Compact moduli of plane curves

Abstract

We construct a compactification $\mathcal{M}$_d of the moduli space of plane curves of degree d. We regard a plane curve C ⊂ℙ² as a surface-divisor pair (ℙ²,C), and we define $\mathcal{M}$_d as a moduli space of pairs (X,D), where X is a degeneration of the plane. We show that, if d is not divisible by 3, the stack $\mathcal{M}$_d is smooth and the degenerate surfaces X can be described explicitly.