Moduli of elliptic surfaces of Kodaira dimension one fibered over rational curves

TL;DR

This paper constructs and analyzes infinite families of moduli spaces of elliptic surfaces with Kodaira dimension one, providing explicit descriptions of their boundary components and stable reduction processes.

Contribution

It introduces a new approach to studying KSB moduli spaces of elliptic surfaces via wall-crossing and twisted stable maps, revealing infinite irreducible components.

Findings

Constructed infinite irreducible components of KSB moduli spaces.

Explicit description of boundary components of these moduli spaces.

Described stable reduction steps combinatorially.

Abstract

In this article, we construct an infinite sequence of irreducible components of Koll\'{a}r--Shepherd-Barron (KSB-) moduli spaces of surfaces of arbitrarily large volumes, and describe the boundary of each component completely. Moreover, we describe the stable reduction steps in finding the KSB-limits in an explicit combinatorial way. Our main approach is to study the moduli spaces of elliptic surfaces with Kodaira dimension one, fibered over rational curves, using the techniques of wall-crossing for KSBA moduli and twisted stable maps.