An inflectionary tangent to the Kummer variety and the Jacobian condition

In this paper we deal with a degenerate version of the trisecant conjecture (see Welters [W1]). Let (X , [Θ]) be an indecomposable principally polarized abelian variety and let Θ be a symmetric representative of the polarization. We shall denote by θ a non-zero section of the sheaf O X (Θ). The linear system |2Θ| is base-point-free and it is independent of the choice of Θ. The image of the morphism K : X −→ |2Θ| * associated with the base-point-free linear system |2Θ| is a projective variety which is called the Kummer variety of (X , [Θ]). Welters conjectured in [W1] 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 , [Θ]) , then (X , [Θ]) is a Jacobian provided that there are no set-theoretical D-invariant components of the scheme DΘ := Θ∩{Dθ = 0} , where D is an invariant vector field on X associated to l. Observe that Welters' conjecture divides naturally into the three possible cases which correspond to the three hypotheses: i) the Kummer variety K (X) has an honest trisecant, i.e. there exists a line l in |2Θ| * meeting K (X) at three distinct points K (a), K (b), K (c); ii) there exists a line l in |2Θ| * which is tangent to K (X) at a smooth point u , and which meets K (X) at some other point; iii) K (X) has an inflectionary tangent at a smooth point. The cases i) and ii) were considered by Debarre, see [D1] and [D2] where he gives an affermative answer to the problem, provided that an extra hypothesis holds. Namely, in case i) the extra hypothesis is that Θ cannot contain invariant divisors under the translations t a−b , t b−c , and in case ii) the extra hypothesis is that Θ