Smooth limits of plane curves of prime degree and Markov numbers

DeVleming, Kristin; Stapleton, David

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

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 ${\mathbb{P}}^3$. In this paper, we propose a conjecture explaining the one-dimensional examples using Hacking and Prokhorov’s work on $\mathbb{Q}$-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.

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 ${\mathbb{P}}^3$. Dans cet article, nous proposons une conjecture expliquant les exemples unidimensionnels en utilisant les travaux de Hacking et Prokhorov sur les limites $\mathbb{Q}$-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.

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

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