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 arbitrary field $k$ , generalizing counts that ${\mathbf {C}}$ there are $27$ lines, and {R}}$ number hyperbolic minus elliptic is $3$ . In general, defined extension $L$ have associated type $\alpha$ in $L^*/(L^*)^2$ There equality Grothendieck–Witt group $\operatorname {GW}(k)$ \[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot 1 + 12 -1 \rangle, \] where {Tr}_{L/k}$ denotes trace {GW}(L) \to \operatorname Taking rank signature recovers results To do this, we develop elementary theory Euler $\mathbf {A}^1$ -homotopy for algebraic vector bundles. expect further enumerative complex real geometry can be obtained with similar methods.