Rank 2 vector bundles and degrees of points of del Pezzo surfaces
Claire Voisin
TL;DR
The paper advances the understanding of points and $0$-cycles on del Pezzo surfaces, focusing on cubic surfaces, by combining a Schwarzenberger construction of rank-$2$ vector bundles with Hilbert-scheme techniques. It proves that the third punctual Hilbert scheme $X^{[3]}$ of a smooth cubic hypersurface $X$ with $ ext{dim }X\ge 2$ is unirational, and uses this to eliminate Coray’s degree-$10$ obstruction for cubic surfaces, while delivering sharper degree bounds for effective $0$-cycles on cubic and low-degree del Pezzo surfaces. The work extends these methods to establish the unirationality of the third symmetric product in the cubic case and derives not-stably-rational results in certain dimensions. Across cubic, degree-$2$, and degree-$1$ del Pezzo surfaces, the authors derive explicit decompositions of $0$-cycles in CH${}_0$ in terms of lower-degree effective cycles plus multiples of canonical basis cycles $h_d$, thereby strengthening previous results of Colliot-Thélène and Coray and providing concrete criteria for the existence of $K$-points or small-degree points.
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-Thélène. 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.