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.
Keywords
Cite
@article{arxiv.2509.17996,
title = {Rank 2 vector bundles and degrees of points of del Pezzo surfaces},
author = {Claire Voisin},
journal= {arXiv preprint arXiv:2509.17996},
year = {2026}
}
Comments
Many typos corrected. Final version to appear in Crelle's bicentennial issue