Semicontinuity of Gauss maps and the Schottky problem

We show that the degree of Gauss maps on abelian varieties is semicontinuous in families, and we study its jump loci. As an application we obtain that in the case of theta divisors this degree answers the Schottky problem. Our proof computes the degree of Gauss maps by specialization of Lagrangian cycles on the cotangent bundle. We also get similar results for the intersection cohomology of varieties with a finite morphism to an abelian variety; it follows that many components of Andreotti–Mayer loci, including the Schottky locus, are part of the stratification of the moduli space of ppav’s defined by the topological type of the theta divisor.

Summands of theta divisors on Jacobians

We show that the only summands of the theta divisor on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the powers of the curve and the Fano surface of lines on the threefold. The proof only uses the decomposition theorem for perverse sheaves, some representation theory and the notion of characteristic cycles.

Division by 1–ζ on Superelliptic Curves and Jacobians

Yuri Zarhin gave formulas for “dividing a point on a hyperelliptic curve by 2”. Given a point $P$ on a hyperelliptic curve $\mathcal{C}$ of genus $g$, Zarhin gives the Mumford representation of an effective degree $g$ divisor $D$ satisfying $2(D - g \infty ) \sim P - \infty $. The aim of this paper is to generalize Zarhin’s result to superelliptic curves; instead of dividing by 2, we divide by $1 - \zeta $. There is no Mumford representation for divisors on superelliptic curves, so instead we give formulas for functions that cut out a divisor $D$ satisfying $(1 - \zeta ) D \sim P - \infty $. Additionally, we study the intersection of $(1 - \zeta )^{-1} \mathcal{C}$ and the theta divisor $\Theta $ inside the Jacobian $\mathcal{J}$. We show that the intersection is contained in $\mathcal{J}[1 - \zeta ]$ and compute the intersection multiplicities.

2-torsion points on theta divisors

In this note we prove a sharp bound for the number of 2-torsion points on a theta divisor and show that this is achieved only in the case of products of elliptic curves. This settles in the affirmative a conjecture of Marcucci and Pirola.

On the closed embedding of the product of theta divisors into product of Jacobians and Chow groups

In this paper, we generalize the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of [Formula: see text] into [Formula: see text] for [Formula: see text], where [Formula: see text] is a smooth projective curve, to symmetric powers of a smooth projective variety of higher dimension. We also prove the analog of this theorem for product of symmetric powers of smooth projective varieties. As an application we prove the injectivity of the push-forward homomorphism at the level of Chow groups, induced by the closed embedding of self-product of theta divisor into the self-product of the Jacobian of a smooth projective curve.

Tangent cones to generalised theta divisors and generic injectivity of the theta map

Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^{0}(C,E)=r+1$. For $g\geqslant (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the generalised theta divisor $\unicode[STIX]{x1D6E9}_{E}$ at ${\mathcal{O}}_{C}$. We use this to give a constructive proof and a sharpening of Brivio and Verra’s theorem that the theta map $\mathit{SU}_{C}(r){\dashrightarrow}|r\unicode[STIX]{x1D6E9}|$ is generically injective for large values of $g$.