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An arithmetic count of the lines on a smooth cubic surface
Published online by Cambridge University Press: 08 April 2021
Abstract
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
$k$,
\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]
$\operatorname {Tr}_{L/k}$ 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.
MSC classification
- Type
- Research Article
- Information
- Copyright
- © The Author(s) 2021
References
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