Towards the geometry of double Hurwitz numbers.

*(English)*Zbl 1086.14022Hurwitz numbers count the number of connected coverings of the Riemann sphere of given genus and degree, with prescribed ramification over \(\infty\) and simple branching at all other branch points. Their study goes back to A. Hurwitz [Math. Ann. XXXIX. 1–61 (1891; JFM 23.0429.01)]. The most important advance in the recent study of Hurwitz numbers has been the so-called ELSV formula (proved in [T. Ekedahl, S. Lando, M. Shapiro and A. Vainstein, Invent. Math. 146, No. 2, 297–327 (2001; Zbl 1073.14041)]), which expresses Hurwitz numbers as integrals of tautological classes over the moduli space of stable curves with marked points.

The paper under review goes a step further and deals with connected coverings with prescribed ramification over two points. Specifically, consider all possible connected coverings of genus \(g\) and degree \(d\) of the Riemann sphere, with fixed branch locus containing \(0\) and \(\infty\). Suppose that the branching over \(0\) and \(\infty\) is specified, respectively, by a partition \(\alpha\) and a partition \(\beta\) of \(d\), and that the branching at all other branch points is simple. Then the authors define the double Hurwitz number \(H^g_{\alpha,\beta}\) as the number of such covers, together with a labelling of the branch points over \(0\) and \(\infty\) (hence, the integer \(H^g_{\alpha,\beta}\) is always a multiple of \(| \text{Aut}(\alpha)| | \text{Aut}(\beta)| \)). The aim of this paper is to investigate the structure of double Hurwitz numbers. In doing this, the authors are guided by the intuition that they should be related by an analogue of the ELSV formula to the intersection theory on a suitable compactification of the universal Picard variety, i.e., a moduli space of curves of genus \(g\) with \(n\) marked points and a given line bundle. This compactified Picard variety is expected to be related to the one constructed in [L. Caporaso, J. Am. Math. Soc. 7, No. 3, 589–660 (1994; Zbl 0827.14014)]. Moreover, the authors announce that in a later paper they will exploit the structure of genus 0 double Hurwitz numbers to understand top intersections on the moduli space of smooth curves and to prove the intersection number conjecture in [C. Faber, in: Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Aspects Math. E33, 109–129 (1999; Zbl 0978.14029)], for every genus and for up to three marked points.

If we fix the length \(m\) (respectively, \(n\)) of \(\alpha\), resp., \(\beta\), and we write \(\alpha=(\alpha_1,\dots,\alpha_m)\), \(\beta=(\beta_1,\dots,\beta_n)\), the double Hurwitz numbers \(H^g_{\alpha,\beta}\) define a function in the \(\alpha_i\), \(\beta_j\). Using ribbon graphs, the authors prove that the functions so obtained are piecewise polynomial of degree \(\leq 4g-3+m+n\). The proof is based on an interpretation of double Hurwitz numbers as the number of lattice points of certain polytopes, such that the position of the faces depends on \(\alpha\) and \(\beta\). In the special case \(g=0\), the authors prove that the degree of the polynomials is bounded below by \(2g-3+m+n\); they conjecture that this should hold also for positive genus. By considering the example \((g,m,n)=(0,2,2)\), the authors show that double Hurwitz numbers are not polynomial in general.

The central part of the paper deals with the case of covers with complete ramification over \(0\), i.e., double Hurwitz numbers of the form \(H^g_{(d),\beta}\), called here one-part double Hurwitz numbers. Using techniques from algebra, geometry and representation theory, the authors give explicit formulas (Theorem 3.1) for these Hurwitz numbers, which imply, among other things, that they are polynomials in the parts of the partition \(\beta\). Furthermore, the authors conjecture the form of an analogue of the ELSV formula for one-part Hurwitz numbers, expressing them as integrals of some (undefined as yet) “geometrically natural classes” on a suitable compactification of the Picard variety. The proprieties which the compactified Picard variety and the classes of it should satisfy are listed. For genus \(0\) and genus \(1\) the conjecture is proved, by choosing the compactified Picard variety to be, respectively, \(\overline{{\mathcal M}}_{0,n}\) and \({\overline{\mathcal M}_{1,n+1}}\). The authors also define a symbol \(\langle\langle \cdot \rangle\rangle_g\), which, in view of the conjecture, is expected to be the analogue of Witten’s correlation function \(\langle\cdot\rangle_g\). Indeed, the symbol \(\langle\langle\cdot\rangle\rangle_g\) satisfies properties similar to those of \(\langle\cdot\rangle_g\). In particular, the authors prove that the generating series of \(\langle\langle\cdot\rangle\rangle_g\) satisfies the string and dilaton equation, and an analogue of Itzykson-Zuber’s ansatz for intersection numbers on the moduli space of curves ((5.32) in [Int. J. Mod. Phys. A 7, No. 23, 5661–5705 (1992; Zbl 0972.14500)]). Moreover, they give analogues of the formulas in [C. Faber, R. Pandharipande, Ann. Math. (2) 157, No. 1, 97–124 (2003; Zbl 1058.14046)].

Next, the authors investigate the generating function of double Hurwitz numbers and express it in terms of Schur symmetric functions. As a corollary, one gets formulas for \(H^g_{\alpha,\beta}\) as a function of \(g\), when \(\alpha\) and \(\beta\) are fixed. This extends results of Kuleshov and Shapiro for covers of degree \(3,4,5\). Finally, when the length \(m\) of the partition \(\alpha\) is fixed, the authors define a symmetrized generating function for these numbers, and prove that it satisfies a topological recursion relation. This leads to closed expressions for the generating function for small \((g,m)\).

The paper under review goes a step further and deals with connected coverings with prescribed ramification over two points. Specifically, consider all possible connected coverings of genus \(g\) and degree \(d\) of the Riemann sphere, with fixed branch locus containing \(0\) and \(\infty\). Suppose that the branching over \(0\) and \(\infty\) is specified, respectively, by a partition \(\alpha\) and a partition \(\beta\) of \(d\), and that the branching at all other branch points is simple. Then the authors define the double Hurwitz number \(H^g_{\alpha,\beta}\) as the number of such covers, together with a labelling of the branch points over \(0\) and \(\infty\) (hence, the integer \(H^g_{\alpha,\beta}\) is always a multiple of \(| \text{Aut}(\alpha)| | \text{Aut}(\beta)| \)). The aim of this paper is to investigate the structure of double Hurwitz numbers. In doing this, the authors are guided by the intuition that they should be related by an analogue of the ELSV formula to the intersection theory on a suitable compactification of the universal Picard variety, i.e., a moduli space of curves of genus \(g\) with \(n\) marked points and a given line bundle. This compactified Picard variety is expected to be related to the one constructed in [L. Caporaso, J. Am. Math. Soc. 7, No. 3, 589–660 (1994; Zbl 0827.14014)]. Moreover, the authors announce that in a later paper they will exploit the structure of genus 0 double Hurwitz numbers to understand top intersections on the moduli space of smooth curves and to prove the intersection number conjecture in [C. Faber, in: Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Aspects Math. E33, 109–129 (1999; Zbl 0978.14029)], for every genus and for up to three marked points.

If we fix the length \(m\) (respectively, \(n\)) of \(\alpha\), resp., \(\beta\), and we write \(\alpha=(\alpha_1,\dots,\alpha_m)\), \(\beta=(\beta_1,\dots,\beta_n)\), the double Hurwitz numbers \(H^g_{\alpha,\beta}\) define a function in the \(\alpha_i\), \(\beta_j\). Using ribbon graphs, the authors prove that the functions so obtained are piecewise polynomial of degree \(\leq 4g-3+m+n\). The proof is based on an interpretation of double Hurwitz numbers as the number of lattice points of certain polytopes, such that the position of the faces depends on \(\alpha\) and \(\beta\). In the special case \(g=0\), the authors prove that the degree of the polynomials is bounded below by \(2g-3+m+n\); they conjecture that this should hold also for positive genus. By considering the example \((g,m,n)=(0,2,2)\), the authors show that double Hurwitz numbers are not polynomial in general.

The central part of the paper deals with the case of covers with complete ramification over \(0\), i.e., double Hurwitz numbers of the form \(H^g_{(d),\beta}\), called here one-part double Hurwitz numbers. Using techniques from algebra, geometry and representation theory, the authors give explicit formulas (Theorem 3.1) for these Hurwitz numbers, which imply, among other things, that they are polynomials in the parts of the partition \(\beta\). Furthermore, the authors conjecture the form of an analogue of the ELSV formula for one-part Hurwitz numbers, expressing them as integrals of some (undefined as yet) “geometrically natural classes” on a suitable compactification of the Picard variety. The proprieties which the compactified Picard variety and the classes of it should satisfy are listed. For genus \(0\) and genus \(1\) the conjecture is proved, by choosing the compactified Picard variety to be, respectively, \(\overline{{\mathcal M}}_{0,n}\) and \({\overline{\mathcal M}_{1,n+1}}\). The authors also define a symbol \(\langle\langle \cdot \rangle\rangle_g\), which, in view of the conjecture, is expected to be the analogue of Witten’s correlation function \(\langle\cdot\rangle_g\). Indeed, the symbol \(\langle\langle\cdot\rangle\rangle_g\) satisfies properties similar to those of \(\langle\cdot\rangle_g\). In particular, the authors prove that the generating series of \(\langle\langle\cdot\rangle\rangle_g\) satisfies the string and dilaton equation, and an analogue of Itzykson-Zuber’s ansatz for intersection numbers on the moduli space of curves ((5.32) in [Int. J. Mod. Phys. A 7, No. 23, 5661–5705 (1992; Zbl 0972.14500)]). Moreover, they give analogues of the formulas in [C. Faber, R. Pandharipande, Ann. Math. (2) 157, No. 1, 97–124 (2003; Zbl 1058.14046)].

Next, the authors investigate the generating function of double Hurwitz numbers and express it in terms of Schur symmetric functions. As a corollary, one gets formulas for \(H^g_{\alpha,\beta}\) as a function of \(g\), when \(\alpha\) and \(\beta\) are fixed. This extends results of Kuleshov and Shapiro for covers of degree \(3,4,5\). Finally, when the length \(m\) of the partition \(\alpha\) is fixed, the authors define a symmetrized generating function for these numbers, and prove that it satisfies a topological recursion relation. This leads to closed expressions for the generating function for small \((g,m)\).

Reviewer: Orsola Tommasi (Mainz)

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

05E05 | Symmetric functions and generalizations |

14K30 | Picard schemes, higher Jacobians |

14H30 | Coverings of curves, fundamental group |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

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\textit{I. P. Goulden} et al., Adv. Math. 198, No. 1, 43--92 (2005; Zbl 1086.14022)

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