## 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.## References

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## Additional Information

**Padmavathi Srinivasan**- Affiliation: School of Mathematics, University of Georgia, 452 Boyd Graduate Studies, 1023 D. W. Brooks Drive, Athens, Georgia 30602.
- MR Author ID: 1193003
- Email: Padmavathi.Srinivasan@uga.edu
**Kirsten Wickelgren**- Affiliation: Department of Mathematics, Duke University, 120 Science Drive, Room 117 Physics, Box 90320, Durham, North Carolina 27708-0320.
- MR Author ID: 776836
- Email: kirsten.wickelgren@duke.edu
**Borys Kadets**- MR Author ID: 1159529
- ORCID: 0000-0003-3520-345X
**Dennis Tseng**- MR Author ID: 986806
- ORCID: 0000-0002-7616-2386
- Received by editor(s): October 8, 2018
- Received by editor(s) in revised form: April 27, 2020, and August 25, 2020
- Published electronically: February 23, 2021
- Additional Notes: The second author was partially supported by National Science Foundation Awards DMS-1552730 and DMS-2001890.
- © Copyright 2021 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**374**(2021), 3427-3451 - MSC (2020): Primary 14N15, 14F42; Secondary 55M25
- DOI: https://doi.org/10.1090/tran/8307
- MathSciNet review: 4237952