The Coble hypersurfaces

Abstract

Let A be an indecomposable principally polarized abelian variety of dimension g. Third order theta functions embed A in a projective space P(V 3) of dimension 3 g −1, while second order theta functions embed the Kummer variety X= A/{±1} in a projective space P(V 2) of dimension 2 g −1. Coble observed that for g=2 there is a unique cubic hypersurface in P(V 3) that is singular along A, and for g=3 a unique quartic hypersurface in P(V 2) singular along X. We explain these facts by a simple analysis of the representations of the corresponding Heisenberg group. To cite this article: A. Beauville, C. R. Acad. Sci. Paris, Ser. I 337 (2003).