KSBA moduli compactifications and weighted stable hyperplane arrangements

We begin with an overview of the framework of KSBA (Kollár—Shepherd-Barron—Alexeev) moduli spaces and stable degenerations, emphasizing some phenomena that distinguish the higher-dimensional cases from the Deligne–Mumford moduli of stable curves. We then survey several existing explicit constructions of KSBA compactifications. As a detailed example, we discuss the construction of KSBA weighted stable hyperplane arrangements via geometric invariant theory (GIT) and matroid tilings due to Valery Alexeev. In particular, here GIT is applied on the construction of the family/stable objects rather than on the construction of the compactified moduli space. Finally, we discuss our ongoing joint work with Hanlong Fang, Patricio Gallardo, and Luca Schaffler on an explicit KSBA wall-crossing morphism to the Losev–Manin space.