A guide to wall crossing for moduli of varieties

TL;DR

This paper provides an accessible overview of wall crossing phenomena in the moduli spaces of varieties, highlighting recent developments, tools, and explicit examples in the context of log pairs and stability conditions.

Contribution

It offers a semi-expository introduction to wall crossing theory in moduli of varieties, including new examples and computational techniques.

Findings

Summarizes recent advances in moduli space construction.

Introduces tools for explicit wall crossing computations.

Provides new examples illustrating wall crossing phenomena.

Abstract

There have been major developments in the theory of moduli of varieties in the past decade, essentially settling the construction of moduli spaces of log canonically polarized slc pairs and moduli spaces of K-polystable log Fano pairs. Given the construction of these moduli spaces of pairs $(X, D)$, it is natural to ask how the moduli spaces vary as the coefficients of $D$ are perturbed. This phenomenon is known as wall crossing, the theory of which has been developed in several important cases in the past five years. This semi-expository article is an introduction to moduli of varieties and wall crossing, capturing a portion of the theory developed in the past several years. It also introduces tools and techniques used in explicit computations and examples, applying them in new examples.