Log canonical models of elliptic surfaces

We classify the log canonical models of elliptic surface pairs (f:X→C,S+F_A) where f:X→C is an elliptic fibration, S is a section, and F_A is 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 compactifications of moduli spaces of elliptic surfaces using the minimal model program.