On Hermite's invariant for binary quintics

Abstract

The Hermite invariant H is the defining equation for the hypersurface of binary quintics in involution. This paper analyses the geometry and invariant theory of H. We determine the singular locus of this hypersurface and show that it is a complete intersection of a linear covariant of quintics. The projective dual of this hypersurface can be identified with itself via an involution. It is shown that the Jacobian ideal of H is perfect of height two, and we describe its SL_2-equivariant minimal 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.