Revistes Catalanes amb Accés Obert (RACO)

On the Jacobian ideal of the binary discriminant

Carlos D' Andrea, Jaydeep Chipalkatti


Let $\Delta$ denote the discriminant of the generic binary $d$-ic. We show that for $d\geq 3$, the Jacobian ideal of $\Delta$ is perfect of height 2. Moreover we describe its $SL_2$-equivariant minimal resolution and the associated differential equations satisfied by $\Delta$. A similar result is proved for the resultant of two forms of orders $d, e$ whenever $d \geq e – 1$. If $\Phi_n$ denotes the locus of binary forms with total root multiplicity $\geq d-n$, then we show that the ideal of $\Phi_n$ 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.

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