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

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## 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].

*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

*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*.

*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.

*C*. By definition for each

*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

*C*and the Equality (2) return

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