Smooth limits of plane curves of prime degree and Markov numbers

Kristin DeVleming; David Stapleton

doi:10.5802/jep.263

Smooth limits of plane curves of prime degree and Markov numbers
[Limites lisses de courbes planes de degré premier et nombres de Markov]

Kristin DeVleming ¹ ; David Stapleton ²

¹ Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA
² Mathematics Department, University of Michigan, Ann Arbor, MI 48109, USA

Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 683-731.

Résumé
Abstract

En dimension 3 au moins, Mori a demandé si toute limite lisse propre d’une famille d’hypersurfaces de degré premier est toujours une hypersurface. En dimensions 1 et 2, ce n’est pas le cas. Par exemple, il est bien connu que les courbes planes quintiques peuvent dégénérer en des courbes hyperelliptiques, et Horikawa a construit des limites lisses de surfaces quintiques qui ne se plongent pas dans $ℙ^{3}$ . Dans cet article, nous proposons une conjecture expliquant les exemples unidimensionnels en utilisant les travaux de Hacking et Prokhorov sur les limites $ℚ$ -Gorenstein du plan projectif et nous prouvons la conjecture pour le degré $5$ . Comme conséquence de la première conjecture, nous conjecturons que, si $p$ est un nombre premier qui n’est pas un nombre de Markov, toute limite projective lisse de courbes planes de degré $p$ est une courbe plane. Nous prouvons cette conjecture pour les courbes de degré 7 et justifions la conjecture en exhibant des limites lisses non planes de familles de courbes de degré $d$ pour tout entier $d$ qui est un multiple d’un nombre de Markov.

In dimension at least 3, Mori asked if every smooth proper limit of a family of prime degree hypersurfaces is still a hypersurface. In dimensions 1 and 2, this is not the case. For example, it is well known that quintic plane curves can degenerate to hyperelliptic curves, and Horikawa constructed smooth limits of quintic surfaces that do not embed in $ℙ^{3}$ . In this paper, we propose a conjecture explaining the one-dimensional examples using Hacking and Prokhorov’s work on $ℚ$ -Gorenstein limits of the projective plane and prove the conjecture for degree $5$ . As a consequence of the first conjecture, we conjecture that, if $p$ is a prime number that is not a Markov number, any smooth projective limit of plane curves of degree $p$ is a plane curve. We prove this conjecture for degree $7$ curves and provide evidence for the conjecture by exhibiting non-planar smooth limits of families of degree $d$ curves for any $d$ that is a multiple of a Markov number.

Reçu le : 2022-11-14
Accepté le : 2024-05-31
Publié le : 2024-06-28

DOI : 10.5802/jep.263

Classification : 14H10, 14H50, 14J10
Keywords: Moduli of curves, hypersurfaces, moduli of stable pairs, Markov numbers
Mot clés : Espaces de modules de courbes, hypersurfaces, espaces de modules de paires stables, nombres de Markov

Affiliations des auteurs :

Kristin DeVleming ¹ ; David Stapleton ²

¹ Department of Mathematics, University of Massachusetts, Amherst, MA 01003, USA
² Mathematics Department, University of Michigan, Ann Arbor, MI 48109, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Kristin DeVleming; David Stapleton. Smooth limits of plane curves of prime degree and Markov numbers. Journal de l’École polytechnique — Mathématiques, Tome 11 (2024), pp. 683-731. doi : 10.5802/jep.263. https://jep.centre-mersenne.org/articles/10.5802/jep.263/

Bibliographie
Cité par

[1] K. Ascher, D. Bejleri, G. Inchiostro & Z. Patakfalvi - “Wall crossing for moduli of stable log pairs”, Ann. of Math. (2) 198 (2023) no. 2, p. 825-866 | DOI | Zbl

[2] K. Ascher, K. DeVleming & Y. Liu - “Wall crossing for K‐moduli spaces of plane curves”, Proc. London Math. Soc. (3) 128 (2024), article ID e12615 | DOI

[3] J. Fernández de Bobadilla, I. Luengo, A. Melle Hernández & A. Némethi - “Classification of rational unicuspidal projective curves whose singularities have one Puiseux pair”, in Real and complex singularities, Trends Math., Birkhäuser, Basel, 2007, p. 31-45 | DOI | Zbl

[4] M. Borodzik & C. Livingston - “Heegaard Floer homology and rational cuspidal curves”, Forum Math. Sigma 2 (2014), article ID e28, 23 pages | DOI | Zbl

[5] E. Brieskorn - “Ein Satz über die komplexen Quadriken”, Math. Ann. 155 (1964), p. 184-193 | DOI | Zbl

[6] E. Brieskorn & H. Knörrer - Plane algebraic curves, Modern Birkhäuser Classics, Birkhäuser/Springer Basel AG, Basel, 1986 | DOI

[7] K. DeVleming - “Moduli of surfaces in $ℙ^{3}$ ”, Compositio Math. 158 (2022) no. 6, p. 1329-1374 | DOI | Zbl

[8] D. Ferrand - “Conducteur, descente et pincement”, Bull. Soc. math. France 131 (2003) no. 4, p. 553-585 | DOI | Numdam | Zbl

[9] T. Fujita - Classification theories of polarized varieties, London Math. Soc. Lect. Note Series, vol. 155, Cambridge University Press, Cambridge, 1990 | DOI

[10] E. E. Griffin - “Families of quintic surfaces and curves”, Compositio Math. 55 (1985) no. 1, p. 33-62 | Numdam | Zbl

[11] P. Hacking - “Compact moduli of plane curves”, Duke Math. J. 124 (2004) no. 2, p. 213-257 | DOI | Zbl

[12] P. Hacking & Y. Prokhorov - “Smoothable del Pezzo surfaces with quotient singularities”, Compositio Math. 146 (2010) no. 1, p. 169-192 | DOI | Zbl

[13] B. Hassett - “Stable log surfaces and limits of quartic plane curves”, Manuscripta Math. 100 (1999) no. 4, p. 469-487 | DOI | Zbl

[14] E. Horikawa - “On deformations of quintic surfaces”, Proc. Japan Acad. 49 (1973), p. 377-379 | Zbl

[15] J.-M. Hwang - “Nondeformability of the complex hyperquadric”, Invent. Math. 120 (1995) no. 2, p. 317-338 | DOI | Zbl

[16] B. V. Karpov & D. Y. Nogin - “Three-block exceptional sets on del Pezzo surfaces”, Izv. Ross. Akad. Nauk Ser. Mat. 62 (1998) no. 3, p. 3-38 | DOI

[17] S. Kobayashi & T. Ochiai - “Characterizations of complex projective spaces and hyperquadrics”, J. Math. Kyoto Univ. 13 (1973), p. 31-47 | DOI | Zbl

[18] J. Kollár - “Maps between local Picard groups”, Algebraic Geom. 3 (2016) no. 4, p. 461-495 | DOI | Zbl

[19] J. Kollár - Families of varieties of general type, Cambridge Tracts in Math., vol. 231, Cambridge University Press, Cambridge, 2023 | DOI | MR

[20] J. Kollár & S. Mori - Birational geometry of algebraic varieties, Cambridge Tracts in Math., vol. 134, Cambridge University Press, Cambridge, 1998 | DOI

[21] T. Liu - On planar rational cuspidal curves, Ph. D. Thesis, M.I.T., Cambridge, 2014, https://dspace.mit.edu/handle/1721.1/90190

[22] M. Manetti - “Normal degenerations of the complex projective plane”, J. reine angew. Math. 419 (1991), p. 89-118 | DOI | Zbl

[23] A. Markoff - “Sur les formes quadratiques binaires indéfinies”, Math. Ann. 17 (1880) no. 3, p. 379-399 | DOI | Zbl

[24] T. Matsuoka & F. Sakai - “The degree of rational cuspidal curves”, Math. Ann. 285 (1989) no. 2, p. 233-247 | DOI | Zbl

[25] T. K. Moe - Rational cuspidal curves, Ph. D. Thesis, Universitetet i Oslo, 2008, https://www.duo.uio.no/handle/10852/10759

[26] S. Mori - “On a generalization of complete intersections”, J. Math. Kyoto Univ. 15 (1975) no. 3, p. 619-646 | DOI | Zbl

[27] S. Y. Orevkov - “On rational cuspidal curves. I. Sharp estimate for degree via multiplicities”, Math. Ann. 324 (2002) no. 4, p. 657-673 | DOI | Zbl

[28] J. C. Ottem & S. Schreieder - “On deformations of quintic and septic hypersurfaces”, J. Math. Pures Appl. (9) 135 (2020), p. 140-158 | DOI | Zbl

[29] N. J. A. Sloane & The OEIS Foundation Inc. - “The on-line encyclopedia of integer sequences, Seq. A178444”, 2020, https://oeis.org/A178444

[30] Stacks Project Authors - “The Stacks Project”, 2019, https://stacks.math.columbia.edu

[31] D. Zagier - “On the number of Markoff numbers below a given bound”, Math. Comp. 39 (1982) no. 160, p. 709-723 | DOI | Zbl

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