Deformations of isolated even double points of corank one

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.