An inflectionary tangent to the Kummer variety and the Jacobian condition (Q1374731)

We deal with a degenerate version of the trisecant conjecture. For \((X, [\Theta])\) an indecomposable principally polarized abelian variety, the linear system \(|2\Theta|\) is base-point-free, it is independent of the choice of \(\Theta\), and it defines a morphism \(K: X\to |2\Theta|^*\) whose image is called the Kummer variety of \((X,[\Theta])\). Welters conjectured that the existence of one trisecant line to the Kummer variety characterizes the Jacobians (it is well known that the Kummer variety of the Jacobian has a rich geometry in terms of trisecants and flexes). We prove the following theorem. Theorem. Let \((X,[\Theta])\) be an indecomposable principally polarized abelian variety. Denote by \(\theta\) a non-zero section of the sheaf \({\mathcal O}_X (\Theta)\), and denote by \(K: X\to|2\Theta|^*\) the Kummer morphism. Assume that \(K(X)\) has an inflectionary trisecant, i.e. that there exists a line \(l\) in the projective space \(|2\Theta|^*\) which meets \(K(X)\) at a smooth point \(K(u)\) with at least multiplicity 3, and assume that the scheme \(\Theta\cap \{D\theta=0\}\) does not contain set-theoretical \(D\)-invariant components, where \(K_*(D)\) is tangent to \(l\) at \(K(u)\). Then, \((X,[\Theta])\) is the Jacobian of a smooth curve \({\mathcal C}\).