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The Gauss map and secants of the Kummer variety. (English) Zbl 1420.14060

The Kummer variety of the Jacobian of a smooth complex projective curve admits a four-dimensional family of trisecant lines. This is a consequence of Fay’s trisecant formula J. D. Fay [Theta functions on Riemann surfaces. Cham: Springer (1973; Zbl 0281.30013)], E. Arbarello and C. de Concini [Ann. Math. (2) 120, 119–140 (1984; Zbl 0551.14016)].
The main focus of the present article is the concept of theta trisecant lines. These are lines which intersect the image of the (symmetric) theta divisor in the Kummer variety in at least three points. The key insight of the article under review is to relate these trisecant lines to the Gauss map.
The authors’ first result establishes existence of theta trisecant lines. Their second result gives a criteria for points on a given trisecant line to have equal image under the Gauss map. The more general concept of theta multisecant lines is also investigated.
Finally, the authors summarize a conjecture of A. Beauville and O. Debarre [Lect. Notes Math. 1399, 27–39 (1989; Zbl 0699.14056)] which gives a sufficient condition for a principally polarized abelian variety to be a Jacobian.

MSC:

14H40 Jacobians, Prym varieties
14H42 Theta functions and curves; Schottky problem
14K25 Theta functions and abelian varieties
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