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An arithmetic count of the lines on a smooth cubic surface

Published online by Cambridge University Press:  08 April 2021

Jesse Leo Kass
Department of Mathematics, University of South Carolina, 1523 Greene Street, Columbia, SC29208,
Kirsten Wickelgren
Department of Mathematics, Duke University, 120 Science Dr, Durham, NC27708,


We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field $k$, generalizing the counts that over ${\mathbf {C}}$ there are $27$ lines, and over ${\mathbf {R}}$ 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 $L^*/(L^*)^2$. There is an equality in the Grothendieck–Witt group $\operatorname {GW}(k)$ of $k$,

\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]
where $\operatorname {Tr}_{L/k}$ denotes the trace $\operatorname {GW}(L) \to \operatorname {GW}(k)$. Taking the rank and signature recovers the results over ${\mathbf {C}}$ and ${\mathbf {R}}$. To do this, we develop an elementary theory of the Euler number in $\mathbf {A}^1$-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.

Research Article
© The Author(s) 2021

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