Compact moduli of elliptic K3 surfaces

TL;DR

This paper constructs modular compactifications of elliptic K3 surface moduli spaces using minimal model program techniques, describing boundary surfaces and relating them to classical compactifications.

Contribution

It introduces new modular compactifications of elliptic K3 surfaces and explicitly characterizes the boundary surfaces and their relation to existing compactifications.

Findings

Constructed various modular compactifications using minimal model program

Described boundary surfaces parametrized by these compactifications

Established morphisms to the Satake-Baily-Borel compactification

Abstract

We construct various modular compactifications of the space of elliptic K3 surfaces using tools from the minimal model program, and explicitly describe the surfaces parametrized by their boundaries. The coarse spaces of our constructed compactifications admit morphisms to the Satake-Baily-Borel compactification. Finally, we show that one of our spaces is smooth with coarse space the GIT quotient of pairs of Weierstrass K3 surfaces with a chosen fiber.