An arithmetic count of the lines on a cubic surface

A celebrated 19th century result of Cayley and Salmon is that a
smooth cubic surface over the complex numbers contains exactly 27 lines. By
contrast, over the real numbers, the number of real lines depends on the
surface. A classification was obtained by Segre, but it is a recent
observation of Benedetti-Silhol, Finashin-Kharlamov, Horev-Solomon and
Okonek-Teleman that a certain signed count of lines is always 3. We extend
this count to an arbitrary field k using an Euler number in A1-homotopy
theory. The resulting count is valued in the Grothendieck-Witt group of
non-degenerate symmetric bilinear forms. (No knowledge of A1-homotopy theory will be assumed in the talk.) This is joint work with Jesse Kass.