Box Graphs and Resolutions I

Understand

Box graphs succinctly and comprehensively characterize singular fibers of elliptic fibrations in codimension two and three, as well as flop transitions connecting these, in terms of representation theoretic data.

• We develop a framework that provides a systematic map between a box graph and a crepant algebraic resolution of the singular elliptic fibration, thus allowing an explicit construction of the fibers from a singular Weierstrass or Tate model.
• The key tool is what we call a fiber face diagram, which shows the relevant information of a (partial) toric triangulation and allows the inclusion of more general algebraic blowups.
• We shown that each such diagram defines a sequence of weighted algebraic blowups, thus providing a realization of the fiber defined by the box graph in terms of an explicit resolution.