;\ i \ne i_1\}$ for some $i_1 = 0, \dots , r$ and $j_i = 1, \dots , n_i$. Following the proof of Lemma 6.6 (ii), we proceed to do cofactor expansion and see that $|\det ({\bar{P}}_A)| = 1$. This proves the claim. The supplement follows directly from the Definition of $M'(P)$. $\square$

The preceding proposition settles the discussion of maximal minors of ${\hat{P}}$ for the cases (pe), (pe) and (pp). The remainder of this section is devoted to the case (ee).

Lemma 6.8

Let $P$ be of type (ee). Let $A \subseteq \hat{\mathcal N}$ with $|A| = |\hat{\mathcal {F}}| = n$ and $\det ({\hat{P}}_A) \ne 0$. Then we have

$$\begin{aligned} |L(A)| = (r+1) - |A \cap \tilde{\mathcal N}|. \end{aligned}$$

In particular, $0 \le |L(A)| \le r+1$.

Proof

Let $\pi _1 :\mathbb {Z}\times \mathbb {Z}\rightarrow \mathbb {Z}$ be the projection onto the first coordinate. Lemma 6.6 (ii) implies $|\pi _1(L(A))| = |L(A)|.$ Furthermore, we have

$$\begin{aligned} |A \cap \hat{\mathcal N}_i| = {\left\{ \begin{array}{ll} n_i-1, &{} i \notin \pi _1(L(A)) \\ n_i, &{} i \in \pi _1(L(A)) \end{array}\right. }. \end{aligned}$$

Since, $|A| = n$, this implies the claim. $\square$

Definition 6.9

Let $P$ be of type (ee). For $k = 0, \dots , r+1$, we define

$$\begin{aligned} M^{\textrm{red}}_k({\hat{P}}):= \{ |\det ({\hat{P}}^{\textrm{red}}_A)|\;\ |L(A)| = k \}. \end{aligned}$$

Remark 6.10

Let $A \subseteq \hat{\mathcal {F}}$ with $|A| = n$ and $L(A) = \emptyset$. Lemma 6.8 implies that $A \cap \tilde{\mathcal {N}} = \tilde{\mathcal {N}}$. We obtain $\det ({\hat{P}}_A^{\textrm{red}}) = {\hat{\mu }}$. Hence $M^{\textrm{red}}_0({\hat{P}}) = \{|{\hat{\mu }}|\}$.

Proposition 6.11

Let $P$ be of type (ee). Let $A \subseteq \hat{\mathcal N}$ with $|A|=n$ such that $|{\hat{P}}_A^{\textrm{red}}| \ne 0$. Write $\pi _1 :\mathbb {Z}\times \mathbb {Z}\rightarrow \mathbb {Z}$ for the projection onto the first coordinate.

• (i) Assume ${\tilde{u}} \notin A$. Then for all $i = 1, \dots , r$, we have ${\tilde{e}}_i \in A$ or $i \in \pi _1(L(A))$. In this case,

  $$\begin{aligned} |\det ({\hat{P}}_A^{\textrm{red}})| = \left( \prod _{(i,j) \in L(A)} |{\hat{\nu }}(i,j)| \right) \left( \prod _{\begin{array}{c} 0 \le i \le r \\ i \notin \pi _1(L(A)) \end{array}} l_{in_i}\right) . \end{aligned}$$

• (ii) Assume ${\tilde{u}} \in A$. Then there exists at most one $i_1 = 1, \dots , r$ with ${\tilde{e}}_{i_1} \notin A$ and $i_1 \notin \pi _1(L(A))$. If there exists such an $i_1$, we have

  $$\begin{aligned} |\det ({\hat{P}}_A^{\textrm{red}})| = \left| d_{i_1n_{i_1}} \left( \prod _{(i,j) \in L(A)} {\hat{\nu }}(i,j) \right) \left( \prod _{\begin{array}{c} 0 \le i \le r \\ i \notin \pi _1(L(A)) \cup \{i_1\} \end{array}} l_{in_i} \right) \right| . \end{aligned}$$

  If there exists no such $i_1$, we have

  $$\begin{aligned} |\det ({\hat{P}}_A^{\textrm{red}})| = \left| \sum _{\begin{array}{c} 0 \le i' \le r \\ i' \notin \pi _1(L(A)) \end{array}} \pm d_{i'n_{i'}} \left( \prod _{(i,j) \in L(A)} {\hat{\nu }}(i,j) \right) \left( \prod _{\begin{array}{c} 0 \le i \le r \\ i \notin \pi _1(L(A)) \cup \{i'\} \end{array}} l_{in_i} \right) \right| . \end{aligned}$$

Proof

For part (i), assume that there is some $i = 1, \dots , r$ such that ${\tilde{e}}_i \notin A$ and $i \notin \pi _1(L(A))$. This implies $\hat{e}_{ij} \notin \hat{\mathcal {N}}_A^{\textrm{red}}$ and ${\hat{u}}_{ij} \notin \hat{\mathcal {N}}_A^{\textrm{red}}$ for all $j = 1, \dots , n_i-1$. Since also ${\tilde{u}} \notin A$, we obtain ${\hat{P}}_A^{\textrm{red}}({\hat{f}}_{in_i}) = 0$. Hence $\det (\hat{P}_A^{\textrm{red}}) = 0$, a contradiction. It follows that if $i \notin \pi _1(L(A))$, either $i = 0$ or ${\tilde{e}}_i \in A$. The formula for $\det (\hat{P}_A^{\textrm{red}})$ now follows from cofactor expansion.

For part (ii), assume that there exist $1 \le i_0 < i_1 \le r$ such that ${\tilde{e}}_{i_0}, {\tilde{e}}_{i_1} \notin A$ and $i_0, i_1 \notin \pi _1(L(A))$. We obtain ${\hat{P}}_A^{\textrm{red}}(\hat{f}_{i_0n_{i_0}}) = d_{i_0n_{i_0}} {\tilde{u}}$ and $\hat{P}_A^{\textrm{red}}({\hat{f}}_{i_1n_{i_1}}) = d_{i_1n_{i_1}} {\tilde{u}}$. Hence $\det ({\hat{P}}_A^{\textrm{red}}) = 0$, a contradiction. The formulas for $\det ({\hat{P}}_A^{\textrm{red}})$ again follow from cofactor expansion. $\square$

Definition 6.12

Let $P$ be of type (ee). Let $\pi _1 :\mathbb {Z}\times \mathbb {Z}\rightarrow \mathbb {Z}$ be the projection onto the first coordinate. We call a subset $L \subseteq \mathcal L$ valid, if $|\pi _1(L)| = |L|$. For $k = 1, \dots , r$, we define

$$\begin{aligned} M'_k({\hat{P}}):= \left\{ \left( \prod _{(i,j) \in L} |{\hat{\nu }}(i,j)| \right) \left( \prod _{\begin{array}{c} 0 \le i \le r \\ i \notin \pi _1(L) \cup \{i_1\} \end{array}} l_{in_i} \right) \;\ \begin{aligned}&L \subseteq \mathcal L \text { valid},\ |L| = k, \\&0 \le i_1 \le r,\ i_1 \notin \pi _1(L), \end{aligned} \right\} . \end{aligned}$$

Example 6.13

Continuing Example 6.3, we have

$$\begin{aligned}{} & {} M_0^{\textrm{red}}({\hat{P}})\ =\ \{|{\hat{\mu }}|\}, \quad M_1^{\textrm{red}}(\hat{P})\ =\ \{d_{11}l_{21}|{\hat{\nu }}(0,1)|,\ l_{11}d_{21}|{\hat{\nu }}(0,1)|,\ l_{11}l_{21}|{\hat{\nu }}(0,1)|\}\\{} & {} M_1'({\hat{P}})\ =\ \{l_{11}|{\hat{\nu }}(0,1)|,\ l_{21}|{\hat{\nu }}(0,1)|\},\qquad M_2'({\hat{P}}) = M_2^{\textrm{red}}(\hat{P}) = M_3^{\textrm{red}}({\hat{P}}) = \emptyset . \end{aligned}$$

Proposition 6.14

Let $P$ be of type (ee). We have

• (i) $\gcd (M^{\textrm{red}}_k({\hat{P}})) = \gcd (M'_k({\hat{P}}))$ for all $k = 1, \dots , r$,

• (ii) $\gcd (M^{\textrm{red}}_{r+1}({\hat{P}}) \cup M'_r({\hat{P}})) = \gcd (M'_r({\hat{P}}))$,

• (iii) $\gcd \left( \bigcup _{k=1}^{r+1} M_k^{\textrm{red}}({\hat{P}})\right) = \gcd \left( \bigcup _{k=1}^{r} M_k'({\hat{P}})\right)$.

Proof

We show (i). Proposition 6.11 implies that every element of $M_k^{\textrm{red}}({\hat{P}})$ is a $\mathbb {Z}$-linear combination of elements of $M'_k({\hat{P}})$. This shows that $\gcd (M'_k(\hat{P}))$ divides $\gcd (M_k^{\textrm{red}}({\hat{P}}))$. For the converse, it suffices to show that $\gcd (M_k^{\textrm{red}}({\hat{P}})) \mid x$ holds for all $x \in M'_k(\hat{P})$. So, let

$$\begin{aligned} x = \left( \prod _{(i,j) \in L} \nu (i,j) \right) \left( \prod _{\begin{array}{c} 0 \le i \le r \\ i \notin \pi _1(L) \cup \{i_1\} \end{array}} l_{in_i} \right) \in M'_k({\hat{P}}) \end{aligned}$$

be arbitraty, where $L \subseteq \mathcal L$ is a valid subset with $|L| = k$ and $0 \le i_1 \le r$ with $i_1 \notin \pi _1(L)$.

Case 1: $i_1 \ne 0$ and $0 \in \pi _1(L)$. Choose a subset $A \subseteq \hat{\mathcal N}$ with $|A| = n$ such that $L(A) = L$ and

$$\begin{aligned} A \cap \tilde{\mathcal N} = \{{\tilde{e}}_i\;\ 1 \le i \le r,\ i \notin \pi _1(L) \}. \end{aligned}$$

Since $0 \in \pi _1(L)$, we have $|A \cap \tilde{\mathcal N}| = r - (|L| - 1) = (r+1) - |L|$. This means we can choose $A$ such that $\det ({\hat{P}}^{\textrm{red}}_A) \ne 0$. Now set

$$\begin{aligned} A':= (A \backslash \{{\tilde{e}}_{i_1}\}) \cup \{{\tilde{u}}\}. \end{aligned}$$

Proposition 6.11 implies that $\det ({\hat{P}}_A^{\textrm{red}}) = l_{i_1n_{i_1}} x$ and $\det ({\hat{P}}_{A'}^{\textrm{red}}) = d_{i_1n_{i_1}} x$. Hence we have

$$\begin{aligned} \gcd (M_k^{\textrm{red}}({\hat{P}})) \mid \gcd (l_{i_1n_{i_1}} x, d_{i_1n_{i_1}} x) = x. \end{aligned}$$

Case 2: $i_1 \ne 0$ and $0 \notin \pi _1(L)$. Since $k \ge 1$, we find some $i_0 \in \pi _1(L)$. Now choose $A \subseteq \hat{\mathcal N}$ with $|A| = n$ such that $L(A) = L$ and

$$\begin{aligned} A \cap \tilde{\mathcal N} = \{{\tilde{e}}_i\;\ 1 \le i \le r,\ i \notin \pi _1(L) \} \cup \{ {\tilde{e}}_{i_0} \}. \end{aligned}$$

Again, we have $|A \cap \tilde{\mathcal N}| = (r+1) - |L|$, hence we can pick $A$ such that $\det ({\hat{P}}_A^{\textrm{red}}) \ne 0$. Proceeding in the same way as in Case 1, we arrive at $\gcd (M_k^{\textrm{red}}({\hat{P}})) \mid x$.

Case 3: $i_1 = 0$. As in Case 2, we can pick some $i_0 \in \pi _1(L)$ as well as a subset $A \subseteq \hat{\mathcal N}$ with $|A| = n$ such that $L(A) = L$ and

$$\begin{aligned} A \cap \tilde{\mathcal N} = \{{\tilde{e}}_i\;\ 1 \le i \le r,\ i \notin \pi _1(L) \} \cup \{ {\tilde{e}}_{i_0} \}. \end{aligned}$$

For all $1 \le i \le r$ with $i \notin \pi _1(L)$, set $A_i:= (A \backslash \{{\tilde{e}}_i\}) \cup \{ {\tilde{u}} \}.$ Then Proposition 6.11 implies that $\det (P_A^{\textrm{red}}) = l_{0n_{0}} x$ and

$$\begin{aligned} \det (P_{A_{i_0}}^{\textrm{red}}) = d_{0n_0} x + \sum _{\begin{array}{c} 1 \le i' \le r \\ i' \notin \pi _1(L) \end{array}} \pm \det (P_{A_{i'}}^{\textrm{red}}). \end{aligned}$$

This implies $\gcd (M_k^{\textrm{red}}({\hat{P}})) \mid \gcd (l_{0n_0} x, d_{0n_0} x) = x$.

Part (ii) follows from the fact that every element of $M^{\textrm{red}}_{r+1}({\hat{P}})$ is an integer multiple of an element of $M'_r({\hat{P}})$. Part (iii) is a consequence of (i) and (ii). $\square$

Definition 6.15

Let $P$ be of type (ee). For $k = 1, \dots , r$, we define the set

$$\begin{aligned} M''_k({\hat{P}}):= \left\{ \left( \prod _{(i,j) \in L} |{\hat{\nu }}(i,j)| \right) \left( \prod _{\begin{array}{c} 0 \le i \le r \\ i \notin \pi _1(L) \cup \{i_1\} \end{array}} l_{ij_i} \right) \;\ \begin{aligned}&L \subseteq \mathcal L \text { valid},\ |L| = k \\&0 \le i_1 \le r,\ i_1 \notin \pi _1(L) \\&1 \le j_i \le n_i \text { for all } i \end{aligned} \right\} . \end{aligned}$$

Lemma 6.16

Let $i = 0, \dots , r$ and $j_i = 1, \dots , n_i-1$. Then we have

$$\begin{aligned} \gcd (l_{in_i}, {\hat{\nu }}(i,j)) \mid l_{ij}. \end{aligned}$$

Proof

By definition we have ${\hat{\nu }}(i,j) = l_{in_i} d_{ij} - l_{ij} d_{in_i}$. Since $l_{in_i}$ and $d_{in_i}$ are coprime, we find $x,y \in \mathbb {Z}$ such that $x l_{in_i} + y d_{in_i} = 1$. Then we have

$$\begin{aligned} (x l_{ij} + y d_{in_i}) l_{in_i} - y {\hat{\nu }}(i,j) = l_{ij}(x l_{in_i} + y d_{in_i}) = l_{ij}. \end{aligned}$$

This implies the claim. $\square$

Proposition 6.17

Let $P$ be of type (ee). We have

• (i) $\gcd (M'_k({\hat{P}}) \cup M''_{k+1}({\hat{P}})) = \gcd (M''_k(\hat{P}))$ for all $k = 1, \dots , r-1$,

• (ii) $\gcd \left( \bigcup _{k=1}^r M'_k({\hat{P}})\right) = \gcd (M''_1({\hat{P}}))$.

Proof

We show (i). Since $M'_k({\hat{P}}) \subseteq M''_k({\hat{P}})$ and elements of $M''_{k+1}({\hat{P}})$ are $\mathbb {Z}$-linear combinations of elements of $M''_k({\hat{P}})$, we have $\gcd (M''_k({\hat{P}})) \mid \gcd (M'_k({\hat{P}}) \cup M''_{k+1}({\hat{P}}))$. For the converse, let

$$\begin{aligned} x = \left( \prod _{(i,j) \in L} {\hat{\nu }}(i,j) \right) \left( \prod _{\begin{array}{c} 0 \le i \le r \\ i \notin \pi _1(L) \cup \{i'\} \end{array}} l_{ij_i} \right) \in M''_k({\hat{P}}), \end{aligned}$$

where $L \subseteq \mathcal L$ is a valid subset with $|L| = k$ and $0 \le i' \le r$ with $i' \notin \pi _1(L)$ and $1 \le j_i \le n_i$ for all $i \notin \pi _1(L) \cup \{i'\}$. Let us write

$$\begin{aligned} \{i_1, \dots , i_{r-k} \}:= \{0, \dots , r\} \backslash (\pi _1(L) \cup \{i'\}). \end{aligned}$$

We define numbers

$$\begin{aligned} \begin{array}{lcl} x_0 &{}:= &{} l_{i_1n_{i_1}} l_{i_2n_{i_2}} \dots l_{i_{r-k}n_{i_{r-k}}} \\ x_1 &{}:= &{} l_{i_1j_{i_1}} l_{i_2n_{i_2}} \dots l_{i_{r-k}n_{i_{r-k}}} \\ &{} \vdots &{} \\ x_{r-k} &{}:= &{} l_{i_1j_{i_1}} l_{i_2j_{i_2}} \dots l_{i_{r-k}j_{i_{r-k}}} \\ \end{array} \end{aligned}$$

as well as

$$\begin{aligned} \begin{array}{lcl} y_1 &{}:= &{} {\hat{\nu }}(i_1, j_{i_1}) l_{i_2n_{i_2}} l_{i_3n_{i_3}} \dots l_{i_{r-k}n_{i_{r-k}}} \\ y_2 &{}:= &{} l_{i_1j_{i_1}} {\hat{\nu }}(i_2, j_{i_2}) l_{i_3n_{i_3}} \dots l_{i_{r-k}n_{i_{r-k}}} \\ &{} \vdots \\ y_{r-k} &{}:= &{} l_{i_1j_{i_1}} \dots l_{i_{r-k-1}j_{i_{r-k-1}}} {\hat{\nu }}(i_{r-k}, j_{i_{r-k}}). \end{array} \end{aligned}$$

Then Lemma 6.16 implies $\gcd (x_{m-1},y_m) \mid x_m$ for all $m = 1, \dots , r-k$. In particular, we obtain $\gcd (x_0, y_1, \dots , y_{r-k}) \mid x_{r-k}$. Now set $c:= \prod _{(i,j) \in L} {\hat{\nu }}(i,j)$. Then we have $cx_{r-k} = x$ as well as $x_0c \in M'_k({\hat{P}})$ and $y_mc \in M''_{k+1}({\hat{P}})$ for all $m = 1, \dots r-k$. Together, we have

$$\begin{aligned} \gcd (M'_k({\hat{P}}) \cup M''_{k+1}({\hat{P}})) \mid \gcd (x_0c, y_1c, \dots , y_{r-k}c) \mid x_{r-k}c = x. \end{aligned}$$

Part (ii) follows from repeated application of (i), together with the fact that $M'_r({\hat{P}}) = M''_r({\hat{P}})$. $\square$

7 Proof of Theorem 1.1 and examples

In this section, we prove the formula for the Picard index of a $\mathbb {K}^*$-surface given in Theorem 1.1. We then give two examples where the formula fails: The first one is a toric threefold, the second one is the $D_8$-singular log del Pezzo surface of Picard number one.

Proposition 7.1

Let $P$ be a defining matrix and ${\hat{P}}$ be as in Construction 5.4. Write $M(P)$ and $M({\hat{P}})$ for the set of maximal minors of $P$ and ${\hat{P}}$ respectively. Then we have

$$\begin{aligned} \gcd (M({\hat{P}})) = \gcd (M(P)). \end{aligned}$$

Proof

By Construction 6.2, we have $\gcd (M(\hat{P})) = \gcd (M^{\textrm{red}}({\hat{P}}))$. On the other hand, Proposition 4.11 says that $\gcd (M(P)) = \gcd (M'(P))$. In the cases (pe), (ep) and (pp), Proposition 6.7 gives the result. In the case (ee), combining Remark 6.10, Proposition 6.14 (iii) and Proposition 6.17 (ii), we get

$$\begin{aligned} \gcd (M^{\textrm{red}}({\hat{P}}))&= \gcd \left( \{|{\hat{\mu }}|\} \cup \bigcup _{k=1}^{r+1} M^{\textrm{red}}_k({\hat{P}})\right) \\&= \gcd \left( \{|{\hat{\mu }}|\} \cup \bigcup _{k=1}^r M'_k({\hat{P}})\right) \\&= \gcd \left( \{|{\hat{\mu }}|\} \cup M''_1(\hat{P})\right) . \end{aligned}$$

By definition, we have $\{|{\hat{\mu }}|\} \cup M''_1({\hat{P}}) = M'(P)$, hence we arrive at the claim. $\square$

Proof of Theorem 1.1

By Theorem 3.7, we can assume $X = X(P) \subseteq Z$ as in Construction 3.4. Note that $|\textrm{Cl}(X,x)| \ne 1$ only holds for fixed points of the $\mathbb {K}^*$-action, which we denote by $X^\textrm{fix}$. By Remark 3.6 and Proposition 3.8 (iii), we get

$$\begin{aligned} \prod _{x \in X} |\textrm{Cl}(X,x)| = \prod _{x \in X^{\textrm{fix}}} |\textrm{Cl}(X,x)| = \prod _{\sigma \in \Sigma _{\max }} |\textrm{Cl}(Z,z_{\sigma })|. \end{aligned}$$

Combining Proposition 3.8 (iv) and Proposition 2.7, we get

$$\begin{aligned} \iota _{\textrm{Pic}}(X) = \iota _{\textrm{Pic}}(Z) = \frac{1}{|{\hat{K}}|} \prod _{\sigma \in \Sigma _{\max }} |\textrm{Cl}(Z,z_{\sigma })|, \end{aligned}$$

By Proposition 3.8 (i), it remains to show that $|{\hat{K}}| = |\textrm{Cl}(Z)^{\textrm{tors}}|$. Since $\textrm{Cl}(Z)$ is isomorphic to the cokernel of $P^*$, we have

$$\begin{aligned} |\textrm{Cl}(Z)^{\textrm{tors}}| = \gcd (M(P^*)) = \gcd (M(P)), \end{aligned}$$

where $M(A)$ denotes the set of maximal minors of a matrix $A$. On the other hand, Proposition 2.3 along with Remark 5.2 says that ${\hat{K}}$ is the cokernel of ${\hat{P}}^*$, hence

$$\begin{aligned} |{\hat{K}}| = \gcd (M({\hat{P}}^*)) = \gcd (M({\hat{P}})). \end{aligned}$$

Proposition 7.1 now implies the claim. $\square$

Example 7.2

Recall that the $\mathbb {K}^*$-surface $X = X(P)$ from Example 3.1 has three fixed points $x^+,x^-$ and $x_{01}$. By Proposition 3.8 (iii), we can compute their local class groups via toric geometry:

$$\begin{aligned} \textrm{Cl}(X,x^+)\ =\ \mathbb {Z}/20\mathbb {Z}, \qquad \textrm{Cl}(X,x^-)\ =\ \mathbb {Z}/12\mathbb {Z}, \qquad \textrm{Cl}(X,x_{01})\ =\ \{0\}. \end{aligned}$$

Recall that $\textrm{Cl}(X) = \mathbb {Z}\times \mathbb {Z}/4\mathbb {Z}$. By Theorem 1.1, we again arrive at $\iota _{\textrm{Pic}}(X) = 60$.

Specializing Theorem 1.1 to toric surfaces, we get the following:

Corollary 7.3

Let $Z = Z_{\Sigma }$ be a projective toric surface. Then

$$\begin{aligned} \iota _{\textrm{Pic}}(Z) = \frac{1}{|\textrm{Cl}(Z)^{\textrm{tors}}|}\prod _{\sigma \in \Sigma _{\max }} |\textrm{Cl}(Z,z_{\sigma })|. \end{aligned}$$

Remark 7.4

Let $Z = \mathbb {P}(w_0, w_1, w_2)$ be a weighted projective plane. We can assume the weights $w_i$ to be pairwise coprime. The divisor class group of a weighted projective space is torsion-free and the orders of the local class groups at the toric fixed points are equal to the weights $w_i$. By Corollary 7.3, we obtain

$$\begin{aligned} \iota _{\textrm{Pic}}(Z) = w_0 w_1 w_2. \end{aligned}$$

In Proposition 8.1, we will generalize this formula to fake weighted projective planes. For weighted projective planes, there is also a direct way to compute the Picard index: The subgroup of divisor classes that are principle on the $i$-th standard affine chart $U_i = Z \backslash V(x_i)$ is generated by $w_i = [V(x_i)] \in \textrm{Cl}(Z) \cong \mathbb {Z}$. For the Picard group, this means

$$\begin{aligned} \textrm{Pic}(Z)\ =\ \mathbb {Z}w_0 \cap \mathbb {Z}w_1 \cap \mathbb {Z}w_2\ =\ \mathbb {Z}\textrm{lcm}(w_0, w_1, w_2)\ \subseteq \ \mathbb {Z}\ \cong \ \textrm{Cl}(Z). \end{aligned}$$

Since the weights are pairwise coprime, we have $\textrm{lcm}(w_0, w_1, w_2) = w_0 w_1 w_2$.

The following example shows that Corollary 7.3 does not hold for higher dimensional toric varieties.

Example 7.5

Consider the three-dimensional weighted projective space $Z = \mathbb {P}(2,2,3,5)$. Note that the weights are well-formed, i.e. any three weights have no common factor. The Picard group is given by

$$\begin{aligned} \textrm{Pic}(Z)\ =\ 2 \mathbb {Z}\ \cap \ 2 \mathbb {Z}\ \cap \ 3 \mathbb {Z}\ \cap \ 5 \mathbb {Z}\ =\ 30 \mathbb {Z}\ \subseteq \ \mathbb {Z}\ \cong \ \textrm{Cl}(Z). \end{aligned}$$

Hence we have $\iota _{\textrm{Pic}}(Z) = 30$. On the other hand, the product of the orders of the local class groups is $60$.

We now consider the $D_8$-singular log del Pezzo surface of Picard number one, which does not admit a $\mathbb {K}^*$-action. Using the description of its Cox Ring [14, Theorem 4.1], we construct the surface via its canonical ambient toric variety, see also [2, Sections 3.2 and 3.3].

Example 7.6

Consider the integral matrix

$$\begin{aligned} P\,=\ \begin{bmatrix} v_1&v_2&v_3&v_4 \end{bmatrix}\,=\ \begin{bmatrix} 1 &{} 0 &{} 1 &{} -3 \\ 0 &{} 1 &{} 1 &{} -2 \\ 0 &{} 0 &{} 2 &{} -2 \end{bmatrix}. \end{aligned}$$

Let $Z = Z_{\Sigma }$ be the toric variety whose fan $\Sigma$ has the following maximal cones:

$$\begin{aligned}{} & {} \sigma _{12}\,=\ \textrm{cone}(v_1, v_2), \qquad \sigma _{23}\,=\ \textrm{cone}(v_2, v_3), \qquad \sigma _{24}\,=\ \textrm{cone}(v_2,v_4),\\{} & {} \sigma _{134}\,=\ \textrm{cone}(v_1,v_3,v_4). \end{aligned}$$

Let $p :{\hat{Z}} \rightarrow Z$ be Cox’s quotient presentation of $Z$, where ${\hat{Z}} \subseteq {\bar{Z}}:= \mathbb {K}^4$. Consider the polynomial

$$\begin{aligned} f:= T_1^2 - T_2 T_3 T_4^2 + T_3^4 + T_4^4. \end{aligned}$$

We obtain a commutative diagram

Here, $\iota :X \rightarrow Z$ is the canonical toric embedding in the sense of [2, Sec. 3.2.5]. This implies that $X$ has non-empty intersection with the toric orbits $\mathbb {T}^3 \cdot z_{\sigma }$ for $\sigma \in \Sigma _{\max }$. We obtain a decomposition into pairwise disjoint pieces

$$\begin{aligned} X = \bigcup _{\sigma \in \Sigma _{\max }} X(\sigma ), \qquad \text {where}\ X(\sigma ):= X \cap \mathbb {T}^3 \cdot z_{\sigma }. \end{aligned}$$

By [2, Proposition 3.3.1.5], we have $\textrm{Cl}(X,x) \cong \textrm{Cl}(Z,z_{\sigma })$ for $x \in X(\sigma )$. Note that $\sigma _{12}, \sigma _{23}$ and $\sigma _{24}$ are regular and $|\textrm{Cl}(Z, z_{\sigma _{134}})| = |\det (v_1, v_3, v_4)| = 2$. This shows that

$$\begin{aligned} \prod _{x \in X} |\textrm{Cl}(X,x)|\ =\ \prod _{\sigma \in \Sigma _{\max }} |\textrm{Cl}(z, z_{\sigma })|\ =\ 2. \end{aligned}$$

We turn to the Picard index. In the notation of Construction 2.2, we have

$$\begin{aligned} N_{\sigma _{ij}}\ =\ \mathbb {Z}v_i + \mathbb {Z}v_j, \qquad \qquad N_{\sigma _{134}}\ =\ N\ =\ \mathbb {Z}^3. \end{aligned}$$

Under the lattice bases $\{v_i, v_j\}$ of $N_{\sigma _{ij}}$, we can view $P_{\sigma _{ij}}$ as the identity matrix and $P_{\sigma _{134}} = \begin{bmatrix} v_1&v_3&v_4 \end{bmatrix}$. Computing matrix representations of the maps involved in Construction 2.2, we obtain

$$\begin{aligned} \begin{array}{rclcrcl} \alpha &{} = &{} \begin{bmatrix} 1 &{} 0 &{} 0 &{} 1 &{} 0 &{} -3 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 1 &{} 1 &{} -2 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 2 &{} 0 &{} -2 &{} 0 &{} 0 &{} 1 \end{bmatrix}, &{} \qquad &{} \beta &{} = &{} \begin{bmatrix} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 1 \end{bmatrix}, \\ \gamma &{} = &{} \begin{bmatrix} -1 &{} 0 &{} 3 &{} -1 &{} 0 &{} 0 \\ -1 &{} -1 &{} 2 &{} 0 &{} -1 &{} -1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ -2 &{} 0 &{} 2 &{} 0 &{} 0 &{} 0 \end{bmatrix}, &{} \qquad &{} \delta &{} = &{} \begin{bmatrix} 0 &{} 0 &{} -1 &{} 0 &{} 0 \\ -1 &{} 0 &{} 0 &{} -1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} -1 \\ 0 &{} -1 &{} 0 &{} 0 &{} 0 \end{bmatrix}, \end{array} \end{aligned}$$

$$\begin{aligned} {\hat{P}} = \begin{bmatrix} 0 &{} 0 &{} 0 &{} 0 &{} 1 \\ 1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 3 &{} 1 &{} 0 &{} -1 \\ 0 &{} 2 &{} 0 &{} 0 &{} -1 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0 \end{bmatrix}. \end{aligned}$$

We see that ${\hat{P}}^*$, is surjective, hence its cokernel $\hat{K}$ is trivial. Using Proposition 2.7, we conclude $\iota _{\textrm{Pic}}(X) = \iota _{\textrm{Pic}}(Z) = 2$. On the other hand, we have

$$\begin{aligned} \textrm{Cl}(X)\ \cong \ \textrm{Cl}(Z)\ \cong \ \mathbb {Z}^3 / \textrm{im}(P^*)\ \cong \ \mathbb {Z}\oplus \mathbb {Z}/ 2 \mathbb {Z}. \end{aligned}$$

This shows that the formula from Theorem 1.1 does not hold for $X$.

8 Log del Pezzo $\mathbb {K}^{*}$-surfaces of Picard number one

We contribute to the classification of log del Pezzo $\mathbb {K}^*$-surfaces of Picard number one, which in the toric case are fake weighted projective planes. As a special case of fake weighted projective spaces, these have been studied by several authors [8, 16]. Our classification relies on Theorem 1.1 to produce bounds for the number of log del Pezzo $\mathbb {K}^*$-surfaces with fixed Picard index. For a related classification of fake weighted projective spaces by their Gorenstein index, we refer to [3].

We begin with the toric case. Consider an integral $2 \times 3$ matrix

$$\begin{aligned} P = \begin{bmatrix} v_0&v_1&v_2 \end{bmatrix}, \end{aligned}$$

where the columns are primitive vectors generating $\mathbb {Q}^3$ as a convex cone. Then we obtain a fake weighted projective plane $Z = Z(P)$ as the toric surface associated to the unique complete lattice fan $\Sigma$ with ray generators $v_i$. We obtain a splitting of the divisor class group:

$$\begin{aligned} \textrm{Cl}(Z)\ \cong \ \mathbb {Z}\times \mathbb {Z}/n\mathbb {Z}. \end{aligned}$$

Under this splitting, we define $(w_i,\eta _i):= \omega _i:= [D_i]$, where $D_i$ is the torusinvariant Weil divisor associated to $v_i$, for $i=0,1,2$. Note that any two of the $\omega _i$ generate $\textrm{Cl}(Z)$. In particular, the $w_i$ are pairwise coprime integers.

Proposition 8.1

Let $Z = Z(P)$ be a fake weighted projective plane with divisor class group $\textrm{Cl}(Z) \cong \mathbb {Z}\times \mathbb {Z}/n\mathbb {Z}$. Write $\textbf{z}(i)$ for the toric fixed points associated to $\sigma _i:= \textrm{cone}(v_j\;\ j\ne i)$. Then the orders of the local class groups are given by

$$\begin{aligned} \vert \textrm{Cl}(Z,\textbf{z}(i)) \vert = nw_i = \vert \det (\sigma _i) \vert . \end{aligned}$$

Moreover, the Picard index is $\iota _{\textrm{Pic}}(Z) = n^2 w_0 w_1 w_2.$

Proof

We have $\textrm{Cl}(Z,\textbf{z}(i))\ \cong \ K / \mathbb {Z}\omega _i$, hence the order equals $nw_i$. On the other hand, $\textrm{Cl}(Z,\textbf{z}(i)) \cong \textrm{Cl}(U_i)$, where $U_i$ is the affine toric chart associated to $\sigma _i$, hence the order equals $\vert \det (\sigma _i) \vert$. The formula for the Picard Index follows from Corollary 7.3. $\square$

Remark 8.2

Let $Z(P)$ be a fake weighted projective plane. Then we have $Z(P) \cong Z(P')$ if and only if $P' = A \cdot P \cdot S$ holds with a unimodular matrix $A$ and a permutation matrix $S$.

Proposition 8.3

Let $Z(P)$ be a fake weighted projective plane. Then there exists an integer $0 \le x < nw_2$ with $\gcd (x, nw_2) = 1$ such that $Z(P) \cong Z(P')$, where

$$\begin{aligned} P' = \begin{bmatrix} 1 &{} x &{} - \frac{w_0 + x w_1}{w_2} \\ 0 &{} nw_2 &{} -nw_1 \end{bmatrix}. \end{aligned}$$

Proof

We write $P = \begin{bmatrix} v_0&v_1&v_2 \end{bmatrix}$, where $v_i = (x_i, y_i) \in \mathbb {Z}^2$ are primitive vectors. Applying a unimodular matrix from the left achieves $v_0 = (1,0)$ and $0 \le x_1 < y_1$. By Proposition 8.1, we have

$$\begin{aligned} nw_2= & {} \det (v_0, v_1) = y_1, \qquad \qquad -nw_1 = \det (v_0, v_2) = y_2.\\ nw_0= & {} \det (v_1, v_2) = -nw_1x_1 - nw_2x_2. \end{aligned}$$

This shows the claim. $\square$

Algorithm 8.4

Input: A positive integer $\iota$, the prospective Picard Index. Algorithm:

• Set $L:= \emptyset$.

• For each quadruple $(n,w_0,w_1,w_2)$ of positive integers with $n^2w_0w_1w_2 = \iota$ such that $(w_0,w_1,w_2)$ are pairwise coprime, do:

  • For each $0 \le x < nw_2$ with $\gcd (x,nw_2) = 1$, do:

    • Set

      $$\begin{aligned} P:= \begin{bmatrix} 1 &{} x &{} - \frac{w_0 + x w_1}{w_2} \\ 0 &{} nw_2 &{} -nw_1 \end{bmatrix}. \end{aligned}$$

    • If for all permutation matrices $S$, the Hermite normal form of $P \cdot S$ differs from the Hermite normal form of all $P' \in L$, add $P$ to $L$.

  • end do.

• end do.

Output: The set $L$. Then every fake weighted projective plane $Z$ with $\iota _{\textrm{Pic}}(Z) = \iota$ is isomorphic to precisely one $Z(P)$ with $P \in L$.

Proof

Propositions 8.1 and 8.3 show that the algorithm produces all generator matrices of fake weighted projective planes sharing $\iota$ as their Picard index. By Remark 8.2, the matrices in $P \in L$ belong to pairwise non-isomorphic surfaces. $\square$

Using this Algorithm, we arrive at the following classification.

Theorem 8.5

There are $15\,086\,426$ isomorphy classes of fake weighted projective planes with Picard index at most $1\,000\,000$. Of those, $68\,053$ have Picard index at most $10\,000$. The number of isomorphy classes for given Picard index develops as follows:

We turn to non-toric rational projective $\mathbb {K}^*$-surfaces of Picard number one. These arise from defining matrices as in Construction 3.4, see also [8, Section 4].

Given a rational projective $\mathbb {K}^*$-surface $X = X(P) \subseteq Z$ as in Construction 3.4, the columns $v_{ij},v^{\pm }$ define torusinvariant Weil divisors $D_{ij},D^{\pm }$ on $Z$. If $X$ is of Picard number one, the divisor class group splits as

$$\begin{aligned} \textrm{Cl}(X)\ \cong \ \textrm{Cl}(Z)\ \cong \ \mathbb {Z}\times \textrm{Cl}(Z)^{\textrm{tors}}. \end{aligned}$$

We define $(w_{ij},\eta _{ij}):= \omega _{ij}:= [D_{ij}]$ and $(w^{\pm },\eta ^{\pm }):= \omega ^{\pm }:= [D^{\pm }]$.

Proposition 8.6

Let $X = X(P)$, where $P = \begin{bmatrix} v_{01}&v_{02}&v_1&\dots&v_r\end{bmatrix}$ is a defining matrix of type (ee) with $n_0 = 2$ and $n_1 = \dots = n_r = 1$. Then $X$ is of Picard number one. We have two elliptic fixed points $x^{\pm }$ and one hyperbolic fixed point $x_{01}$. With $\lambda := |\textrm{Cl}(X)^{\textrm{tors}}|$, the orders of the local class groups are

$$\begin{aligned}{} & {} |\textrm{Cl}(X,x^+)|\ =\ \lambda w_{02} = |\det (\sigma ^+)|, \qquad |\textrm{Cl}(X,x^-)|\ =\ \lambda w_{01} = |\det (\sigma ^-)|.\\{} & {} |\textrm{Cl}(X,x_{01})| = \det \begin{bmatrix} -l_{01} &{} -l_{02} \\ d_{01} &{} d_{02} \end{bmatrix} = d_{01} l_{02} - d_{02} l_{01}. \end{aligned}$$

With $M:= |\textrm{Cl}(X,x_{01})|$, the Picard index is $\iota _{\textrm{Pic}}(X) = \lambda w_{01} w_{02} M.$

Proposition 8.7

Let $X = X(P)$, where $P = \begin{bmatrix} v_0&\dots&v_r&v^- \end{bmatrix}$ is a defining matrix of type (ep) with $n_0 = \dots = n_r = 1$. Then $X = X(P)$ is of Picard number one. We have one elliptic fixed point $x^+$ and parabolic fixed points $x_{in_i}$ for $i = 0, \dots , r$. With $\lambda := |\textrm{Cl}(X)^{\textrm{tors}}|$, the orders of the local class groups are

$$\begin{aligned} |\textrm{Cl}(X,x^+)| = \lambda w^- = \det (\sigma ^+), \qquad |\textrm{Cl}(X,x_{in_i})| = \det \begin{bmatrix} l_i &{} 0 \\ d_i &{} 1 \end{bmatrix} = l_i \end{aligned}$$

Moreover, the Picard index is $\iota _{\textrm{Pic}}(X) = w^- l_0 \cdots l_r.$

Proof of Propositions 8.6 and 8.7

In both cases, we have $n+m = r+2$, hence $X(P)$ is of Picard number one. For the descriptions of the fixed points, see Remark 3.6. For the orders of the local class groups, we apply [2, Lemma 2.1.4.1] to the toric ambient variety of $X$ and use Proposition 3.8 (iii). Applying Theorem 1.1 gives the expressions for the Picard index. $\square$

Proposition 8.8

[See [8, Propositions 5.9 and 5.10]] Let $X$ be a non-toric rational projective $\mathbb {K}^*$-surface of Picard number one. Assume that $X$ is log terminal. Then we have $X \cong X(P)$ for some defining matrix $P$ as in one of the Propositions 8.6 and 8.7. The possible tuples $(l_{01}, l_{02}, l_1, \dots , l_r)$ for type (ee) and $(l_0, \dots , l_r)$ for type (ep) are the following:

$$\begin{aligned} \begin{array}{cllcll} \mathrm {(eAeA):} &{} (1,1,x_1,x_2), &{} &{} \mathrm {(eAeD):} &{} (1,y,2,2), &{} (1,2,y,2),\\ \mathrm {(eAeE):} &{} (1,z,3,2), &{} (1,3,z,2), &{} \mathrm {(eDeD):} &{} (2,2,y,2), &{} (y_1,y_2,2,2)\\ &{} (1,2,z,3), &{} &{} &{} (1,1,y,2,2), &{} \\ \mathrm {(eDeE):} &{} (2,3,z,2), &{} (2,z,3,2) &{} \mathrm {(eEeE):} &{} (2,2,z,3) &{} (z_1,z_2,3,2), \\ &{} &{} &{} &{} (3,3,z,2) &{} (1,1,z,3,2), \\ \mathrm {(eDp):} &{} (y,2,2), &{} &{} \mathrm {(eEp):} &{} (z,3,2), &{} \\ \end{array} \end{aligned}$$

where $2 \le x_1, x_2, y, y_1, y_2$ and $3 \le z, z_1 z_2 \le 5$ and the notation “eA, eD, eE” refers to log terminal $x^{\pm } \in X$ of type $A_n, D_n$ or $E_6, E_7, E_8$. Moreover, any non-toric, rational, log terminal, projective $\mathbb {K}^*$-surface of Picard number one is del Pezzo.

Using the formula for the Picard index in Proposition 8.6, we can go through all possible quadruples of positive integers $(\lambda , w_{01}, w_{02}, M)$ whose product equals a given Picard index. Going through all the possible tuples $(l_{01}, l_{02}, l_1, \dots , l_r)$ in Proposition 8.8, we can use the equations for the orders of the local class groups as well as the fact that $(w_{01},w_{02},w_{11},\dots ,w_{r1}) \in \ker (P)$ to derive bounds for all remaining entries of $P$. This yields an efficient classification of all non-toric rational projective log terminal $\mathbb {K}^*$-surfaces of Picard number one and type (ee) by their Picard index. The type (ep) is handled analogously. We arrive at the following classification.

Theorem 8.9

There are $1\,347\,433$ families of non-toric, log del Pezzo $\mathbb {K}^*$-surfaces of Picard number one and Picard index at most $10\,000$. The numbers of families for given Picard index develop as follows:

Remark 8.10

Comparing the numbers of surfaces per given Picard index in the different subcases of our classification, we observe that the non-toric $\mathbb {K}^*$-surfaces quickly outnumber the toric ones. The fastest growing subcase seems to be “eDeD”, which is responsible for the cone shape in the upper part of the plot in Theorem 8.9. The following table gives an impression of the proportion of the different cases:

$\iota _{\textrm{Pic}}$	toric	eAeA	eAeD	eAeE	eDeD	eDeE	eEeE	eDp	eEp
$\le 10$	14	5	4	10	1	0	0	1	0
$\le 100$	243	260	129	39	117	4	15	28	5
$\le 1\,000$	4 205	7 425	2 209	206	11 622	32	103	521	51
$\le 10\,000$	68 053	157 482	31 561	1 011	1 148 587	197	569	7 520	506

Up to Picard index 2 500, the complete list of defining matrices is available at [9].