An arithmetic count of the lines meeting four lines in P^3

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 $k$, this enrichment counts the number of lines meeting four lines defined over $k$ in ${\mathbb{P}}_k^3$, 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 ${\mathbb{A}}^1$-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms. In the appendix, the condition that the four lines each be defined over $k$ is relaxed to the condition that the set of four lines being defined over $k$.

Accepted for publication in Transactions of the AMS

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Cited by in corpus (4)

• Quadratic types and the dynamic Euler number of lines on a quintic threefold
• Bézoutians and the ${\mathbb{A}}^1$-degree
• Conics meeting eight lines over perfect fields
• Quadratic counts of highly tangent lines to hypersurfaces