A self-map on the space of plane quartics

Let C be a plane quartic. The equianharmonic contravariant of C is another plane quartic H(C) in the dual projective plane. The assignment

C --> H(C)

determines a rational self-map of the space of plane quartics to itself. Dolgachev asked whether H is generically finite and, if so, to compute its degree.

In joint work with Marco Pacini, we show that the map H is indeed generically finite and that its degree is 15.

The argument involves a little bit of classical invariant theory, a tiny amount of Lie algebras and a direct computation. I plan to explain gently all the constructions that I will use!