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

Abstract

In this article, we construct an infinite sequence of irreducible components of Kollár--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.

40 pages, 2 figures; comments are very welcome! v2: The final section of v1 has been split into two separate sections with additional details added. This version also answers a question regarding the non-emptiness of certain strata and fixes a gap in the proof of the isomorphism between stacks using a wall-crossing argument