Title: Orthogonal bundles and skew-Hamiltonian matrices

Abstract: Let M(r,n) be the moduli space of stable vector bundles of rank r
\ge 2 on P^2, with Chern classes (c_1, c_2) = (0, n). An element E of M(r,n) is orthogonal
(symplectic) if it is isomorphic to its dual via a symmetric
(skew-symmetric) isomorphism.
In 1980 Hulek proved that M(r,n) is irreducible. Following the lines of
Hulek's result, Ottaviani proved in 2007 that the moduli space of
symplectic bundles is irreducible as well. In this talk I will focus on the orthogonal case.
Though quite a lot is known on the classi cation of such bundles on curves, this is not
the case for surfaces. Indeed, already on the projective plane they require a much
more involved argument than their symplectic and general counterparts. In particular when
n is even, some interesting and unexpected properties of skew-Hamiltonian
matrices come into play, as well as the study of highly non-generic hyperplane
sections of some determinantal variety. This is a joint project, and work in progress, with
R.Abuaf.