Moduli of generalized syzygy bundles

Abstract

Given a vector bundle $F$ on a variety $X$ and $W\subset H^0(F)$ such that the evaluation map $W\otimes {\mathcal{O}}_X\to F$ is surjective, its kernel $S_{F,W}$ is called generalized syzygy bundle. Under mild assumptions, we construct a moduli space ${\mathcal{G}}_U^0$ of simple generalized syzygy bundles, and show that the natural morphism $\alpha$ to the moduli of simple sheaves is a locally closed embedding. If moreover $H^1(X,{\mathcal{O}}_X)=0$, we find an explicit open subspace ${\mathcal{G}}_V^0$ of ${\mathcal{G}}_U^0$ where the restriction of $\alpha$ is an open embedding. In particular, if $\dim X\ge 3$ and $H^1({\mathcal{O}}_X)=0$, starting from an ample line bundle (or a simple rigid vector bundle) on $X$ we construct recursively open subspaces of moduli spaces of simple sheaves on $X$ that are smooth, rational, quasiprojective varieties.