Fibrations with few rational points

Abelian fibrations and rational points on symmetric products

Let X be an algebraic variety defined over a number field K and X(K) its set of K-rational points. We are interested in properties of X(K) imposed by the global geometry of X. We say that rational points on X are potentially dense if there exists a finite field extension L/K such that X(L) is Zariski dense. It is expected at least for surfaces that if there are no finite étale covers of X domin...

Twelve Points on the Projective Line, Branched Covers, and Rational Elliptic Fibrations

The following divisors in the space Sym12 P1 of twelve points on P 1 are actually the same: (A) The possible locus of the twelve nodal fibers in a rational elliptic fibration (i.e. a pencil of plane cubic curves); (B) degree 12 binary forms that can be expressed as a cube plus a square; (C) the locus of the twelve tangents to a smooth plane quartic from a general point of the plane; (D) the bra...

Rational Fibrations of M5,1 and M6,1

We construct rational maps from M5,1 and M6,1 to lower-dimensional moduli spaces. As a consequence, we identify geometric divisors that generate extremal rays of the effective cones for these spaces.

Distribution of Rational Points: A Survey

Lattice 3-Polytopes with Few Lattice Points

This paper is intended as a first step in a program for a full algorithmic enumeration of lattice 3-polytopes. The program is based in the following two facts, that we prove: • For each n there is only a finite number of (equivalence classes of) 3polytopes of lattice width larger than one, where n is the number of lattice points. Polytopes of width one are infinitely many, but easy to classify....