Abstracts of Talks

Diophantine Subsets of Function Fields of Curves

János Kollár (Princeton University)

We consider diophantine subsets of function fields of curves and show, roughly speaking, that they are either very small or very large. In particular, this implies that the ring of polynomials k[t] is a not a diophantine subset of the field of rational functions k(t) for many fields k.

Speculations about Rational Curves on Varieties over Countable Fields

Bjorn Poonen (Berkeley)

Let k be an algebraic closure of Q. Bogomolov asked whether the rational curves on a K3 surface over k necessarily cover all the k-points. I will present related conjectures that would imply a negative answer to this and other such questions. (But I will not prove much.)

Hilbert's Tenth Problem for Function Fields of Varieties over C

Kirsten Eisentraeger (Pennsylvania State University)

In 1970 Matiyasevich proved that Hilbert's Tenth Problem is undecidable. That is, there does not exist an algorithm that, given an arbitrary multivariable polynomial equation f(x1,...,xn)=0 with integer coefficients, decides whether there is a solution over the integers. Since Matiyasevich's result, analogues of Hilbert's Tenth Problem have been investigated over other rings and fields. In this talk I will discuss the result by Kim and Roush that Hilbert's Tenth Problem for C(t1,t2) is undecidable, and show how their theorem can be generalized to function fields K/C with tr.deg.(K/C) at least 2.

Rational Points of Varieties over Function Fields beyond Rational (higher) Connectedness

Jason Starr (MIT)

Rational connectedness is a beautiful property which is exactly the right notion (in a precise sense) for proving existence of rational points of varieties over function fields of curves. For varieties over function fields of surfaces there is a related property, rational simple connectedness, which again implies existence of rational points under some additional hypotheses. But consideration of hypersurfaces in Grassmannians suggests there is some unknown weaker property implying existence of rational points. I will discuss what A. J. de Jong and I have observed about rational points of hypersurfaces in Grassmannians. And then I would like to open the floor to discussion about what the unknown weaker property might be.

Vanishing of Quantum Cohomology

Izzet Coskun (MIT)

One way to show that a variety is uniruled (respectively, rationally connected) is to exhibit a non-vanishing genus zero Gromov-Witten invariant involving the condition of passing through a fixed point (respectively, through two fixed points). Unfortunately, proving the non-vanishing of a genus zero Gromov-Witten invariant on an arbitrary rationally connected variety seems hard. I will talk about some simple vanishing and non-vanishing results for the small quantum cohomology of partial flag varieties. As a consequence, I will show how to obtain various equalities among the structure constants of the ordinary cohomology of partial flag varieties.

Restriction of Sections for Families of Abelian Varieties

Tom Graber (California Institute of Technology)

I will describe joint work with Jason Starr showing that the Mordell-Weil group of an abelian scheme over a higher dimensional algebraic variety agrees with that of its restriction to an appropriate curve.

Congruence for Rational Points over Finite Fields and Coniveau over Local Fields

Chenyang Xu (Princeton University)

If the l-adic cohomology of a projective smooth variety, defined over a local field K with finite residue field k, is supported in codimension >1, then every model over the ring of integers of K has a k-rational point. If the model X is regular, one has a congruence |X(k)| = 1 modulo |k| for the number of k-rational points. The congruence is violated if one drops the regularity assumption. (Joint with Esnault)