Pl\"ucker formulas using equivariant cohomology of coincident root strata

Abstract

We give a new method to calculate the universal cohomology classes of coincident root loci. We show a polynomial behavior of them and apply this result to prove that generalized Pl\"ucker formulas are polynomials in the degree, just as the classical Pl\"ucker formulas counting the bitangents and flexes of a degree $d$ generic plane curve. We establish an upper bound for the degrees of these polynomials, and we calculate the leading terms of those whose degrees reach this upper bound. We believe that the paper is understandable without detailed knowledge of equivariant cohomology. It may serve as a demonstration of the use of equivariant cohomology in enumerative geometry through the examples of coincident root strata. We also explain how the equivariant method can be "translated" into the traditional non-equivariant method of resolutions.