Non representability of twice the minimal class in a generic Jacobian (Q1337764)

Let \(A\) be a principally polarized abelian variety of dimension \(a\), and let \(\Theta\) be a divisor representing the polarization. We shall say \(A\) is \(m\)-representable, for an integer \(m\), if there exists an integral curve \(C \subset A\) with cohomological class \(m \cdot {[\Theta]^{a - 1} \over (a - 1)!}\). For example, \(A\) is the Jacobian of a curve if and only if \(A\) is 1-representable (Matsusaka's theorem). In this paper we prove that a generic Jacobian of dimension \(\geq 6\) is not 2-representable.