Smooth limits of plane curves of prime degree and Markov numbers

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.