Genus \(0\) and \(1\) Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations (Q2723462)

Let \(\alpha\) denote a permutation in the symmetric group \(S_d\) and let \(l(\alpha)\) denote the number of cycles in \(\alpha\). The Hurwitz numbers \(G^g_\alpha\) count the number of smooth degree \(d\) covers of the Riemann sphere with ramification above \(\infty\) given by \(\alpha\) (and where the ramification points are labeled), simple branching at \(d+l(\alpha)+2g-2\) other specified points, and no other branching. The computation of Hurwitz numbers is essentially equivalent to counting the number of transitive factorizations of permutations into transpositions. Hurwitz gave a formula for \(G^0_\alpha\), and his ideas have been extended recently by \textit{V. Strehl} [Sémin. Lothar. Comb. 37, B37c (1996; Zbl 0886.05006)], who gave a complete proof.NEWLINENEWLINENEWLINEThe author gives a closed-form expression for all genus 1 Hurwitz numbers. First, divisor theory on the moduli stack of stable maps for degree \(d\) stable maps from genus \(g\) curves with labeled points to the projective line is used to derive recursions for genus 0 and genus 1 Hurwitz numbers. This extends ideas of \textit{R. Pandharipande} [Trans. Am. Math. Soc. 351, No. 4, 1481-1505 (1999; Zbl 0911.14028)]. Then, the author shows that these recursions are also satisfied by the solution to a certain graph-counting problem. Finally, a closed-form solution to the graph-counting problem is obtained. The formula for \(G^1_\alpha\) has also been obtained by \textit{I. P. Goulden} and \textit{D. M. Jackson} [J. Comb. Theory, Ser. A 88, No.2, 246-258 (1999; Zbl 0939.05006)] by combinatorial methods. Some results on higher genus Hurwitz numbers have been obtained by \textit{T. Ekedahl, S. Lando, M. Shapiro}, and \textit{A. Vainshtein} [C. R. Acad. Sci., Paris, Sér. I, Math. 328, No. 12, 1175-1180 (1999; Zbl 0953.14006)], and by \textit{I. P. Goulden, M. Jackson}, and \textit{R. Vakil} [``The Gromov-Witten potential of a point, Hurwitz numbers, and Hodge integrals'', Proc. Lond. Math. Soc., III. Ser. 83, No. 3, 563-581 (2001; Zbl 1074.14520)].