Smooth limits of plane curves of prime degree and Markov numbers

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.