On the abelian fivefolds attached to cubic surfaces

To a family of smooth projective cubic surfaces one can canonically associate a family of abelian fivefolds. In characteristic zero, we calculate the Hodge groups of the abelian varieties which arise in this way. In arbitrary characteristic we calculate the monodromy group of the universal family of abelian varieties, and thus show that the Galois group of the 27 lines on a suitably general cubic surface in positive characteristic is as large as possible.Comment: Fixed classification of Hodge groups; other small edit

Superspecial rank of supersingular abelian varieties and Jacobians

An abelian variety defined over an algebraically closed field k of positive characteristic is supersingular if it is isogenous to a product of supersingular elliptic curves and is superspecial if it is isomorphic to a product of supersingular elliptic curves. In this paper, the superspecial condition is generalized by defining the superspecial rank of an abelian variety, which is an invariant of its p-torsion. The main results in this paper are about the superspecial rank of supersingular abelian varieties and Jacobians of curves. For example, it turns out that the superspecial rank determines information about the decomposition of a supersingular abelian variety up to isomorphism; namely it is a bound for the maximal number of supersingular elliptic curves appearing in such a decomposition.Comment: V2: New coauthor, major rewrit

Generic Newton polygons for curves of given p-rank

We survey results and open questions about the $p$-ranks and Newton polygons of Jacobians of curves in positive characteristic $p$. We prove some geometric results about the $p$-rank stratification of the moduli space of (hyperelliptic) curves. For example, if $0 \leq f \leq g-1$, we prove that every component of the $p$-rank $f+1$ stratum of ${\mathcal M}_g$ contains a component of the $p$-rank $f$ stratum in its closure. We prove that the $p$-rank $f$ stratum of $\overline{\mathcal M}_g$ is connected. For all primes $p$ and all $g \geq 4$, we demonstrate the existence of a Jacobian of a smooth curve, defined over $\overline{\mathbb F}_p$, whose Newton polygon has slopes $\{0, \frac{1}{4}, \frac{3}{4}, 1\}$. We include partial results about the generic Newton polygons of curves of given genus $g$ and $p$-rank $f$.Comment: 16 pages, to appear in Algebraic Curves and Finite Fields: Codes, Cryptography, and other emergent applications, edited by H. Niederreiter, A. Ostafe, D. Panario, and A. Winterho