Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

An arithmetic count of the lines meeting four lines in $ \mathbf{P}^3$


Authors: Padmavathi Srinivasan and Kirsten Wickelgren; with an appendix by Borys Kadets; Padmavathi Srinivasan, Ashvin A. Swaminathan; Libby Taylor; Dennis Tseng
Journal: Trans. Amer. Math. Soc. 374 (2021), 3427-3451
MSC (2020): Primary 14N15, 14F42; Secondary 55M25
DOI: https://doi.org/10.1090/tran/8307
Published electronically: February 23, 2021
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2020): 14N15, 14F42, 55M25

Retrieve articles in all journals with MSC (2020): 14N15, 14F42, 55M25


Additional Information

Padmavathi Srinivasan
Affiliation: School of Mathematics, University of Georgia, 452 Boyd Graduate Studies, 1023 D. W. Brooks Drive, Athens, Georgia 30602.
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.
Email: kirsten.wickelgren@duke.edu

DOI: https://doi.org/10.1090/tran/8307
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.
Article copyright: © Copyright 2021 American Mathematical Society