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

TL;DR

This paper proves the existence and irreducibility of certain moduli spaces of sheaves on Enriques surfaces, advancing understanding of their geometric properties and filling gaps in prior research.

Contribution

It establishes non-emptiness of moduli spaces for positive rank sheaves, confirms the stable locus is non-empty for positive Mukai square, and proves irreducibility in specific cases.

Findings

Non-emptiness of $M_{H,Y}(v)$ for positive rank sheaves.

Stable locus $M^s_{H,Y}(v)$ is non-empty when $v^2>0$.

Irreducibility of $M_{H,Y}(v)$ when $v^2=0$ and $v$ is primitive.

Abstract

We prove the non-emptiness of $M_{H,Y}(v)$, the moduli space of Gieseker-semistable sheaves on an unnodal Enriques surface $Y$ with Mukai vector $v$ of positive rank with respect to a generic polarization $H$. This completes the chain of progress initiated by H. Kim in \cite{Kim98}. We also show that the stable locus $M^s_{H,Y}(v)\neq\varnothing$ for $v^2>0$. Finally, we prove irreducibility of $M_{H,Y}(v)$ in case $v^2=0$ and $v$ primitive.