Rational lines on smooth cubic surfaces

TL;DR

This paper investigates how the arithmetic properties of the base field influence the enumeration of lines on smooth cubic surfaces, providing criteria for possible line counts based on Galois theory.

Contribution

It introduces a criterion linking Galois theory of the base field to the possible numbers of lines on cubic surfaces, extending classical enumerative results.

Findings

Classifies possible line counts over various fields

Provides Galois-theoretic criteria for line configurations

Connects arithmetic properties to geometric configurations

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.