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Revista Matemática Complutense

, Volume 32, Issue 2, pp 575–577 | Cite as

Correction to: Ulrich bundles on non-special surfaces with \(p_g=0\) and \(q=1\)

  • Gianfranco CasnatiEmail author
Correction

1 Correction to: Rev Mat Complut  https://doi.org/10.1007/s13163-017-0248-z

The proof of Theorem 1.2 in [2] contains a gap which can be filled by simply assuming that the base field k is uncountable. In particular all the results in [2] remain true if we assume such a restriction in Sections 4 and 5. We shortly explain below the due changes in the paper and in the proof. For all the definitions and the notation, we refer to [2].

Recall that S is a smooth, irreducible surface with \(p_g(S)=0\) and \(q(S)=1\) endowed with a very ample line bundle \(\mathcal {O}_S(h)\) which is non-special (i.e. \(h^1\big (S,\mathcal {O}_S(h)\big )=0\)). If \(K_S\) is the canonical class of S, the Riemann–Roch theorem implies that \(\mathcal {O}_S(h)\) gives an embedding \(S\subseteq {\mathbb {P}^{N}}\) where
$$\begin{aligned} N:=h^0\big (S,\mathcal {O}_S(h)\big )-1=\frac{h^2-hK_S}{2}-1, \end{aligned}$$
because \(h^2\big (S,\mathcal {O}_S(h)\big )=h^0\big (S,\mathcal {O}_S(K_S-h)\big )\le p_g(S)=0\).
Each general hyperplane section \(C=S\cap H\) is a smooth irreducible curve with genus
$$\begin{aligned} \pi (\mathcal {O}_S(h)):=\frac{h^2+hK_S}{2}+1. \end{aligned}$$
The Hilbert scheme \(\mathcal H_C\) of 0-dimensional subschemes of degree \(N+1\) on C has dimension \(N+1\) and contains an open non-empty subset \(\mathcal {R}\subseteq \mathcal {H}_C\) corresponding to reduced subschemes of C consisting of \(N+1\) points in general linear position in H.
If \(Z\subseteq C\subseteq S\) corresponds to a general point in \(\mathcal {R}\) and \(\mathcal {O}_S(\eta )\in {\text {Pic}}^0(S){\setminus }\{\ \mathcal {O}_S\ \}\) is such that
$$\begin{aligned} h^0\big (S,\mathcal {O}_S(K_S\pm \eta )\big )=h^1\big (S,\mathcal {O}_S(h\pm \eta )\big )=0. \end{aligned}$$
Theorem 1.1 of [2] guarantees the existence of an Ulrich bundle \(\mathcal {E}\) of rank 2 fitting into an exact sequence of the form
$$\begin{aligned} 0\longrightarrow \mathcal {O}_S(h+K_S+\eta )\longrightarrow \mathcal {E}\longrightarrow \mathcal {I}_{Z\vert S}(2h+\eta )\longrightarrow 0. \end{aligned}$$
(1)
Assume that k is uncountable: Theorem 1.2 in [2] then holds in the following form.

Theorem 1.2

Let S be a surface with \(p_g(S)=0\), \(q(S)=1\) and endowed with a very ample non-special line bundle \(\mathcal {O}_S(h)\).

If \(\pi (\mathcal {O}_S(h))\ge 2\), then the bundle \(\mathcal {E}\) constructed in Theorem 1.1 of [2] from a very general set \(Z\subseteq C\subseteq S\) of \(h^0\big (S,\mathcal {O}_S(h)\big )\) points is stable.

Proof

We know how to construct a rank 2 Ulrich bundle fitting into Sequence (1) starting from C, Z and \(\mathcal {O}_S(\eta )\) as above.

Let \(\mathcal {D}\subseteq {\text {Pic}}(S)\) be the set of Ulrich line bundles. If \(\mathcal {O}_S(D)\in \mathcal {D}\), then
$$\begin{aligned} Dh=\frac{1}{2}(3h^2+hK_S) \end{aligned}$$
(2)
(see Proposition 2.1 of [2]). Equality (2) yields \((h+\eta -D)h\le -1\) because \(\pi (\mathcal {O}_S(h))\ge 2\). Thus
$$\begin{aligned} h^0\big (S,\mathcal {I}_{C\vert S}(2h+\eta -D)\big )=h^0\big (S,\mathcal {O}_S(h+\eta -D)\big )=0, \end{aligned}$$
i.e. there are no divisors \(A\in \vert 2h+\eta -D\vert \) containing C. By definition for each
$$\begin{aligned} Z\in \mathcal {Z}_D:=\{Z\in \mathcal {R}\ \vert \ h^0\big (S,\mathcal {I}_{Z\vert S}(2h+\eta -D)\big )\ge 1\} \end{aligned}$$
there is \(A\in \vert 2h+\eta -D\vert \) such that \(Z\subseteq A\). Thus \(Z=C\cap A\), because \((2h+\eta -D)h=N+1\) and \(C\not \subseteq A\). If \(i:C\rightarrow S\) denotes the inclusion map, then Z identifies a unique element in \(\vert i^*\mathcal {O}_S(2h+\eta -D)\vert \). Such a construction gives an injective morphism \(\mathcal {Z}_D\rightarrow \vert i^*\mathcal {O}_S(2h+\eta -D)\vert \), hence
$$\begin{aligned} \dim (\mathcal {Z}_D)\le h^0\big (C,i^*\mathcal {O}_S(2h+\eta -D)\big )-1. \end{aligned}$$
If \(i^*\mathcal {O}_S(2h+\eta -D)\) is special, then the Clifford theorem and the Equality (2) imply
$$\begin{aligned} h^0\big (C,i^*\mathcal {O}_S(2h+\eta -D)\big )\le \frac{(2h+\eta -D)h}{2}+1=\frac{N+3}{2}\le N, \end{aligned}$$
because \(N\ge 3\). If \(i^*\mathcal {O}_S(2h+\eta -D)\) is non-special, the Riemann–Roch theorem on C and the Equality (2) return
$$\begin{aligned} h^0\big (C,i^*\mathcal {O}_S(2h+\eta -D)\big )=N+2-\pi (\mathcal {O}_S(h))\le N, \end{aligned}$$
because \(\pi (\mathcal {O}_S(h))\ge 2\). Thus \(\dim (\mathcal {Z}_D)\le N-1\) for each Ulrich line bundle \(\mathcal {O}_S(D)\).

Let \(\nu :{\text {Pic}}(S)\rightarrow {\text {NS}}(S)\) be the natural map onto the Néron–Severi group. Thanks to the discussion above the dimension of \( \widehat{\mathcal {Z}}_\delta :=\bigcup _{D\in \nu ^{-1}(\delta )}\mathcal {Z}_D\) is at most N for each \(\delta \in \nu (\mathcal {D})\), because the fibres of \(\nu \) have dimension 1. Since \({\text {NS}}(S)\) is a finitely generated abelian group, \(\dim (\mathcal {R})=N+1\) and k is uncountable, it follows the existence of \(Z\in \mathcal {R}{\setminus } \bigcup _{\delta \in \nu (\mathcal {D})}\widehat{\mathcal {Z}}_\delta \) (see Exercise V.4.15 (c) of [3]). We show below that the corresponding bundle \(\mathcal {E}\) is stable.

If not, then it is also not \(\mu \)-stable. Theorem 2.9 of [1] guarantees that \(\mathcal {E}\) is \(\mu \)-semistable; hence there exists a line bundle \(\mathcal {O}_S(D)\subseteq \mathcal {E}\) such that \(\mu (\mathcal {E})=\mu (\mathcal {O}_S(D))\). Again the aforementioned theorem also implies that \(\mathcal {O}_S(D)\in \mathcal {D}\).

On the one hand, the choice of Z implies that \(h^0\big (S,\mathcal {I}_{Z\vert S}(2h+\eta -D)\big )=0\); hence \(\mathcal {O}_S(D)\) must be contained in the kernel \(\mathcal {O}_S(h+K_S+\eta )\) of the map \(\mathcal {E}\rightarrow \mathcal {I}_{Z\vert S}(2h+\eta )\) in Sequence (1). On the other hand, Equality (2) implies \((h+K_S+\eta -D)h\le -2\) because \(N\ge 3\), hence \(h^0\big (S,\mathcal {O}_S(h+K_S+\eta -D)\big )=0\). We deduce that such an \(\mathcal {O}_S(D)\) cannot exist; hence the bundle \(\mathcal {E}\) is necessarily stable. \(\square \)

Notes

References

  1. 1.
    Casanellas, M., Hartshorne, R., Geiss, F., Schreyer, F.O.: Stable Ulrich bundles. Int. J. Math. 23, 1250083 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Casnati, G.: Ulrich bundles on non-special surfaceswith \(p_g=0\) and \(q=1\). Rev. Mat. Complut. (2019)Google Scholar
  3. 3.
    Hartshorne, R.: Algebraic Geometry. G.T.M. 52. Springer, Berlin (1977)CrossRefGoogle Scholar

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© Universidad Complutense de Madrid 2019

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematichePolitecnico di TorinoTurinItaly

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