A characterization of hyperelliptic Jacobians
Let \(C\) be a smooth curve of genus \(g\), \(J(C)\) its Jacobian and \(\Theta\) the Theta divisor on \(J(C)\). It is a well known result that the dimension of the singularity locus \(\text{sing} (\Theta)\) is \(g-4\) for nonhyperelliptic \(C\) and \(g-3\) for hyperelliptic curves \(C\). This statement is part of a criterion characterizing Jacobians of hyperelliptic curves among all principally polarized abelian varieties. The main theorem of the paper under consideration is: A principally polarized abelian variety \((X,\Theta)\) of dimension \(g\) is the Jacobian of a hyperelliptic curve if and only if \(\dim \text{sing} (\Theta) \leq g-3\) and there exists a global section of the sheaf \({\mathcal O}_ \Theta (\Theta)\) whose zero locus admits nonreduced components.
characterizing Jacobians of hyperelliptic curves among all principally polarized abelian varieties
scientific article; zbMATH DE number 492752