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Non-Jacobians in the Schottky loci. (English) Zbl 0636.14020

The author formerly showed that in the moduli space of principally polarized abelian varieties with a marked, non-zero point of order 2, the Schottky locus contains the Jacobian locus as an irreducible component. In the present paper, he shows that for the case genus \(\geq 5,\) the Schottky locus contains other components. An example is the locus \(\mathcal{RC}^o\) of intermediate Jacobians of cubic threefolds with an “even” point of order 2 (for genus 5).
In section 3 is proved that \(\mathcal {RC}^o\) is contained in the Schottky locus, using theta maps and Prym varieties. In section 5 it is proved that \(\mathcal{RC}^o\) is a component of the Schottky locus, on the basis of the results on degeneration of cubic threefolds (Collino, van Geemen and Welters) and on Prym varieties (by the author, Beauville and Mumford). In section 6 several open problems are discussed.

MSC:

14K30 Picard schemes, higher Jacobians
14K10 Algebraic moduli of abelian varieties, classification
14H40 Jacobians, Prym varieties
14K25 Theta functions and abelian varieties
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