An arithmetic count of the lines meeting four lines in 𝐏³

Abstract

We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field k k , this enrichment counts the number of lines meeting four lines defined over k k in P k 3 \mathbf {P}^3_k , with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in A 1 \mathbf {A}^1 -homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms.