Mathematics

The discriminant of a space of functions is the closed subset consistingof the functions which are singular in some sense. For instance, forcomplex polynomials in one variable the discriminant is the locus ofpolynomials with multiple roots. In this special case, it is known bywork of Arnol'd that the cohomology of the complement of thediscriminant stabilizes when the degree of the polynomials grows, in thesense that the k-th cohomology group of the space of polynomials withoutmultiple roots is independent of the degree of the polynomials considered. A more general set-up is to consider the space of non-singular sectionsof a very ample line bundle on a fixed non-singular variety. In thiscase, Vakil and Wood proved a stabilization behaviour for the class ofcomplements of discriminants in the Grothendieck group of varieties. Inthis talk, I will discuss a topological approach for obtaining thecohomological counterpart of Vakil and Wood's result and describe stablecohomology explicitly for the space of complex homogeneous polynomialsin a fixed number of variables and for spaces of smooth divisors on analgebraic curve.