Curves of genus 2 and Desargues configurations

Abstract

A Desargues configuration is the configuration of 10 points and 10 lines of the classical theorem of Desargues in the complex projective plane. For a precise definition see Section 2. The Greek mathematician C. Stephanos showed in 1883 (see [10]) that one can associate to every Desargues configuration a curve of genus 2 in a canonical way. Moreover he proved that the induced map from the moduli space of Desargues configurations to the moduli space of curves of genus 2 is birational. Our main motivation for writing this paper was to understand the last result. Stephanos needed about a hundred pages of classical invariant theory to prove it. We apply instead a simple argument of Schubert calculus to prove a slightly more precise version of his result. Let MD denote the (coarse) moduli space of Desargues configurations. It is a threedimensional quasiprojective variety. On the other hand, let M 6 denote the moduli space of stable binary sextics. There is a canonical isomorphism H2nD1 GM 6 (see [1]). Here H2 denotes the moduli space of stable curves of genus 2 in the sense of Deligne–Mumford and D1 the boundary divisor parametrizing two curves of genus 1 intersecting transversely in one point. So instead of curves of genus 2, we may speak of binary sextics. The main result of the paper consists of the following two statements: