On the Schottky problem for genus five Jacobians with a vanishing theta null

Abstract

We give a solution to the weak Schottky problem for genus five Jacobians with a vanishing theta null, answering a question of Grushevsky and Salvati Manni. More precisely, we show that if a principally polarized abelian variety of dimension five has a vanishing theta null with a quadric tangent cone of rank at most three, then it is in the Jacobian locus, up to extra irreducible components. We employ a degeneration argument, together with a study of the ramification loci for the Gauss map of a theta divisor.

The Dubrovin threefold of an algebraic curve

“…. , a g , b g for the first homology group H 1 (C, Z) of a compact Riemann surface C of genus g. For any point p ∈ C, we choose three normalized differentials of the second type, denoted Ω (1) , Ω (2) and Ω (3) . These are meromorphic differentials on C, with poles only at p, and with local expansions at p of the form…”

“…We denote the first and second derivative of this function by Ḣk (z) and Ḧk (z). Our differentials of the second type are Ω (1) = −ω (1) , Ω (2) = −2ω (2) , Ω (3) = −3ω (3) ,…”

“…We can now use these matrices to construct two-phase solutions of the KP equation (2). First we compute the following Riemann matrix for the curve C = {y 2 = x 6 − 1} in Example 1.1:…”

“…The Dubrovin threefold D C comprises triples (U, V, W ) for which the following condition holds: there exist c, d ∈ C such that, for all vectors D ∈ C g , the function (6) satisfies the equation ( 4). This implies that (3) satisfies (2). A celebrated result due to Igor Krichever [10,Theorem 3.1.3] states that such solutions exist and can be constructed from every smooth point on the curve C. This ensures that D C is a threefold.…”

“…The function τ = τ(x, y, t) is known as the τ-function in the theory of integrable systems. A sufficient condition for (2) to hold is the quadratic PDE (τ xxxx τ−4τ xxx τ x +3τ 2 xx ) + 4(τ x τ t − ττ xt ) + 6c(τ xx τ − τ 2 x ) + 3(ττ yy − τ 2 y ) + 8dτ 2 = 0. (4) This is known as Hirota's bilinear form.…”