Rational lines on smooth cubic surfaces
Algebraic Geometry
2025-03-04 v4
Abstract
We prove that the enumerative geometry of lines on smooth cubic surfaces is governed by the arithmetic of the base field. In 1949, Segre proved that the number of lines on a smooth cubic surface over any field is 0, 1, 2, 3, 5, 7, 9, 15, or 27. Over a given field, each of these line counts may or may not be realized by some cubic surface. We give a sufficient criterion for each of these line counts in terms of the Galois theory of the base field.
Cite
@article{arxiv.2101.08217,
title = {Rational lines on smooth cubic surfaces},
author = {Stephen McKean},
journal= {arXiv preprint arXiv:2101.08217},
year = {2025}
}
Comments
Substantial revision, with some results removed due to an error in the previous version. 28 pages, 3 figures