K-moduli of log del Pezzo pairs and variations of GIT

TL;DR

This paper explores the moduli spaces of certain algebraic surfaces called log del Pezzo pairs, revealing their structure and relationships with GIT variations for degrees 2, 3, and 4, thus advancing understanding of K-stability and moduli theory.

Contribution

It establishes isomorphisms between K-moduli spaces of log del Pezzo pairs and GIT variations for degrees 2, 3, and 4, generalizing previous absolute case results.

Findings

K-moduli spaces are isomorphic to GIT variations for degrees 2, 3, 4

Provides a natural parameter-dependent framework for these moduli spaces

Generalizes known results from absolute cases to log pairs

Abstract

We study the K-moduli of log del Pezzo pairs formed by a del Pezzo surface of degree $d$ and an anti-canonical divisor. These moduli spaces naturally depend on one parameter, providing a natural problem in variations of K-moduli spaces. For degrees 2, 3, 4, we establish an isomorphism between the K-moduli spaces and variations of Geometric Invariant Theory compactifications, which generalizes the isomorphisms in the absolute cases established by Odaka--Spotti--Sun and Mabuchi--Mukai.