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 \(\mathcal{G}^0_U\) 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 \(\mathcal{G}^0_V\) of \(\mathcal{G}^0_U\) 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.