LetA be an abelian variety over a field k, D a symmetric divisor onA, s and d the sum and difference maps fromA×A intoA, andp1 andp2 the projections onto the first and second factors. The theorem of the square and the seesaw principle [M1, Secs. 5, 6] guarantee that there exists a function f(u, v) on A×A (determined up to constant multiples) with divisor s∗D+ d ∗D− 2p∗ 1D− 2p∗2D. Since this fun...

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

Let P be the Prym variety associated with a covering π : Y → X between non-singular irreducible projective curves. If P̃ is a principally polarized Prym-Tyurin variety associated with P , we prove that the induced Abel-Prym morphism ρ̃ : Y → P̃ is birational onto its image for genus gX > 2 and deg π 6= 2. We use such result to prove that the Picard bundle over the Prym variety is simple and moreov...

We consider Weierstrass functions and divisor functions arising from q-series. Using these we can obtain new identities for divisor functions. Farkas [3] provided a relation between the sums of divisors satisfying congruence conditions and the sums of numbers of divisors satisfying congruence conditions. In the proof he took logarithmic derivative to theta functions and used the heat equation. ...

A new relation between Prym varieties of arbitrary morphisms of algebraic curves and integrable systems is discovered. The action of maximal commutative subalgebras of the formal loop algebra of GLn defined on certain infinite-dimensional Grassmannians is studied. It is proved that every finite-dimensional orbit of the action of traceless elements of these commutative Lie algebras is isomorphic...

For most curves of genus 4 and characteristic > 3 the second osculating cone of the theta divisor is the cone over the canonical curve. Let G be a nonhyperelliptic complete smooth curve of genus 4. The canonical morphism G —> P3 is an embedding and the image is defined as the intersection of a quadric and a cubic. In this paper we will discuss a geometric way of constructing such equations from...

Consider, for every g ≥ 4, a non-singular, complex, canonical curve of genus g, that is, a curve X embedded in P by its complete canonical series |ωX |. Each such a curve possesses 2 2g line bundles L of degree g − 1 such that L = ωX , the theta-characteristics of X, and precisely Ng := ( 2 2 ) of them are odd (i.e. h(X,L) is odd). To a non-zero section σ of a theta-characteristic one associate...

In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with the probability measure Z−1 n exp (−n (tr(V (M1) +W (M2)− τM1M2)) dM1dM2, in the case where W = y 4 4 and V is a general even polynomial. We studied the correlation kernel for the eigenvalues of the matrix M1 in the limit as n → ∞. We extended the results of Duits and Kuijlaars in [14] to the case when the ...