A splitting criterion for an isolated singularity at x=0 in a family of even hypersurfaces

作者：Smith Roy^*; Varley Robert

来源：Manuscripta Mathematica, 2012, 137(1-2): 233-245.

DOI：10.1007/s00229-011-0468-3

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 %26quot;splits in codimension one%26quot;, 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, I similar to) with dim(A) = g a parts per thousand yen 4, a singularity of even multiplicity on I similar to, 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.

• 出版日期2012-1