Abstract
Assume given a family of even local analytic hypersurfaces, whose central fiber has an isolated singularity at x = 0 which is not an ordinary double point. We prove that if the family is sufficiently general, for instance if the general fiber is smooth and the general singular fiber has only ordinary double points, then the singularity at x = 0 “splits in codimension one”, i.e., the local discriminant divisor has an irreducible component, over which a general fiber has more than one singularity specializing to the original one. As a corollary, we deduce the result by Grushevsky and Salvati Manni (Singularities of the theta divisor at points of order two, IMRN, 2007, Proposition 8) that on a principally polarized abelian variety (A, Θ) with dim(A) = g ≥ 4, a singularity of even multiplicity on Θ, isolated or not, at a point of order two and not an ordinary double point, must be a limit of two distinct ordinary double points {x, −x} on nearby theta divisors.
Similar content being viewed by others
References
Andreotti A., Mayer A.: On period relations for abelian integrals on algebraic curves. Ann. Scuola Norm. Sup. Pisa 21, 189–238 (1967)
Beauville A.: Prym varieties and the Schottky problem. Invent. Math. 41, 149–196 (1977)
Ciliberto C., van der Geer G.: Andreotti-Mayer Loci and the Schottky Problem. Documenta Mathematica 13, 453–504 (2008)
Debarre O.: Le lieu des variétés abéliennes dont le diviseur thêta est singulier a deux composantes. Ann. Sci. Éc. Norm. Sup. 4e série, tome 25(6), 687–708 (1992)
Debarre, O.: Erratum pour “Le lieu des variétés abéliennes dont le diviseur thêta est singulier a deux composantes”, 15 avril 2008, http://www.dma.ens.fr/~debarre/23erratum.pdf.
Ein L., Lazarsfeld R.: Singularities of theta divisors and the birational geometry of irregular varieties. J. Amer. Math. Soc. 10(1), 243–258 (1997)
Farkas, H.: Vanishing theta nulls and Jacobians. In: Muñoz Porras, J., Popescu, S., Rodríguez, R. (eds.) The Geometry of Riemann Surfaces and Abelian Varieties, III Iberoamerican Congress on Geometry in Honor of Professor Sevín Recillas Pishmish’s 60th Birthday. Contemporary Mathematics, Amer. Math. Soc. 397, 37–53 (2006)
Kas A., Schlessinger M.: On the versal deformation of a complex space with an isolated singularity. Math. Ann. 196, 23–29 (1972)
Grushevsky S., Salvati Manni R.: Jacobians with a vanishing theta null in genus 4. Israel J. Math. 164, 303–315 (2008)
Grushevsky, S., Salvati Manni, R.: Singularities of the theta divisor at points of order two, Int. Math. Res. Not. (IMRN), 2007, no.15, Art. ID, rnm 045, 15 pages
Gunning R., Rossi H.: Analytic functions of several complex variables. Prentice Hall, Englewood Cliffs (1967)
Mumford D.: Algebraic geometry I, Complex projective varieties. Springer Verlag, New York (1976)
Mumford D.: Tata lectures on theta II. Birkhaúser, Boston (1984)
Smith, R., Varley, R.: Deformations of isolated even double points of corank one. In: Proceedings of the American Mathematical Society (2011) (to appear)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Smith, R., Varley, R. A splitting criterion for an isolated singularity at x = 0 in a family of even hypersurfaces. manuscripta math. 137, 233–245 (2012). https://doi.org/10.1007/s00229-011-0468-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00229-011-0468-3