Publications [#355952] of Kirsten G. Wickelgren

Papers Published

1. Leo Kass, J; Wickelgren, K, An arithmetic count of the lines on a smooth cubic surface, Compositio Mathematica, vol. 157 no. 4 (April, 2021), pp. 677-709 [doi]
   (last updated on 2024/04/25)

   Abstract:
   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.