Algebraic Geometry Seminar Zero-cycles on geometrically rational surfaces

Abstract

Let k be a field of characteristic zero. In 1974, D. Coray showed that on a smooth cubic surface over k, with a point over a field extension K of k of degree prime to 3, there exists such a point over a field extension K over k of degree 1, 4 or 10 (as of today unsurpassed result). We show how a combination of generisation, specialisation, Bertini theorems and "large" fields avoids lengthy considerations of degeneracy cases in Coray's proof, and leads to more results. For smooth cubic surfaces with a rational point, we then show that any zero-cycle of degree at least 10 is Chow-rationally equivalent to an effective cycle. Using the k-birational classification of geometrically rational k-surfaces (Enriques, Manin, Iskovskikh, Mori), by a case by case discussion, we show how these two results have analogues for arbitrary smooth, projective, geometrically rational surfaces.

Speaker

Jean-Louis Colliot-Thélène (born 2 December 1947), is a French mathematician. He is a Directeur de Recherches at CNRS at the Université Paris-Saclay in Orsay. He studies mainly number theory and arithmetic geometry.

Homepage：

https://handwiki.org/wiki/Biography:Jean-Louis_Colliot-Th%C3%A9l%C3%A8ne