Mathematics

Given a morphism F between vector bundles on P^N=P(V), its (first) degeneracy locus X is made up of points in which the morphism has not maximal rank; locally, X is cut by the maximal minors of the matrix representing F. Interesting questions arise when one wants to give a parametrization of such degeneracy loci, by looking at the union of the irreducible components of the Hilbert Scheme H_X containing the general X's. I will focus on the case in which the morphism is given by m global sections of Omega(2), the twisted cotangent sheaf on P(V). In this case, making use of the Kempf-Weyman's method for computing syzygies via resolutions of singularities, H_X can be proved to be birational to the Grassmannian Gr(m, Lambda^2 (V)), when 3 < m < N. If m=3, the Grassmannian is birational to a subscheme of H_X, a nice geometric description of which can be given.