Rational lines on smooth cubic surfaces

TL;DR

The paper connects the enumerative geometry of lines on smooth cubic surfaces to the arithmetic of the base field, showing that the nine possible line counts $\{0,1,2,3,5,7,9,15,27\}$ occur according to Galois-theoretic data. It develops a constructive, $k$-rational blow-up framework anchored on six points on the cuspidal cubic to realize each count when $|k|\geq 23$, and analyzes the role of the inverse Galois problem via the Weyl group $W(E_6)$ to explain how field extensions control line configurations. The paper also clarifies characteristic-2 subtleties, connects to real and $p$-adic cases, and proves corollaries for finitely generated fields and finite transcendental extensions, showing that arithmetic governs the geometry across broad classes of fields. Together, these results provide a unified, constructive bridge between the arithmetic of base fields and the geometry of cubic surfaces, with implications for inverse Galois theory and explicit realizations of line counts.

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.

Figures (3)

• Figure 1: Impermissible graph of seven lines
• Figure 2: Impermissible graphs of nine lines
• Figure 3: Intersection graphs

Theorems & Definitions (67)

• Theorem 1.1: Segre
• Theorem 1.2: Segre
• Theorem 1.3
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Corollary 1.7
• Definition 2.1
• Proposition 2.2
• proof