ar X iv : m at h / 04 05 05 2 v 1 [ m at h . A G ] 4 M ay 2 00 4 On the ring of invariants of ordinary quartic curves in characteristic 2

Traditionally, non-singular curves of a fixed genus g are classified in two categories, those which are hyperelliptic and those which are not. The first ones are usually considered to be the simplest, since their geometry relies on their Weierstra points and is condensed on a line. Thus, the hyperelliptic locus Mg of the corresponding moduli space Mg is easily described and well understood. Seen from an invariant theoretical viewpoint, this is reflected by the fact that one only needs to deal with binary forms. To the contrary, non-hyperelliptic curves are more involved. Even in the simplest case, non-singular quartic curves over C, no complete description of the relevant invariant ring is known. More precisely, for the invariant ring of the natural action of SL3(C) on the vector space of homogeneous polynomials of degree 4 in 3 variables a set of primary invariants, see [6], but no complete algebra generating set is known; a conjecture of Shioda says that the invariant ring is generated as an algebra by 13 elements.