Prioritary omalous bundles on Hirzebruch surfaces

TL;DR

This paper proves the unirationality of the stack of prioritary omalous bundles on Hirzebruch surfaces, providing explicit monad descriptions and showing the moduli space's rationality for these bundles.

Contribution

It establishes the unirationality of the stack of prioritary omalous bundles on Hirzebruch surfaces and describes these bundles explicitly via monads.

Findings

The stack of prioritary omalous bundles is unirational.

Explicit monad descriptions of omalous bundles are provided.

The moduli space of stable bundles is shown to be rational.

Abstract

An irreducible algebraic stack is called \emph{unirational} if there exists a surjective morphism, representable by algebraic spaces, from a rational variety to an open substack. We prove unirationality of the stack of prioritary omalous bundles on Hirzebruch surfaces, which implies also the unirationality of the moduli space of omalous $H$-stable bundles for any ample line bundle $H$ on a Hirzebruch surface. To this end, we find an explicit description of the duals of omalous rank-two bundles with a vanishing condition in terms of monads. Since these bundles are prioritary, we conclude that the stack of prioritary omalous bundles on a Hirzebruch surface different from $\mathbb P^1\times \mathbb P^1$ is dominated by an irreducible section of a Segre variety, and this linear section is rational \cite{I}. In the case of the space quadric, the stack has been explicitly described by N. Buchdahl. As a main tool we use Buchdahl's Beilinson-type spectral sequence. Monad descriptions of omalous bundles on hypersurfaces in $\mathbb P^4$, Calabi-Yau complete intersection, blowups of the projective plane and Segre varieties have been recently obtained by A. A. Henni and M. Jardim~\cite{HJ}, and monads on Hizebruch surfaces have been applied in a different context in~\cite{BBR}.