Revista Matemática Complutense

, Volume 32, Issue 2, pp 575–577

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

Correction

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

The proof of Theorem 1.2 in  contains a gap which can be filled by simply assuming that the base field k is uncountable. In particular all the results in  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 .

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  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  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  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 ). 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 ). We show below that the corresponding bundle $$\mathcal {E}$$ is stable.

If not, then it is also not $$\mu$$-stable. Theorem 2.9 of  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. $$\square$$

## References

1. 1.
Casanellas, M., Hartshorne, R., Geiss, F., Schreyer, F.O.: Stable Ulrich bundles. Int. J. Math. 23, 1250083 (2012)
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) 