Triangulations of branched affine surfaces

TL;DR

This paper proves a foundational decomposition theorem for branched affine surfaces into affine triangles and cylinders, shows these decompositions are connected by flips, and explores compactifications of their moduli spaces.

Contribution

It provides the first proof of Veech's decomposition theorem, introduces an invariant for degeneracy control, and studies compactifications of moduli spaces.

Findings

Any branched affine surface decomposes into affine triangles and cylinders.

Decompositions can be connected by a chain of flips.

Introduces an invariant controlling structure degeneracy.

Abstract

A branched affine structure on a compact topological surface with marked points is a complex affine structure outside the marked points. We give a proof of an unpublished foundational theorem of Veech, stating that any branched affine surface can be decomposed into affine triangles and some annulus-shaped cylinders. Then, we prove that any pair of such decompositions can be connected by a chain of flips. As a first step toward a compactification of the moduli spaces of branched affine structures, we introduce invariant $\alpha$ of a branched affine surface that controls degeneracy of the structure. Finally, we consider some examples of compactification of spaces of branched affine surfaces by stable curves.