Series of rational moduli components of stable rank two vector bundles on $\pmb {\mathbb {P}}^3$

Abstract

We study the problem of rationality of an infinite series of components, the so-called Ein components, of the Gieseker–Maruyama moduli space M(e, n) of rank 2 stable vector bundles with the first Chern class $e=0$ or $-1$ and all possible values of the second Chern class n on the projective space ${{\mathbb {P}}^{3}}$. We show that, in a wide range of cases, the Ein components are rational, and in the remaining cases they are at least stably rational. As a consequence, the union of the spaces M(e, n) over all $n\ge 1$ contains new series of rational components in the case $e=0$, exteding and improving previously known results of Vedernikov (Math USSSR-Izv 25:301–313, 1985) on series of rational families of bundles, and a first known infinite series of rational components in the case $e=-\,1$. Explicit constructions of rationality (stable rationality) of Ein components are given. Our approach is based on the study of a correspondence between generalized null correlation bundles constituting open subsets of Ein components and certain rank 2 reflexive sheaves on ${{\mathbb {P}}^{3}}$. This correspondence is obtained via elementary transformations along surfaces. We apply the technique of Quot-schemes and universal spaces of extensions of sheaves to relate the parameter spaces of these two types of sheaves. In the case of rationality, we construct universal families of generalized null correlation bundles over certain open subsets of Ein components showing that these subsets are fine moduli spaces. As a by-product of our construction, for $c_1=0$ and n even, they provide, perhaps the first known, examples of fine moduli spaces not satisfying the condition “n is odd”, which is a usual sufficient condition for fineness.