Rank 2 vector bundles and degrees of points of del Pezzo surfaces

Abstract

We study points and 0-cycles on del Pezzo surfaces defined over a field K of characteristic 0, with emphasis on cubic surfaces. We prove that a cubic surface that admits a point defined over a field extension of K of degree coprime to 3 either has a K-point or has a point defined over a field extension of degree 4. This improves a result of Coray (who allowed also field extensions of degree 10). We also prove that 0-cycles of degree at least 18 on a cubic surface are effective and get similar results for degree 2 and degree 1 del Pezzo surfaces, improving results of Colliot-Th\'{e}l\`{e}ne. In a different direction, we prove that the third symmetric product of a cubic hypersurface of dimension at least 2 is unirational over any field, and that in dimension 2 or 3, it is not stably rational in general.