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An enriched count of the bitangents to a smooth plane quartic curve

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Abstract

Recent work of Kass–Wickelgren gives an enriched count of the 27 lines on a smooth cubic surface over arbitrary fields. Their approach using \(\mathbb {A}^1\)-enumerative geometry suggests that other classical enumerative problems should have similar enrichments, when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. Here, we consider the closely related problem of the 28 bitangents to a smooth plane quartic. However, it turns out that the relevant vector bundle is not relatively orientable and new ideas are needed to produce enriched counts. We introduce a fixed “line at infinity,” which leads to enriched counts of bitangents that depend on their geometry relative to the quartic and this distinguished line.

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References

  1. Benedetti, R., Silhol, R.: \({\rm Spin}\) and \({\rm Pin}^-\) structures, immersed and embedded surfaces and a result of Segre on real cubic surfaces. Topology 34(3), 651–678 (1995)

    Article  MathSciNet  Google Scholar 

  2. Caporaso, L., Sernesi, E.: Recovering plane curves from their bitangents. J. Algebraic Geom. 12(2), 225–244 (2003)

    Article  MathSciNet  Google Scholar 

  3. Eisenbud, D., Harris, J.: 3264 and all that—A Second Course in Algebraic Geometry. Cambridge University Press, Cambridge (2016)

    Book  Google Scholar 

  4. Finashin, S., Kharlamov, V.: Abundance of real lines on real projective hypersurfaces. Int. Math. Res. Not. IMRN 16, 3639–3646 (2013)

    Article  MathSciNet  Google Scholar 

  5. Jacobi, C.G.J.: Beweis des Satzes dass eine Curve n\(^{ten}\) Grades im Allgemeinen \(1/2n(n-2)(n^2-9)\) Doppeltangenten hat. J. Reine Angew. Math. 40, 237–260 (1850)

    MathSciNet  Google Scholar 

  6. Kass, J., Wickelgren, K.: An arithmetic count of lines on a smooth cubic surface. Compositio Mathematica, to appear

  7. Klein, F.: Eine neue Relation zwischen den Singularitäten einer algebraischen Curve. Math. Ann. 10(2), 199–209 (1876)

    Article  MathSciNet  Google Scholar 

  8. Kollár, J.: Rational curves on algebraic varieties, volume 32 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Berlin (1996)

  9. Kummer, M.: Totally real theta characteristics. Ann. Mat. Pura Appl. 198(6), 2141–2150 (2019)

    Article  MathSciNet  Google Scholar 

  10. Larson, H., Vogt, I.: Sage implementation of Algorithm 4.2. https://github.com/ivogt161/RealBitangents, (2019)

  11. McKean, Stephen.: An arithmetic enrichment of Bézout’s theorem. Mathematische Annalen, to appear

  12. Milnor, J.W.: Topology from the differentiable viewpoint. Based on notes by David W. Weaver. The University Press of Virginia, Charlottesville, Va., (1965)

  13. Okonek, C., Teleman, A.: Intrinsic signs and lower bounds in real algebraic geometry. J. Reine Angew. Math. 688, 219–241 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Plaumann, D., Sturmfels, B., Vinzant, C.: Quartic curves and their bitangents. J. Symbol. Comput. 46(6), 712–733 (2011)

    Article  MathSciNet  Google Scholar 

  15. Reichstein, Z.B.: On a property of real plane curves of even degree. Can. Math. Bull. 62(1), 179–182 (2019)

    Article  MathSciNet  Google Scholar 

  16. Ronga, F.: Felix Klein’s paper on real flexes vindicated. In: Singularities Symposium—Lojasiewicz 70 (Kraków, 1996; Warsaw, 1996), volume 44 of Banach Center Publ., pp. 195–210. Polish Acad. Sci. Inst. Math., Warsaw, (1998)

  17. Salmon, G.: A treatise on the higher plane curves: intended as a sequel to “A treatise on conic sections,” 3rd edn. Chelsea Publishing Co., New York (1960)

  18. Viro, O.Ya.: Some integral calculus based on Euler characteristic. In: Topology and geometry—Rohlin Seminar, volume 1346 of Lecture Notes in Math., pp. 127–138. Springer, Berlin, (1988)

  19. Wall, C.T.C.: Duality of real projective plane curves: Klein’s equation. Topology 35(2), 355–362 (1996)

    Article  MathSciNet  Google Scholar 

  20. Wall, C.T.C.: Singular points of plane curves. London Mathematical Society Student Texts, vol. 63. Cambridge University Press, Cambridge (2004)

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Acknowledgements

Thanks to Jesse Kass and Kirsten Wickelgren for many insightful conversations, comments on several drafts of this article, and for advising the \(\mathbb {A}^1\)-enumerative geometry problem session at the 2019 Arizona Winter School. We are grateful to the organizers, funders, and other participants—in particular Ethan Cotterill, Ignacio Darago, and Changho Han—of the Winter School for fostering the stimulating environment that inspired this work. We also thank Sam Payne and Hannah Markwig for introducing us to the grouping of bitangents by avoidance locus, which is a key idea in the proof of Proposition 4.3.

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  1. Stanford University, Stanford, USA

    Hannah Larson

  2. University of Washington, Seattle, USA

    Isabel Vogt

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  1. Hannah Larson
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  2. Isabel Vogt
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Correspondence to Isabel Vogt.

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IV was supported by a NSF GRFP and MSPRF under Grants DGE-1122374 and DMS-1902743. HL was supported by the Hertz Foundation and NSF GRFP under Grant DGE-1656518.

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Larson, H., Vogt, I. An enriched count of the bitangents to a smooth plane quartic curve. Res Math Sci 8, 26 (2021). https://doi.org/10.1007/s40687-021-00260-9

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