The Coble hypersurfaces
Arnaud Beauville
Abstract
Let A be an indecomposable principally polarized abelian variety of dimension g . Third order theta functions embed A in a projective space P(V3), while second order theta functions embed the Kummer variety K=A/<-1> in a projective space P(V2). Coble observed that for g = 2 there is a unique cubic hypersurface in P(V3) that is singular along A, and for g = 3 a unique quartic hypersurface in P(V2) singular along K . We explain these facts by a simple analysis of the representations of the corresponding Heisenberg group.
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