Prioritary omalous bundles on Hirzebruch surfaces

An irreducible algebraic stack is called 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 HH-stable bundles for any ample line bundle HH on a Hirzebruch surface (compare with Costa and Miro-Ŕoig, 2002). 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 P1×P1P1×P1 is dominated by an irreducible section of a Segre variety, and this linear section is rational (Ionescu, 2015). 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 P4P4, Calabi–Yau complete intersection, blowups of the projective plane and Segre varieties have been recently obtained by A.A. Henni and M. Jardim (Henni and Jardim, 2013), and monads on Hirzebruch surfaces have been applied in a different context in Bartocci et al. (2015).