Siegel-Veech Constants for Cyclic Covers of Generic Translation Surfaces
Abstract
We compute the asymptotic number of cylinders, weighted by their area to any non-negative power, on any cyclic branched cover of any generic translation surface in any stratum. Our formulas depend only on topological invariants of the cover and number-theoretic properties of the degree: in particular, the ratio of the related Siegel-Veech constants for the locus of covers and for the base stratum component is independent of the number of branch values. One surprising corollary is that this ratio for Siegel-Veech constants is always equal to the reciprocal of the degree of the cover. A key ingredient is a classification of the connected components of certain loci of cyclic branched covers.
Keywords
Cite
@article{arxiv.2409.06600,
title = {Siegel-Veech Constants for Cyclic Covers of Generic Translation Surfaces},
author = {David Aulicino and Aaron Calderon and Carlos Matheus and Nick Salter and Martin Schmoll},
journal= {arXiv preprint arXiv:2409.06600},
year = {2025}
}
Comments
65 pages, 8 figures, minor revisions following referee report