A holomorphic 1-form on a compact Riemann surface S naturally defines a flat metric on S with cone-type singularities. We present the following surprising phenomenon: having found a geodesic segment (saddle connection) joining a pair of conical points one can find with a nonzero probability another saddle connection on S having the same direction and the same length as the initial one. A similar phenomenon is valid for the families of parallel closed geodesics. We give a complete description of all possible configurations of parallel saddle connections (and of families of parallel closed geodesics) which might be found on a generic flat surface S. We count the number of saddle connections of length less than L on a generic flat surface S; we also count the number of admissible configurations of pairs (triples,...) of saddle connections; we count the analogous numbers of configurations of families of closed geodesics. By the previous result of [EMa] these numbers have quadratic asymptotics $c\xb7\left(\pi {\mathrm{L}}^{2}\right)$. Here we explicitly compute the constant $c$ for a configuration of every type. The constant $c$ is found from a Siegel-Veech formula. To perform this computation we elaborate the detailed description of the principal part of the boundary of the moduli space of holomorphic 1-forms and we find the numerical value of the normalized volume of the tubular neighborhood of the boundary. We use this for evaluation of integrals over the moduli space.

@article{PMIHES_2003__97__61_0, author = {Eskin, Alex and Masur, Howard and Zorich, Anton}, title = {Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants}, journal = {Publications Math\'ematiques de l'IH\'ES}, publisher = {Springer}, volume = {97}, year = {2003}, pages = {61-179}, doi = {10.1007/s10240-003-0015-1}, zbl = {1037.32013}, language = {en}, url = {http://www.numdam.org/item/PMIHES_2003__97__61_0} }

Moduli spaces of abelian differentials : the principal boundary, counting problems, and the Siegel-Veech constants. Publications Mathématiques de l'IHÉS, Volume 97 (2003) pp. 61-179. doi : 10.1007/s10240-003-0015-1. http://www.numdam.org/item/PMIHES_2003__97__61_0/

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