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I. Krichever proved Welter's trisecant conjecture that says that a principally polarized abelian variety (over the complex numbers) is the Jacobian of some curve if and only if its Kummer variety has a trisecant line. O. Debarre showed that Prym varieties are characterized by a one dimensional family of quadrisecant planes.

My question is: Has there been any progress made on singling out certain abelian varieties by higher dimensional secancy conditions?

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