Wall crossings for double Hurwitz numbers

• DOI: 10.1016/j.aim.2011.06.021
• Publication Date: 2011-07-05T08:16:38Z
• Authors: Renzo Cavalieri; Paul Johnson; Hannah Markwig
• Keywords: Ramification; Hyperplane; Hurwitz polynomial; Routh–Hurwitz stability criterion

AbstractDouble Hurwitz numbers count covers of P1 by genus g curves with assigned ramification profiles over 0 and ∞, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification (Goulden et al., 2005) [10], and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g=0 (Shadrin et al., 2008) [15]. This paper gives a unified approach to these results and strengthens them in several ways — the most important being the extension of the results of Shadrin et al. (2008) [15] to arbitrary genus.The main tool is the authorsʼ previous work (Cavalieri et al., 2010) [6] expressing double Hurwitz number as a sum over certain labeled graphs. We identify the labels of the graphs with lattice points in the chambers of certain hyperplane arrangements, which give rise to piecewise polynomial functions. Our understanding of the wall crossing for these functions builds on the work of Varchenko (1987) [17], and could have broader applications.