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.

Duke Scholars

Author Kirsten Graham Wickelgren Mathematics