  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.
