Compact moduli spaces of surfaces of general type

TL;DR

This paper introduces the compactification of the moduli space of surfaces of general type, highlighting the role of Mori's minimal model program and exploring boundary geometry and vector bundle classification.

Contribution

It reviews the construction of compact moduli spaces for surfaces of general type, extending previous work to include surfaces with divisors and connecting boundary geometry to vector bundle classification.

Findings

Compactification constructed via Mori's minimal model program.

Boundary geometry linked to vector bundle classification.

Example of the projective plane with a degree d > 3 curve analyzed.

Abstract

We give an introduction to the compactification of the moduli space of surfaces of general type introduced by Koll\'ar and Shepherd-Barron and generalized to the case of surfaces with a divisor by Alexeev. The construction is an application of Mori's minimal model program for 3-folds. We review the example of the projective plane with a curve of degree d > 3. We explain a connection between the geometry of the boundary of the compactification of the moduli space and the classification of vector bundles on the surface in the case H^{2,0}=H^1=0.