The Coble hypersurfaces

الأصل · EN

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.