On the Jacobian Ideal of the Binary Discriminant

Let ∆ denote the discriminant of the generic binary d-ic. We show that for d ≥ 3, the Jacobian ideal of ∆ is perfect of height 2. Moreover we describe its SL2-equivariant minimal resolution and the associated differential equations satisfied by ∆. A similar result is proved for the resultant of two forms of orders d, e whenever d ≥ e − 1. If Φn denotes the locus of binary forms with total root multiplicity ≥ d − n, then we show that the ideal of Φn is also perfect, and we construct a covariant which characterizes this locus. We also explain the role of the Morley form in the determinantal formula for the resultant. This relies upon a calculation which is done in the appendix by A. Abdesselam. Mathematics Subject Classification(2000): 13A50, 13C40.