Stable log surfaces and limits of quartic plane curves

Abstract:

Let C⊂ℙ² be a smooth plane curve of degree d≥ 4. We regard pairs (ℙ}²,C) as stable log surfaces, higher-dimensional analogs to pointed stable curves. Using the log minimal model program, Kollár, Shepherd-Barron, and Alexeev have constructed projective moduli spaces for stable log surfaces. Unfortunately, few explicit examples of these moduli spaces are known. The purpose of this paper is to give a concrete description of these spaces for plane curves of small degree. In particular, we show that the moduli space of stable log surfaces corresponding to quartic plane curves coincides with the moduli space of stable curves of genus three.