Compactifications of moduli of del Pezzo surfaces via line arrangement and K-stability

TL;DR

This paper explores how to compactify the moduli space of smooth del Pezzo surfaces using K-stability and line arrangements, revealing isomorphisms for degrees 2, 3, 4, and differences for degree 1.

Contribution

It constructs K-moduli spaces for log del Pezzo pairs with boundary divisors and identifies isomorphisms and wall-crossings for different degrees.

Findings

K-moduli spaces for degrees 2, 3, 4 are isomorphic to those of del Pezzo surfaces

For degree 1, the K-moduli spaces differ due to wall-crossings

The approach uses line arrangements and K-stability to study compactifications

Abstract

In this paper, we study compactifications of the moduli of smooth del Pezzo surfaces using K-stability and the line arrangement. We construct K-moduli of log del Pezzo pairs with sum of lines as boundary divisors, and prove that for $d=2,3,4$, these K-moduli of pairs are isomorphic to the K-moduli spaces of del Pezzo surfaces. For $d=1$, we prove that they are different by exhibiting some walls.