Log canonical models of elliptic surfaces

TL;DR

This paper classifies the log canonical models of elliptic surfaces with sections and marked fibers, analyzing how these models vary with weights, and provides a formula for their canonical bundles, aiding moduli space compactification.

Contribution

It introduces a classification of log canonical models of elliptic surfaces based on weight-dependent wall and chamber decompositions, advancing the minimal model program for these surfaces.

Findings

Describes how log canonical models depend on weights.

Provides a wall and chamber decomposition of weight space.

Gives a generalized formula for the canonical bundle.

Abstract

We give a classification of the log canonical models of elliptic surface pairs consisting of an elliptic fibration, a section, and a weighted sum of marked fibers. In particular, we show how the log canonical models depend on the choice of the weights. We describe a wall and chamber decomposition of the space of weights based on how the log canonical model changes. In addition, we give a generalized formula for the canonical bundle of an elliptic surface with section and marked fibers. This is the first step in constructing compactifcations of moduli spaces of elliptic surfaces using the minimal model program.