The Coxeter-Todd abelian sixfold, Gamma00, and big Schottky

Abstract

We recall the construction of the Coxeter-Todd principally polarized abelian sixfold, and study its special properties. We show that the base locus of the Gamma00 linear system on the Coxeter-Todd sixfold consists of 253 points, disproving the stronger version by Donagi and Beauville--Debarre of the Gamma00 conjecture of van Geemen and van der Geer. This also leads us to construct explicit indecomposable 7-dimensional principally polarized abelian varieties that are not Jacobians, but lie in the so-called Big Schottky locus, thus disproving the corresponding conjecture from the Schottky-Jung approach to the Schottky problem. We furthermore show that while no even theta constant vanishes for the Coxeter-Todd sixfold, its associated Scorza quartic is identically zero. Finally, we show that there exist two different theta level (2,4) structures on the Coxeter-Todd sixfold, for which the sets of second order theta constants are projectively equal. This disproves the expected injectivity of the second order theta constant map for all genera starting with 6.