Duality of (1, 5)–polarized abelian surfaces

Abstract

There is not many open questions about the moduli space of (1, 5)-polarized abelian surfaces and level structure, it has been carefully studied by Barth, Hulek and Moore in the 80's. The goal of this paper is to give a description of the moduli space of (1, 5)-polarized abelian surfaces (without level structure) up to duality. We prove that an open set of this moduli space is isomorphic to an open set of the moduli space of sextuplets of points on a complex projective line (up to homographies) and we study what happens on the boundary. IntroductionThe origin of this redaction is the article "On rational plane sextics with six tritangents" by W. Barth and R. Moore (cf. [6]). All the sextics they considered are obtained by projection of a unique twisted sextic K , which is invariant under an action of the icosahedral group A 5 PSL(2, F 5 ) on P 3 and which is endowed with the following property:K admits infinitely many tritangent planes .( * )The reason for studying such a family is related to the Horrocks-Mumford vector bundle F over P 4 : let us project in the projective space PH 0 (F (3)) from a section s of PH 0 (F (3)) the sextic curve K (it parameterizes the sections for which the vanishing locus is a (doubled) elliptic quintic scroll), it is a rational sextic in P 2 with six tritangents. The double covering of P 2 which is ramified along this sextic is birational to the Kummer surface of the dual abelian surface of s −1 (0) ("reconstruction method"). In the first section ("Bring's curve"), we consider the sextic K itself (often called Klein's sextic) making essentially use of SL(2, F 5 )-modules (especially McKay's correspondence). The graph of the (3, 3) correspondence between the curve K and its tritangent planes is a genus 4 curve whose automorphisms group contains A 5 . This last property characterizes the curve (called Bring's curve). We notice that one can illustrate this curve as the graph of the incidence correspondence between a twisted cubic curve and a dual twisted cubic curve with apolar nets of quadrics.In the section "Canonical model for a genus 4 curve", we see the generic genus 4 curve as the graph of the incidence correspondence between a twisted cubic curve and a dual twisted cubic curve, which allows us to construct its canonical model in P 3 as the complete intersection of two surfaces (of degree 2 and 3) entirely determined by the datum of the curve i.e. we do not make any choice (this is not trivial for the cubic surface). The cubic surface so constructed comes with a marked double-six. We end this part describing in which case the cubic surface admits four double points: the genus 4 curve is the incidence correspondence between a rational twisted quintic and its trisecant lines (Sub-section 2.3).The preceding results lead to Theorem 3.1 (page 65):Every rational plane sextic with 6 tritangents is a projection of Klein's sextic K