Prioritary omalous bundles on Hirzebruch surfaces

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}.

Final version, to appear in J. Geom. Phys

References in corpus (2)

• Monad constructions of omalous bundles
• Monads for framed sheaves on Hirzebruch surfaces

Cited by in corpus (1)

• Quantum Sheaf Cohomology on surfaces of general type I: construction of stable omalous bundles