Jeroen Sijsling (Ulm)

02.12.2016 (Friday!) - W01 0-006 (Wechloy), 2 pm

Reconstructing plane quartics from their invariants

Abstract: Up to isomorphism, elliptic curves over the complex numbers are classified by their j-invariant; their coarse moduli space is an affine line with the j-invariant as coordinate. Conversely, it is not difficult to construct an elliptic curve with a specified j-invariant. In higher genus the situation is quite a bit more complicated. The moduli space of smooth genus 2 curves, as determined by Igusa, is already no longer a quasi-affine space, although it is still birational. In this genus Clebsch and Mestre have developed methods to reconstruct curves from their invariants, which also apply to hyperelliptic curves of higher genus. These methods are however very specific to the hyperelliptic case and do not at all generalise.

This talk describes joint work with Reynald Lercier and Christophe Ritzenthaler that describes how reconstruction is possible in the next simplest case: that of non-hyperelliptic curves in genus 3, or in other words smooth plane quartics in the projective plane.