Algebra/Topology Seminar

Orsola Tommasi: Cohomological stabilization of complements of discriminants

The discriminant of the space of complex polynomials of degree d in one variable is the locus of polynomials with multiple roots. Arnol'd proved that the cohomology of the complement of this discriminant stabilizes when the degree of the polynomials grows, in the sense that the k-th cohomology group of the space of polynomials without multiple roots is independent of the degree of the polynomials considered.

In this talk, I will present a similar stability result for the space of non-singular complex homogeneous polynomials in a fixed number of variables and its rational cohomology and discuss an extension to the more general situation of the space of sections of a very ample line bundle on a fixed non-singular variety. This is inspired by work of Vakil and Wood on stabilization behaviour in the Grothendieck group of varieties.