On Hermite’s Invariant for Binary Quintics

Let H ⊆ P5 denote the hypersurface of binary quintics in involution, with defining equation given by the Hermite invariant H. In §2 we find the singular locus of H, and show that it is a complete intersection of a linear covariant of quintics. In §3 we show that the projective dual of H can be canonically identified with itself via an involution. The Jacobian ideal of H is shown to be perfect of height two in §4, moreover we describe its SL2-equivariant minimal free resolution. The last section develops a general formalism for evectants of covariants of binary forms, which is then used to calculate the evectant of H. Mathematics Subject Classification(2000): 13A50, 13C40.