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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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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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