An arithmetic count of the lines meeting four lines in 𝐏³

Srinivasan, Padmavathi; Wickelgren, Kirsten

doi:10.1090/tran/8307

An arithmetic count of the lines meeting four lines in $\mathbf {P}^3$
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by Padmavathi Srinivasan and Kirsten Wickelgren; with an appendix by Borys Kadets; Padmavathi Srinivasan, Ashvin A. Swaminathan; Padmavathi Srinivasan, Libby Taylor; Padmavathi Srinivasan, Dennis Tseng PDF

Trans. Amer. Math. Soc. 374 (2021), 3427-3451 Request permission

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 $\mathbf {P}^3_k$, 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 $\mathbf {A}^1$-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms.

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