On the Jacobian ideal of the binary discriminant

Abstract

Let $\Delta$ denote the discriminant of a generic binary $d$-ic. We show that for $d \ge 3$, the Jacobian ideal of $\Delta$ is perfect of height 2. Moreover, we describe its SL_2-equivariant minimal resolution, and the associated invariant differential equations satisfied by $\Delta$. A similar result is proved for the resultant of two forms of orders $d,e$, whenever $d \ge e-1$. 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.