## An arithmetic count of the lines on a smooth cubic surface

We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field k, generalizing the counts that over C there are 27 lines, and over 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 α in L∗/(L∗)2. There is an equality in the Grothendieck–Witt group GW(k) of k, ∑ TrL/k <α> = 15 · <1> + 12 · <−1>, lines where TrL/k denotes the trace GW(L) → GW(k). Taking the rank and signature recovers the results over C and R. To do this, we develop an elementary theory of the Euler number in A1-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.

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- 0101 Pure Mathematics

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*Compositio Mathematica*,

*157*(4), 677–709. https://doi.org/10.1112/S0010437X20007691

*Compositio Mathematica*157, no. 4 (April 1, 2021): 677–709. https://doi.org/10.1112/S0010437X20007691.

*Compositio Mathematica*, vol. 157, no. 4, Apr. 2021, pp. 677–709.

*Scopus*, doi:10.1112/S0010437X20007691.

## Published In

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## ISSN

## Publication Date

## Volume

## Issue

## Start / End Page

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 0101 Pure Mathematics