# Series of rational moduli components of stable rank 2 vector bundles on $\mathbb{P}^3$

## TL;DR

This paper investigates the rationality of certain components of the moduli space of rank 2 stable vector bundles on projective 3-space, providing explicit constructions and demonstrating the existence of infinitely many rational components.

## Contribution

It establishes conditions under which these moduli components are rational or stably rational, and constructs universal families showing these are fine moduli spaces, including new examples.

## Key findings

- Ein components are rational when c>2a+b-e, b>a, and (e,a)≠(0,0)
- Remaining cases yield at least stably rational components
- Infinite series of rational components exist for e=0 and e=-1

## 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 3-space. The generalized null correlation bundles constituting open dense subsets of these components are defined as cohomology bundles of monads whose members are direct sums of line bundles of degrees depending on nonnegative integers $a,b,c$, where $b\ge a$ and $c>a+b$. We show that, in the wide range when $c>2a+b-e,\ b>a,\ (e,a)\ne(0,0)$, 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 $n\ge1$ contains an infinite series of rational components for both $e=0$ and $e=-1$. Explicit constructions of rationality of Ein components under the above conditions on $e,a,b,c$ and, respectively, of their stable rationality in the remaining cases, are given. 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.