An Arithmetic Count of the Lines on a Smooth Cubic Surface

Abstract

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over $\mathbb{C}$ there are $27$ lines, and over $\mathbb{R}$ the number of hyperbolic lines minus the number of elliptic lines is $3$. In general, the lines are defined over a field extension $L$ and have an associated arithmetic type $\alpha$ in $L^{\ast }/(L^{\ast }{)}^2$. There is an equality in the Grothendieck-Witt group $\mathrm{GW}(k)$ of $k$ $$\sum _{\text{lines}}{\mathrm{Tr}}_{L/k}⟨\alpha ⟩=15\cdot ⟨1⟩+12\cdot ⟨-1⟩,$$ where ${\mathrm{Tr}}_{L/k}$ denotes the trace $\mathrm{GW}(L)\to \mathrm{GW}(k)$. Taking the rank and signature recovers the results over $\mathbb{C}$ and $\mathbb{R}$. To do this, we develop an elementary theory of the Euler number in ${\mathbb{A}}^1$-homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.

34 pages. Accepted for publication in Compositio Mathematica

References in corpus (3)

• The class of Eisenbud--Khimshiashvili--Levine is the local A1-Brouwer degree
• Motivic Euler characteristics and Witt-valued characteristic classes
• An arithmetic enrichment of Bézout's Theorem

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• Avoidance loci and tropicalizations of real bitangents to plane quartics
• Quadratic counts of highly tangent lines to hypersurfaces
• ${\mathbb{A}}^1$-Brouwer degrees in Macaulay2
• Compactly supported ${\mathbb{A}}^1$-Euler characteristics of symmetric powers of cellular varieties