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 $l$.

Marini, G. (1997). An inflectionary tangent to the Kummer variety and the Jacobian condition. MATHEMATISCHE ANNALEN, 309(3), 483-490.

### An inflectionary tangent to the Kummer variety and the Jacobian condition

#### Abstract

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 $l$.
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Settore MAT/03 - Geometria
English
Marini, G. (1997). An inflectionary tangent to the Kummer variety and the Jacobian condition. MATHEMATISCHE ANNALEN, 309(3), 483-490.
Marini, G
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/2108/55538