Smooth limits of plane curves of prime degree and Markov numbers

TL;DR

This paper investigates the limits of smooth plane curves of prime degree, proposing a conjecture that relates the behavior to Markov numbers, and proves it for degree 7 while extending counterexamples to all Markov primes.

Contribution

It introduces a conjecture linking Markov numbers to limits of plane curves and proves it for degree 7, extending known counterexamples to all Markov prime degrees.

Findings

Proved the conjecture for degree 7 curves.

Extended Griffin's counterexamples to all Markov prime degrees.

Suggested a connection between Markov numbers and limits of plane curves.

Abstract

For prime degree hypersurfaces of dimension at least 3, Mori asked if every smooth proper limit is still a hypersurface. Interestingly in dimensions 1 and 2, this is not the case. For example, Griffin constructed explicit families of quintic curves that give counterexamples in dimension 1 (Horikawa constructed similar examples for quintics in dimension 2). In this paper we propose a conjecture explaining these examples using Hacking and Prokhorov's work on Q-Gorenstein limits of the projective plane. In particular, if p is a prime number that is not a Markov number, we conjecture that any smooth projective limit of plane curves of degree p is a plane curve. The main results are to prove this conjecture for degree 7 curves and to extend Griffin's counterexamples to all prime numbers that are also Markov numbers.