# The Dual Complex of a semi-log canonical Surface

## TL;DR

This paper explores the topology of dual complexes associated with semi-log canonical surfaces, providing methods to compute their PL homeomorphism types from normalization data, advancing understanding in moduli theory.

## Contribution

It introduces a technique to determine the PL homeomorphism type of dual complexes for certain three-fold pairs directly from normalization data.

## Key findings

- Computed the PL homeomorphism type of dual complexes from normalization data.
- Linked dual complex topology to boundary normalization in semi-log canonical surfaces.
- Enhanced tools for studying limits of canonical models in moduli spaces.

## Abstract

Semi-log canonical varieties are a higher-dimensional analogue of stable curves. They are the varieties appearing as the boundary $\Delta$ of a log canonical pair $(X,\Delta)$, and also appear as limits of canonically polarized varieties in moduli theory. For certain three-fold pairs $(X,\Delta)$ we show how to compute the PL homeomorphism type of the dual complex of a dlt minimal model directly from the normalization data of $\Delta$.