Multiplicity \(g\) points on theta divisors
Let \((A,\Theta)\) be a principally polarized abelian variety of dimension \(g\). \(\text{Sing}_m (\Theta)\) denotes the subvariety of the theta divisor \(\Theta\) of points of multiplicity \(\geq m\). -- \textit{J. Kollár} [``Shafarevich maps and automorphic forms'' (Princeton 1995; Zbl 0871.14015)] showed that the dimension of \(\text{Sing}_m (\Theta)\) is less or equal to \(g-m\) for \(m\geq 2\). This inequality is sharp as the case of products of elliptic curves shows. In remark 17.13.1 of the cited paper J. Kollár suggests that equality for some \(m\) implies a splitting of \((A,\Theta)\). The paper under review considers the case \(m=g\). It is shown that \(\text{Sing}_g (\Theta)\neq 0\) implies that \((A,\Theta)\) is a polarized product of \(g\) elliptic curves.
scientific article; zbMATH DE number 908692
scientific article