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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Deformations of isolated even double points of corank one


Authors: R. Smith and R. Varley
Journal: Proc. Amer. Math. Soc. 140 (2012), 4085-4096
MSC (2010): Primary 14-xx; Secondary 32-xx
Published electronically: April 10, 2012
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Abstract: We give a local deformation theoretic proof of Farkas' conjecture, first proved by Grushevsky and Salvati Manni, that a complex principally polarized abelian variety (ppav) of dimension 4 whose theta divisor has an isolated double point of rank 3 at a point of order two is a Jacobian of a smooth curve of genus 4. The basis of this proof is Beauville's result that a 4 dimensional ppav is a non-hyperelliptic Jacobian if and only if some symmetric translate of the theta divisor has singular locus which either consists of precisely two distinct conjugate singularities $ \{\pm x\}$ or has an isolated singular point which is a limit of two distinct conjugate singularities. We establish an explicit local normal form for the theta function near an isolated double point of rank 3 at a point of order two, which implies the point is such a limit (after translation to the origin), i.e.  has a small deformation within the family defined by the universal theta function whose nearby singularities include two conjugate ordinary double points (odp's). The existence of such a deformation depends only on the facts that the theta function is even, a general theta divisor is smooth, and a general singular theta divisor has only odp's, also proved by Beauville in dimension 4. The argument yields a similar result, also proved by Grushevsky and Salvati Manni, for ppav's of dimension $ g > 4$ whose theta divisor has an isolated double point of rank $ (g-1)$, i.e.  corank one, at a point of order two.


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Additional Information

R. Smith
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: roy@math.uga.edu

R. Varley
Affiliation: Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email: rvarley@math.uga.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11366-8
PII: S 0002-9939(2012)11366-8
Received by editor(s): December 31, 2008
Received by editor(s) in revised form: April 6, 2009, July 3, 2010, and May 24, 2011
Published electronically: April 10, 2012
Communicated by: Ted Chinburg
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.