An inflectionary tangent to the Kummer variety and the Jacobian condition

In this paper we deal with a degenerate version of the trisecant conjecture Let $ [X,\Theta] $ be an indecomposable principally polarized abelian variety and let $ \Theta $ be a symmetric representative of the polarization. We shall denote by $ \theta $ a non-zero section of the corresponding sheaf. The linear system $ \vert 2 \Theta \vert $ is base-point-free and it is independent of the choice of $ \Theta $. The image of the morphism $ \ K : \ X \rightarrow \vert 2 \Theta \vert^* $ associated with the base-point-free linear system $ \vert 2 \Theta \vert $ is a projective variety which 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 a Jacobian has a rich geometry in terms of trisecants and flexes). We prove that if there exists an inflectionary tangent $ l $ to the Kummer variety associated with $ [X,\Theta] $ then $ [X,\Theta] $ is a Jacobian provided that there are no set-theoretical $ D$-invariant components of the scheme $ D \Theta := \Theta \cap \{D\theta = 0 \} $, where $ D $ is an invariant vector field on $ X $ associated to $ l $.