Moduli of Weierstrass fibrations with marked section

Abstract

We study the moduli space of stable pairs (X,sS+∑aiFi) consisting of a Weierstrass fibration X, its section S, and some fibers Fi. 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≪1. We prove that there are wall-crossing morphisms when the weights s and ai change. When s=1, this recovers the work of La Nave [36]; and a special case of the work of Ascher-Bejleri [11].