A note on the existence of stable vector bundles on Enriques surfaces

Abstract

We prove the non-emptiness and irreducibility of $M_H(v,L)$, the moduli space of Gieseker semistable sheaves on an unnodal Enriques surface Y with primitive Mukai vector v of positive rank and determinant L with respect to a generic polarization H. This completes the chain of progress initiated by Kim (Nagoya Math J 150:85–94, 1998). We also show that the stable locus $M^s_H(v)\ne \varnothing$ for non-primitive v with $v^2>0$.

Appendix: Dimension estimates for Brill–Noether loci on Hilbert schemes of points

We prove in this appendix the dimension estimate we used in the body of the paper. The result concerns bounding the dimension of the locus of 0-cycles with given cohomology with respect to a linear system.

Lemma 8.1

Let L be an effective divisor and $S^i:=\{Z|h^0(I_Z(L))=i\}\subset Y^{(l)}$. Then for $i>0$,

$$\begin{aligned} \mathop {\mathrm {dim}}\nolimits S^i\le \mathop {\mathrm {dim}}\nolimits |L|+l(Z)-(i-1). \end{aligned}$$

In particular, if L is ample, then $\mathop {\mathrm {dim}}\nolimits |L|=\frac{1}{2}L^2$, so

$$\begin{aligned} \mathop {\mathrm {dim}}\nolimits S^i\le \frac{1}{2}L^2+1+l(Z)-i. \end{aligned}$$

Proof

Denote by $S:=\{(Z,{\mathbb {C}}v)|{\mathbb {C}}v\in |I_Z(L)|\}\subset Y^{(l)}\times |L|$. Then the second projection

$$\begin{aligned} p_2: S\rightarrow |L|, \end{aligned}$$

is surjective with fibers of dimension l(Z). Indeed, for any $C\in |L|$ the fiber over C is $C^{(l)}$. Thus $\mathop {\mathrm {dim}}\nolimits S=\mathop {\mathrm {dim}}\nolimits |L|+l(Z)$. The image of S under the first projection is

$$\begin{aligned} p_1(S)=\bigcup _{i>0} S^i, \end{aligned}$$

where the fiber over $Z\in Y^{(l)}$ is $|I_Z(L)|$. Thus $p_1^{-1}(S^i)$ is a ${\mathbb {P}}^{i-1}$ bundle over $S^i$, from which it follows that

$$\begin{aligned} \mathop {\mathrm {dim}}\nolimits S^i\le \mathop {\mathrm {dim}}\nolimits |L|+l(Z)-(i-1). \end{aligned}$$

$\square$