Prioritary omalous bundles on Hirzebruch surfaces

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}.