Moduli of Weierstrass fibrations with marked section

TL;DR

This paper constructs a compactified moduli space for Weierstrass fibrations with marked sections, analyzing boundary objects, extension of fibrations, and wall-crossing phenomena as weights vary.

Contribution

It introduces a DM stack compactification for the moduli of Weierstrass fibrations with marked sections and studies boundary objects and wall-crossing behavior.

Findings

Constructed a DM stack compactification of the moduli space.

Described boundary objects and extension of fibrations.

Identified wall-crossing phenomena as weights change.

Abstract

We study the the moduli space of KSBA stable pairs $(X,sS+\sum a_i F_i)$, consisting of a Weierstrass fibration $X$, its section $S$, and some fibers $F_i$. We find a compactification which is a DM stack, and we describe the objects on the boundary. We show that the fibration in the definition of Weierstrass fibration extends to the boundary, and it is equidimensional when $s \ll 1$. We prove that there are wall-crossing morphisms when the weights $s$ and $a_i$ change. When $s=1$, this recovers the work of La Nave (arXiv:math/0205035); and a special case of the work of Ascher-Bejleri (arXiv:1702.06107).