Triangulations of branched affine surfaces

Guillaume Tahar

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arXiv · 1912.00149

Triangulations of branched affine surfaces

Guillaume Tahar

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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 We give a proof of an unpublished foundational theorem of Veech, stating that any branched affine surface can be decomposed into affine triangles and some annulus-shaped cylinders.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.

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Guillaume Tahar. 2019-11-30. Triangulations of branched affine surfaces. https://arxiv.org/abs/1912.00149

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