Paper detail

Smooth limits of plane curves of prime degree and Markov numbers

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.