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Mathematics, Geometric Topology, Algebraic Geometry, Number Theory
Abstract:
We prove the quasimodularity of generating functions for counting pillowcase
covers, with and without Siegel-Veech weight. Similar to prior work on torus
covers, the proof is based on analyzing decompositions of half-translation
surfaces into horizontal cylinders. It provides an alternative proof of the
quasimodularity results of Eskin-Okounkov and a practical method to compute
area Siegel-Veech constants. A main new technical tool is a quasi-polynomiality
result for 2-orbifold Hurwitz numbers with completed cycles.