Alena Pirutka

Cyclic covers that are not stably rational

Using methods developed by Kollar, Voisin, ourselves and Totaro, we prove that a cyclic cover of , , of prime degree , ramified along a very general hypersurface of degree , is not stably rational if . In dimension 3 we recover double covers of ramified along a very general surface of degree 4 (Voisin) and double covers of ramified along a very general surface of degree 6 (Beauville). We also find double covers of ramified along a very general hypersurface of degree 6. This method also enables us to produce examples over a number field.

R-equivalence on low degree complete intersections

Let k be a function field in one variable over C or the field C((t)). Let X be a k-rationally simply connected variety defined over k. In this paper we show that R-equivalence on rational points of X is trivial and that the Chow group of zero-cycles of degree zero A0(X) is zero. In particular, this holds for a smooth complete intersection of r hypersurfaces in P n of respective degrees d1;:::;dr with r P i=1 d 2 n + 1.

La conjecture de Tate entière pour les cubiques de dimension quatre

We prove the integral Tate conjecture for cycles of codimension

On a Local-Global Principle for H3 of Function Fields of Surfaces over a Finite Field

2

Hypersurfaces quartiques de dimension 3: Non rationalite stable

on smooth cubic fourfolds over an algebraic closure of a field finitely generated over its prime subfield and of characteristic different from

A very general quartic double fourfold is not stably rational

2

Stable rationality of quadric surface bundles over surfaces

or

Revetements cycliques qui ne sont pas stablement rationnels

3

Varieties that are not stably rational, zero-cycles and unramified cohomology

. The proof relies on the Tate conjecture with rational coefficients, proved in that setting by the first author, and on an argument of Voisin coming from complex geometry.

Sur le groupe de Chow de codimension deux des variétés sur les corps finis

Let K be the function field of a smooth projective surface S over a finite field \( \mathbb{F}\). In this article, following the work of Parimala and Suresh, we establish a local-global principle for the divisibility of elements in \( {H}^{3}(K, \mathbb{Z}/\ell)\) by elements in \( {H}^{2}(K, \mathbb{Z}/\ell), {l} \neq car.K \).