Coincident root loci of binary forms

Abstract

Coincident root loci are subvarieties of SC — the space of binary forms of degree d — labeled by partitions of d. Given a partition λ, let Xλ be the set of forms with root multiplicity corresponding to λ. There is a natural action of GL2(C) on SC and the coincident root loci are invariant under this action. We calculate their equivariant Poincare duals, generalizing formulas of Hilbert and Kirwan. In the second part we apply these results to present the cohomology ring of the corresponding moduli spaces (in the GIT sense) by geometrically defined relations. One of the main goals of Geometric Invariant Theory is to calculate the cohomology ring of a geometric quotient. In the case when all semistable point are stable several techniques were developed. But even for very simple representations this condition is not satisfied. In this paper we study the action of GL(2) on the space of binary forms of degree d. In the d odd case methods of [Kir84], [JK95], [Mar99] can be applied, but none of these methods compute the cohomology ring of the moduli space in the d even case. We show how equivariant Poincare-dual calculations lead to relations for the cohomology ring in both the odd and the even case. Closely related rings have been computed earlier. The computation for H∗ G(X ) is wellknown (since Kirwan’s thesis in the case of betti numbers), and the existing procedure is independent of d being even or odd. In the d even case, rational intersection cohomology of the moduli space is also known ([Kir86])—which result we also recover in Remark 4.11. Our Poincare-dual (a.k.a. Thom polynomial) calculations are also interesting on their own right since they generalize formulas of Hilbert and Kirwan on coincident root loci. These calculations do not only lead to explicit relations for these cohomology rings but also identify them with the equivariant Poincare-duals of the simplest unstable coincident root loci.