A. J. de Jong

Smoothness, semi-stability and alterations

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Purity of the stratification by Newton Polygons

In this paper we prove four theorems: one on surface singularities, two on Fcrystals, and one on moduli of p-divisible groups. The reason we put together these results in one paper is that the proofs, as given here, show how these theorems are related. Let us first describe our results. Let (S, 0) be a normal surface singularity over an algebraically closed field of characteristic p. Let S S be a resolution of singularities. Our first result is Theorem 3.2: (1) Any Qp-cohomology class on the link of the singularity extends to the resolution, more precisely

A conjecture on arithmetic fundamental groups

AbstractThe conjecture is the following: Over an algebraic variety over a finite field, the geometric monodromy group of every smooth

Divisor Classes and the Virtual Canonical Bundle for Genus 0 Maps

\overline {\mathbb{F}_\ell ((t))}

The period-index problem for the Brauer group of an algebraic surface

is finite. We indicate how to prove this for rank 2, using results of Drinfeld. We also show that the conjecture implies that certain deformation rings of Galois representations are complete intersection rings.

Homomorphisms of Barsotti-Tate groups and crystals in positive characteristic

Let X be a Fano manifold of pseudo-index ≥ 3 such that c1(X)− 2c2(X) is nef. Irreducibility of some spaces of rational curves on X implies a general point of X is contained in a rational surface.

Every rationally connected variety over the function field of a curve has a rational point

We prove divisor class relations for families of genus 0 curves and used them to compute the divisor class of the “virtual” canonical bundle of the Kontsevich space of genus 0 maps to a smooth target. This agrees with the canonical bundle in good cases. This work generalizes Pandharipande’s results in the special case that the target is projective space, [7] (Pandharipande, Trans. Am. Math. Soc. 351(4), 1481–1505, 1999), [8] (Pandharipande, Trans. Am. Math. Soc. 351(4), 1481–1505, 1999). Our method is completely different from Pandharipande’s.

Families of rationally simply connected varieties over surfaces and torsors for semisimple groups

Let