Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Cubic threefolds and abelian varieties of dimension five


Authors: Sebastian Casalaina-Martin and Robert Friedman
Journal: J. Algebraic Geom. 14 (2005), 295-326
DOI: https://doi.org/10.1090/S1056-3911-04-00379-0
Published electronically: August 11, 2004
MathSciNet review: 2123232
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Abstract | References | Additional Information

Abstract: This paper proves the following converse to a theorem of Mumford: Let $A$ be a principally polarized abelian variety of dimension five, whose theta divisor has a unique singular point, and suppose that the multiplicity of the singular point is three. Then $A$ is isomorphic as a principally polarized abelian variety to the intermediate Jacobian of a smooth cubic threefold. The method of proof is to analyze the possible singularities of the theta divisor of $A$, and ultimately to show that $A$ is the Prym variety of a possibly singular plane quintic.


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Additional Information

Sebastian Casalaina-Martin
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Address at time of publication: Department of Mathematics, SUNY Stony Brook, Stony Brook, New York 11794-3651
Email: casa@math.columbia.edu, casa@math.sunysb.edu

Robert Friedman
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Email: rf@math.columbia.edu

DOI: https://doi.org/10.1090/S1056-3911-04-00379-0
Received by editor(s): July 28, 2003
Published electronically: August 11, 2004
Additional Notes: The first author was supported in part by a VIGRE fellowship from NSF Grant #DMS-98-10750. The second author was supported in part by NSF Grant #DMS-02-00810.

American Mathematical Society