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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
DOI: https://doi.org/10.1090/S0002-9939-2012-11366-8
Published electronically: April 10, 2012
MathSciNet review: 2957198
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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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  • [A-M] A.  Andreotti and A.  Mayer, On period relations for abelian integrals on algebraic curves, Ann.  Scuola Norm.  Sup.  Pisa 21 (1967), 189-238. MR 0220740 (36:3792)
  • [A-G-V] V.  Arnol'd, S.  Gusein-Zade, A.  Varchenko, Singularities of Differentiable Maps. vol. I, Monographs in Mathematics, Birkhäuser, 1985. MR 777682 (86f:58018)
  • [B] A.  Beauville, Prym varieties and the Schottky problem, Invent.  Math.  41 (1977), 149-196. MR 0572974 (58:27995)
  • [CM] S. Casalaina-Martin, Cubic threefolds and abelian varieties of dimension five. II, Math.  Z.  260 (2008), 115-125. MR 2413346 (2009b:14090)
  • [D1] O.  Debarre, Le lieu des variétés abéliennes dont le diviseur thêta est singulier a deux composantes, Ann.  Sc.  École Norm.  Sup.  25 (1992), 687-708. MR 1198094 (94a:14045)
  • [D2] O.  Debarre, Annulation de thêtaconstantes sur les variétés abéliennes de dimension quatre, C.R.  Acad.  Sci.  Paris Sér.  I Math. 305 (1987), 885-888. MR 925290 (89a:14055)
  • [F] H.  Farkas, Vanishing theta nulls and Jacobians, in The Geometry of Riemann Surfaces and Abelian Varieties, Contemporary Mathematics, vol.  397, Amer.  Math.  Soc., 2006, 37-53. MR 2217996 (2007c:14026)
  • [F-K] H.  Farkas and I.  Kra, Riemann Surfaces, 2nd ed., Springer-Verlag, 1992. MR 1139765 (93a:30047)
  • [G-SM1] S.  Grushevsky and R.  Salvati Manni, Jacobians with a vanishing theta null in genus $ 4$, Israel J.  Math. 164 (2008), 303-315. MR 2391151 (2009e:14071)
  • [G-SM2] S.  Grushevsky and R.  Salvati Manni, Singularities of the theta divisor at points of order two, Int.  Math.  Res.  Not.  IMRN, 2007, no. 15, Art.  ID, rnm 045, 15 pages. MR 2348405 (2008g:14077)
  • [G-R] R.  Gunning and H.  Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, 1965. MR 0180696 (31:4927)
  • [H] A.  Hatcher, Algebraic Topology, Cambridge University Press, 2002. MR 1867354 (2002k:55001)
  • [K-S] A.  Kas and M.  Schlessinger, On the versal deformation of a complex space with an isolated singularity, Math.  Ann.  196 (1972), 23-29. MR 0294701 (45:3769)
  • [Ra] H.  Rauch, The vanishing of a theta constant is a peculiar phenomenon, Bull.  Amer.  Math.  Soc.  73 (1967), 339-342. MR 0213544 (35:4404)
  • [Ri] D.  Rim, Equivariant G-structure on versal deformations, Trans.  Amer.  Math.  Soc.  257 (1980), 217-226. MR 549162 (80k:14021)
  • [S-V1] R.  Smith and R.  Varley, On the geometry of $ \mathcal N_0$, Rend.  Sem.  Mat.  Univers.  Politecn.  Torino 42 (1984), 29-37. MR 812628 (87c:14052)
  • [S-V2] R.  Smith and R.  Varley, Components of the locus of singular theta divisors of genus $ 5$, Algebraic Geometry, Sitges 1983, Springer Lecture Notes, 1124, Springer-Verlag, 1985, 338-416. MR 805339 (86k:14030)
  • [S-V3] R.  Smith and R.  Varley, The tangent cone to the discriminant, Proceedings of the 1984 Vancouver Conference in Algebraic Geometry, Amer. Math. Soc., 1986, 443-460. MR 846034 (88h:14019)
  • [S-V4] R.  Smith and R.  Varley, A splitting criterion for an isolated singularity at $ x = 0$ in a family of even hypersurfaces, Manuscripta Math. 137 (2012), 233-245.
  • [V] R.  Varley, Weddle's surfaces, Humbert's Curves, and a certain $ 4$-dimensional abelian variety, Amer.  J. Math.  108 (1986), 931-952. MR 853219 (87g:14050)

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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: https://doi.org/10.1090/S0002-9939-2012-11366-8
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.

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