We study double Hurwitz numbers in genus zero counting the number of covers $\CP^1\to\CP^1$ with two branching points with a given branching behavior. By the recent result due to Goulden, Jackson and Vakil, these numbers are piecewise polynomials in the multiplicities of the preimages of the branching points. We describe the partition of the parameter space into polynomiality domains, called chambers, and provide an expression for the difference of two such polynomials for two neighboring chambers. Besides, we provide an explicit formula for the polynomial in a certain chamber called totally negative, which enables us to calculate double Hurwitz numbers in any given chamber as the polynomial for the totally negative chamber plus the sum of the differences between the neighboring polynomials along a path connecting the totally negative chamber with the given one.
We derive a closed-form expression for all genus 1 Hurwitz numbers, and give a simple new graph-theoretic interpretation of Hurwitz numbers in genus 0 and 1. (Hurwitz numbers essentially count irreducible genus g covers of the sphere, with arbitrary specified branching over one point, simple branching over other specified points, and no other branching. The problem is equivalent to counting transitive factorisations of permutations into transpositions.) These results prove a conjecture of Goulden and Jackson, and extend results of Hurwitz and many others.
We calculate the cycle class of the Hurwitz divisor $D_2$ on the moduli space of stable curves of genus $g=2k$ given by the degree $k+1$ covers of the projective line with simple ramification points, two of which lie in the same fibre. We also study some aspects of the geometry of the natural map of the Hurwitz space of admissible covers of degree $k+1$ and genus $2k$ to the moduli space of stable curves of genus $2k$.
In the Hurwitz space of rational functions on CP^1 with poles of given orders, we study the loci of multisingularities, that is, the loci of functions with a given ramification profile over 0. We prove a recursion relation on the Poincare dual cohomology classes of these loci and deduce a differential equation on Hurwitz numbers.
In this short note we prove the unirationality of Hurwitz spaces of 6-gonal curves of genus $g$ with $5\leq g\leq 28$ or $g=30,31,33,35,36,40,45$. Key ingredient is a liaison construction in $\PP^1 \times \PP^2$. By semicontinuity, the proof of the dominance of this construction is reduced to a computation of a single curve over a finite field.
We study "pure-cycle" Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-0 covers, using a combination of limit linear series theory and group theory to show that these spaces are always irreducible. In the case of four branch points, we also compute the associated Hurwitz numbers. Finally, we give a conditional result in the higher-genus case, requiring at least 3g simply branched points. These results have equivalent formulations in group theory, and in this setting complement results of Conway-Fr…
In recent years, monotone double Hurwitz numbers were introduced as a naturally combinatorial modification of double Hurwitz numbers. Monotone double Hurwitz numbers share many structural properties with their classical counterparts, such as piecewise polynomaility, while the quantitative properties of these two numbers are quite different. We consider real analogues of monotone double Hurwitz numbers and study the asymptotics for these real analogues. The key ingredient is an interpretation of real tropical covers with arbitrary splittings as factorizations in the symmetric group which genera…
Hurwitz numbers count ramified genus $g$, degree $d$ coverings of the projective line with with fixed branch locus and fixed ramification data. Double Hurwitz numbers count such covers, where we fix two special profiles over $0$ and $\infty$ and only simple ramification else. These objects feature insteresting structural behaviour and connections to geometry. In this paper, we introduce the notion of pruned double Hurwitz numbers, generalizing the notion of pruned simple Hurwitz numbers in \cite{DN13}. We show that pruned double Hurwitz numbers, similar to usual double Hurwitz numbers, satisfy…
Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permuations into transpositions), have been extensively studied for over a century. The Gromov-Witten potential F of a point, the generating series for Hodge integrals on the moduli space of curves, has been a central object of study in Gromov-Witten theory. We define a slightly enriched Gromov-Witten potential G (including integrals involving one "lambda-class"), and show that after a non-trivial change of variables, G=H in positive genus, where H is a generating series for Hurwitz numbers.…
We study the real counterpart of double Hurwitz numbers, called real double Hurwitz numbers here. We establish a lower bound for these numbers with respect to their dependence on the distribution of branch points. We use it to prove, under certain conditions, existence of real Hurwitz covers as well as logarithmic equivalence of real and classical Hurwitz numbers. The lower bound is based on the \enquote{tropical} computation of real Hurwitz numbers in arXiv:1412.4235.
We derive an algorithm to produce explicit formulas for certain generating functions of double Hurwitz numbers. These formulas generalize a formula of Goulden, Jackson and Vakil for one part double Hurwitz numbers. Immediate consequences include a new proof that double Hurwitz numbers are piecewise polynomial, an understanding of the chamber structure and wall crossing for these polynomials, and a proof of the Goulden, Jackson and Vakil's Strong Piecewise Polynomiality conjecture. The method is a straightforward application of Okounkov's expression for double Hurwitz numbers in terms of operat…
Università degli studi di Trieste（的里雅斯特大学）；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste（意大利国家核物理研究所（INFN）的里雅斯特分部）；Faculdade de Ciências da Universidade de Lisboa（里斯本大学理学院）
本文通过基本方法研究Hurwitz数在高 genus 下的渐进行为，涵盖单、双及任意固定分支类型，以及简单或包含完整循环等分支类型，总结了主要贡献。
Hurwitz numbers count genus $g$, degree $d$ covers of the complex projective line with fixed branched locus and fixed ramification data. An equivalent description is given by factorisations in the symmetric group. Simple double Hurwitz numbers are a class of Hurwitz-type counts of specific interest. In recent years a related counting problem in the context of random matrix theory was introduced as so-called monotone Hurwitz numbers. These can be viewed as a desymmetrised version of the Hurwitz-problem. A combinatorial interpolation between simple and monotone double Hurwitz numbers was introdu…
Edge-contraction operations form an effective tool in various graph enumeration problems, such as counting Grothendieck's dessins d'enfants and simple and double Hurwitz numbers. These counting problems can be solved by a mechanism known as topological recursion, which is a mirror B-model corresponding to these counting problems. We show that for the case of orbifold Hurwitz numbers, the mirror objects, i.e., the spectral curve and the differential forms on it, are constructed solely from the edge-contraction operations of the counting problem in genus $0$ and one marked point. This forms a pa…
We study rational double Hurwitz cycles, i.e. loci of marked rational stable curves admitting a map to the projective line with assigned ramification profiles over two fixed branch points. Generalizing the phenomenon observed for double Hurwitz numbers, such cycles are piecewise polynomial in the entries of the special ramification; the chambers of polynomiality and wall crossings have an explicit and "modular" description. A main goal of this paper is to simultaneously carry out this investigation for the corresponding objects in tropical geometry, underlining a precise combinatorial duality …
We compute equations for a Hurwitz curve of genus 17 and we conclude that the canonical ideal of any Hurwitz curve of genus 14 or 17 is generated by quadrics.
We prove the irreducibility of the space parametrizing branched covers of a fixed Riemann surface $B$ of degree $d$, with at least 2d branch points, and with monodromy group equal to $S_d$. The result is classical for $g(B)=0$. The result is well-known for $g(B) > 0$, but we could find no reference.
Double Hurwitz numbers count covers of the projective line by genus g curves with assigned ramification profiles over 0 and infinity, 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, and Shadrin, Shapiro and Vainshtein have determined the chamber structure and wall crossing formulas for g=0. 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 Shapiro, Shadrin and Vainshtein to …
We define the dimension 2g-1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P^1 with given ramification over infinity and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localization expresses such classes in terms of localization trees weighted by ``top intersections'' of tautological classes and genus 0 double Hurwitz numbers. This identity of generating series can be inverted, yielding a ``combinatorialization'' of top intersections of psi-classes. As genus 0 double Hur…
In this paper we prove that the Hurwitz space H_{9,8}, which parameterizes 8-sheeted covers of P^1 by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.
Double Hurwitz numbers count branched covers of the projective line with fixed branch points, with simple branching required over all but two points 0 and infinity, and the branching over 0 and infinity specified by partitions of the degree (with m and n parts respectively). Single Hurwitz numbers have a rich structure, explored by authors in many fields. The ELSV formula relates single Hurwitz numbers to intersection theory on the moduli space of curves, and has led to many consequences. We determine the structure of double Hurwitz numbers using geometry, algebra, and representation theory. O…
Hurwitz numbers count genus g, degree d covers of the projective line with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers.
In this paper we find an explicit formula for the number of topologically different ramified coverings $C\to\CP^1$ (C is a compact Riemann surface of genus g) with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.
Motivated by results for the HCIZ integral in Part I of this paper, we study the structure of monotone Hurwitz numbers, which are a desymmetrized version of classical Hurwitz numbers. We prove a number of results for monotone Hurwitz numbers and their generating series that are striking analogues of known results for the classical Hurwtiz numbers. These include explicit formulas for monotone Hurwitz numbers in genus 0 and 1, for all partitions, and an explicit rational form for the generating series in arbitrary genus. This rational form implies that, up to an explicit combinatorial scaling, m…
We investigate the combinatorics of real double Hurwitz numbers with real positive branch points using the symmetric group. Our main focus is twofold. First, we prove correspondence theorems relating these numbers to counts of tropical real covers and study the structure of real double Hurwitz numbers with the help of the tropical count. Second, we express the numbers as counts of paths in a subgraph of the Cayley graph of the symmetric group. By restricting to real double Hurwitz numbers with real positive branch points, we obtain a concise translation of the counting problem in terms of tupl…
We construct several modular compactifications of the Hurwitz space $H^d_{g/h}$ of genus $g$ curves expressed as $d$-sheeted, simply branched covers of genus $h$ curves. These compactifications are obtained by allowing the branch points of the covers to collide to a variable extent. They are very well-behaved if $d = 2, 3$, or if relatively few collisions are allowed. We recover as special cases the spaces of twisted admissible covers of Abramovich, Corti and Vistoli and the spaces of hyperelliptic curves of Fedorchuk.
By associating to a curve C of genus g=2k and a pencil of degree d=k+1 the so-called trace curve (resp. the reduced trace curve) we define a rational map from the Hurwitz space of admissible covers of genus g=2k and degree d=k+1 to a moduli space of stable curves. We study the induced map between the divisor class groups of these moduli spaces of curves.
This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points.
In this short note, we propose a definition of complete Hurwitz schemes (and stacks) in mixed characteristic. We follow an idea of R. Pandharipande, and define the complete Hurwitz stack as a substack of stable maps of degree d of nodal pointed curves of genus g to the universal curve over the moduli stack of stable pointed curves of genus h. This substack is simply the closure of the locus of Hurwitz covers in characteristic 0. We describe in detail the reduction modulo 2 of the complete Hurwitz stack of genus 1 double covers of the projective line. The result is wilder than you would imagine…
In this paper, we collect a number of facts about double Hurwitz numbers, where the simple branch points are replaced by their more general analogues --- completed (r+1)-cycles. In particular, we give a geometric interpretation of these generalised Hurwitz numbers and derive a cut-and-join operator for completed (r+1)-cycles. We also prove a strong piecewise polynomiality property in the sense of Goulden-Jackson-Vakil. In addition, we propose a conjectural ELSV/GJV-type formula, that is, an expression in terms of some intrinsic combinatorial constants that might be related to the intersection …
We consider two applications of the strata of differentials of the second kind (all residues equal to zero) with fixed multiplicities of zeros and poles: Positivity: In genus $g=0$ we show any associated divisorial projection to $\overline{\mathcal{M}}_{0,n}$ is $F$-nef and hence conjectured to be nef. We compute the class for all genus when the divisorial projection forgets only simple zeros and show in these cases the genus $g=0$ projections are indeed nef. Hurwitz spaces: We show the Hurwitz spaces of degree $d$, genus $g$ covers of $\mathbb{P}^1$ with pure branching (one ramified point ove…
As the real counterpart of double Hurwitz number, the real double Hurwitz number depends on the distribution of real branch points. We consider the problem of asymptotic growth of real and complex double Hurwitz numbers. We provide a lower bound for real double Hurwitz numbers based on the tropical computation of real double Hurwitz numbers. By using this lower bound and J. Rau's result ( Math. Ann. 375(1-2): 895-915, 2019), we prove the logarithmic equivalence of real and complex Hurwitz numbers.
We prove two explicit formulae for one-part double Hurwitz numbers with completed 3-cycles. We define "combinatorial Hodge integrals" from these numbers in the spirit of the celebrated ELSV formula. The obtained results imply some explicit formulae and properties of the combinatorial Hodge integrals.
We show that various loci of stable curves of sufficiently large genus admitting degree $d$ covers of positive genus curves define non-tautological algebraic cycles on $\overline{\mathcal{M}}_{g,N}$, assuming the non-vanishing of the $d$-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.
We describe double Hurwitz numbers as intersection numbers on the moduli space of curves. Assuming polynomiality of the Double Ramification Cycle (which is known in genera 0 and 1), our formula explains the polynomiality in chambers of double Hurwitz numbers, and the wall crossing phenomenon in terms of a variation of correction terms to the ψ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle.
Simple Hurwitz numbers enumerate branched morphisms between Riemann surfaces with fixed ramification data. In recent years, several variants of this notion for genus $0$ base curves have appeared in the literature. Among them are so-called monotone Hurwitz numbers, which are related to the HCIZ integral in random matrix theory and strictly monotone Hurwitz numbers which count certain Grothendieck dessins d'enfants. We generalise the notion of Hurwitz numbers to interpolations between simple, monotone and strictly monotone Hurwitz numbers to any genus and any number of arbitrary but fixed ramif…
A Hurwitz curve is a closed Riemann surface of genus $g \geq 2$ whose group of conformal automorphisms has order $84(g-1)$. In 1895, Wiman proved that for $g=3$ there is, up to isomorphisms, a unique Hurwitz curve; this being Klein's plane quartic curve. Moreover, he also proved that there is no Hurwitz curve of genus $g=2,4,5,6$. Later, in 1965, Macbeath proved the existence, up to isomorphisms, of a unique Hurwitz curve of genus $g=7$; this known as the Fricke-Macbeath curve. Equations were also provided; that being the fiber product of suitable three elliptic curves. In the same year, Edge …
This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov-Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov-Witten theory, with a particular eye to the contributions made to the understanding of the Double Ramification Cycle, a cycle in the moduli space of curves that compactifies the double Hurwitz locus. We explore the algebro-combinatorial properties of single and double Hurwitz numbers, and the connections with intersection theoretic problems on appropriate moduli spaces. We survey several res…
Single Hurwitz numbers enumerate branched covers of the Riemann sphere with specified genus, prescribed ramification over infinity, and simple branching elsewhere. They exhibit a remarkably rich structure. In particular, they arise as intersection numbers on moduli spaces of curves and are governed by the topological recursion of Chekhov, Eynard and Orantin. Double Hurwitz numbers are defined analogously, but with prescribed ramification over both zero and infinity. Goulden, Jackson and Vakil have conjectured that double Hurwitz numbers also arise as intersection numbers on moduli spaces. In t…
We study genus $g$ coverings of full moduli dimension of degree $d=[\frac {g+3} 2]$. There is a homomorphism between the corresponding Hurwitz space $\H$ of such covers to the moduli space $\M_g$ of genus $g$ curves. In the case $g=3$, using the signature of such covering we provide an equation for the generic ternary quartic. Further, we discuss the degenerate subloci of the corresponding Hurwitz space of such covers from the computational group theory viewpoint. In the last section, we show that one of these degenerate loci corresponds to the locus of curves with automorphism group $C_3$. We…
In this paper, we prove Hurwitz-Eichler type formulas for Hurwitz class numbers with each level $ M $ when the modular curve $ X_0(M) $ has genus zero. A key idea is to calculate intersection numbers of modular correspondences with the level in two different ways. A generalization of Atkin-Lehner involutions for $ Γ_0(M) $ and its subgroup $ Γ_0^{(M')}(M) $ is introduced to calculate intersection multiplicities of modular correspondences at cusps.
We derive explicit formulae for the generating series of mixed Grothendieck dessins d'enfant/monotone/simple Hurwitz numbers, via the semi-infinite wedge formalism. This reveals the strong piecewise polynomiality in the sense of Goulden-Jackson-Vakil, generalising a result of Johnson, and provides a new explicit proof of the piecewise polynomiality of the mixed case. Moreover, we derive wall-crossing formulae for the mixed case. These statements specialise to any of the three types of Hurwitz numbers, and to the mixed case of any pair.
E. Kani has shown that the Hurwitz functor, which parametrizes the (normalized) genus 2 covers of degree 3 of an elliptic curve, is representable. In this paper the corresponding moduli scheme and the universal family are explicitly calculated and described by short equations.
We study the components of the Hurwitz scheme of ramified coverings of $\mathbb{P}^1$ with monodromy given by the alternating group $A_6$ and elements in the conjugacy class of product of two disjoint cycles. In order to detect the connected components of the Hurwitz scheme, inspired by the case of the spin structures studied by Fried for the $3$-cycles, we use as invariant the lifting to the Valentiner group, triple covering of $A_6$. We prove that the Hurwitz scheme has two irreducible components when the genus of the covering is greater than zero, in accordance with the asymptotic solution …
Let $g$ and $g'$ be two integers and $p$ a prime number. Denote by $\mathscr H_{g, g', p}^c$ the moduli space of morphisms of degree $p$ between smooth curves of genus $g$ and $g'$ and with constant ramification. The purpose of this article is to construct and describe a modular compactification of this space.
