JOURNAL OF ALGEBRA 122, 244249 (1989) The Equations of Strata for Binary Forms JERZV WEYMAN Department of Mathematics, Northeastern University, Boston, Massachusetts 02115 Communicated by D. A. Buchsbaum Received September 1, 1987 Let g be a fixed integer. We consider the space A’, of binary forms of degree g. We write X, = Spec R, where R, = S.(S, V), V being a two- dimensional vector space. Let X,,. c X, be the subset of binary forms having a root of multiplicity 2 p. We consider the ideal J, of polynomial functions vanishing on X,,,. For p 3 [g/2] + 1, XP,g is a stratum in the sense of Hesselink [H]. For p = [g/2] + 1, XP, g is a null-cone, so J, becomes the radical of the ideal generated by SL( V)-invariants of positive degree in R,. We describe explicitly the generators of J, for p > [g/2] + 1. The surprising result is that those generators occur in degrees d 4. We also describe explicitly the Hilbert functions of RR/JP and the decomposition of Rg/JP into representations of SL( V). We denote by SC,,,, V the space Sa-b vo (A2V)b. Let R, = S.(S, V), V-vector space over C, dim V = 2, and let 01, fl be the basis of V. Let Jp be the ideal of elements of R, vanishing on X,, ,(p > [g/2] + 1). By the result of Hesselink [H] the space PP. g = Proj(R,/J,) has the following desingularization Y,, g Yp.g= WWWV’)x~gIfh as a root of multiplicity p at R} I We can think of P(V) as the grassmannian with the tautological sequence O+R-V+Q+O (dimR=dimQ=l). Then Yp,,c[Fp(V)xXg and we can treat OyP,g = S.( Tp) where Tp = Qp @ S, _ p V is a factor of S, V. Then we know again from [H] that the normalization Rg/Jp = nr,S.( T,). One should observe that geometrically the normalization xP,g = 244 0021-8693/89 $3.00 Copyright 0 1989 by Academic Press. Inc. All rights of reproduction m any form reserved brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector