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Tropical invariants for binary quintics and reduction types of Picard curves

Part of: General commutative ring theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  06 November 2023

Paul Alexander Helminck ,
Yassine El Maazouz Open the ORCID record for Yassine El Maazouz [Opens in a new window]  and
Enis Kaya
Paul Alexander Helminck*
Affiliation:
Department of Mathematical Sciences, Durham University, Durham, UK
Yassine El Maazouz
Affiliation:
Department of Statistics, University of California Berkeley, Berkeley, CA, USA
Enis Kaya
Affiliation:
Department of Mathematics, KU Leuven, Heverlee, Belgium
*
Corresponding author: Paul Alexander Helminck; Email: yassine.el-maazouz@berkeley.edu
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Abstract

In this paper, we express the reduction types of Picard curves in terms of tropical invariants associated with binary quintics. We also give a general framework for tropical invariants associated with group actions on arbitrary varieties. The problem of finding tropical invariants for binary forms fits in this general framework by mapping the space of binary forms to symmetrized versions of the Deligne–Mumford compactification $\overline{M}_{0,n}$.

Keywords

Tropical geometryInvariant theoryPicard curvesBinary forms

MSC classification

Primary: 13A50: Actions of groups on commutative rings; invariant theory 11G20: Curves over finite and local fields

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 66 , Issue 1 , January 2024 , pp. 65 - 87
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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References

Abramovich, D., Caporaso, L. and Payne, S., The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4) 48(6) (2015), 765–809.CrossRefGoogle Scholar
Achter, J. D. and Pries, R., The integral monodromy of hyperelliptic and trielliptic curves, Math. Ann. 338(1) (2007), 187–206.CrossRefGoogle Scholar
Amini, O., Baker, M., Brugallé, E. and Rabinoff, J., Lifting harmonic morphisms I: metrized complexes and Berkovich skeleta, Res. Math. Sci. 2 (2015), Art.7,67.CrossRefGoogle Scholar
Baker, M., Payne, S. and Rabinoff, J., On the structure of non-Archimedean analytic curves, in Tropical and non-Archimedean geometry, Contemporary Mathematics, vol. 605 (American Mathematical Society, Providence, RI, 2013), 93–121.CrossRefGoogle Scholar
Chan, M., Lectures on tropical curves and their moduli spaces, in Moduli of curves, Lecture Notes of the Unione Matematica Italiana, vol. 21 (Springer, Cham, 2017), 1–26 CrossRefGoogle Scholar
Cléry, F. and van der Geer, G., Generating Picard modular forms by means of invariant theory, Pure Appl. Math. Q. 19(1) (2023), 95–147.CrossRefGoogle Scholar
Derksen, H. and Kemper, G., Computational invariant theory, vol. 130 (Springer, Heidelberg, 2015). Invariant Theory and Algebraic Transformation Groups, VIII.CrossRefGoogle Scholar
Gordan, P., Beweis, dass jede covariante und invariante einer binären form eine ganze function mit numerischen coefficienten einer endlichen anzahl solcher formen ist, J. Reine Angew. Math. 69 (1868), 323–354.Google Scholar
Gubler, W., A guide to tropicalizations, in Algebraic and combinatorial aspects of tropical geometry, Contemporary Mathematics, vol. 589 (American Mathematical Society, Providence, RI, 2013), 125–189.CrossRefGoogle Scholar
Helminck, P. A., Tropical Igusa Invariants (2021). Preprint available at https://arxiv.org/abs/1604.03987 Google Scholar
Helminck, P. A., Invariants for trees of non-archimedean polynomials and skeleta of superelliptic curves, Math. Z. 301 (2022), 1259–1297.CrossRefGoogle Scholar
Helminck, P. A., Skeletal filtrations of the fundamental group of a non-archimedean curve, Adv. Math. 431 (2023), 109242.CrossRefGoogle Scholar
Kaveh, K. and Manon, C., Gröbner theory and tropical geometry on spherical varieties, Transform. Groups 24(4) (2019), 1095–1145.CrossRefGoogle Scholar
Liu, Q., Courbes stables de genre $2$ et leur schéma de modules, Math. Ann. 295(2) (1993), 201–222.CrossRefGoogle Scholar
Maclagan, D. and Sturmfels, B., Introduction to tropical geometry, Graduate Studies in Mathematics, vol. 161 (American Mathematical Society, Providence, RI, 2015).CrossRefGoogle Scholar
Markwig, H., Tropical curves and covers and their moduli spaces, Jahresber. Dtsch. Math.-Ver. 122(3) (2020), 139–166.CrossRefGoogle Scholar
Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, 3rd edition (Springer, Berlin, 1994).CrossRefGoogle Scholar
Payne, S., Analytification is the limit of all tropicalizations, Math. Res. Lett. 16(3) (2009), 543–556.CrossRefGoogle Scholar
Popoviciu Draisma, M. I., Invariants of binary forms, Doctoral Thesis (University of Basel, Faculty of Science, 2014).Google Scholar
Silverman, J. H., The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, 2nd edition (Springer, Dordrecht, 2009).CrossRefGoogle Scholar
Sturmfels, B., Algorithms in invariant theory, Texts and Monographs in Symbolic Computation, 2nd edition (Springer, Vienna, 2008).Google Scholar
The Sage Developers, SageMath, the Sage Mathematics Software System (Version 9.5) (2022).Google Scholar
van der Geer, G., Siegel modular forms of degree two and three and invariant theory, in The art of doing algebraic geometry, Trends in Mathematics (Birkhäuser/Springer, Cham, 2023), 217–239.CrossRefGoogle Scholar
Mathematical Research-Data Repository, Tropical invariants for binary forms and reduction types of Picard curves (2023). https://mathrepo.mis.mpg.de/TropicalInvariantsPicardCurves/index.html Google Scholar