Hodge integrals with one λ-class

Hodge integrals over moduli space of stable curves play an important roles in understanding the topological properties of moduli space.ELSV formula connects the Hodge integrals with Hurwitz numbers,and the generating function of Hurwitz numbers satisfies the cut-and-join equation.Therefore,it is natural to consider how to use the cut-and-join equation for Hurwitz numbers to compute Hodge integrals which appear in ELSV formula.In this paper,at first,we will review the method introduced in Goulden et al.’s paper to get the λ g conjecture for Hodge integral.Through some variables transformation,the generating function of Hurwitz number becomes a symmetric polynomial which satisfies a symmetrized cut-and-join equation.By comparing the coefficients of the lowest degree term of both sides in this equation,we can get the λ g conjecture.Then,in a similar way,we obtain our main result in this paper:a recursive formula for Hodge integral of type contains only one λ g 1-class.We also point out that our results are closely related to the degree 0 Virasoro conjecture for a curve.     

KP hierarchy for Hodge integrals

Starting from the ELSV formula, we derive a number of new equations on the generating functions for Hodge integrals over the moduli space of complex curves. This gives a new simple and uniform treatment of certain known results on Hodge integrals like Witten's conjecture, Virasoro constrains, Faber's λg-conjecture, etc. Among other results we show that a properly arranged generating function for Hodge integrals satisfies the equations of the KP hierarchy.     

Polynomiality of monotone Hurwitz numbers in higher genera

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of these branched covers, related to the expansion of complete symmetric functions in the Jucys–Murphy elements, and have arisen in recent work on the asymptotic expansion of the Harish-Chandra–Itzykson–Zuber integral. In previous work we gave an explicit formula for monotone Hurwitz numbers in genus zero. In this paper we consider monotone Hurwitz numbers in higher genera, and prove a number of results that are reminiscent of those for classical Hurwitz numbers. These include an explicit formula for monotone Hurwitz numbers in genus one, and an explicit form for the generating function in arbitrary positive genus. From the form of the generating function we are able to prove that monotone Hurwitz numbers exhibit a polynomiality that is reminiscent of that for the classical Hurwitz numbers, i.e.  , up to a specified combinatorial factor, the monotone Hurwitz number in genus g$g$ with ramification specified by a given partition is a polynomial indexed by g$g$ in the parts of the partition.     

Explicit formulae for one-part double Hurwitz numbers with completed 3-cycles

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.     

Polynomiality of Hurwitz numbers,Bouchard–Mariño conjecture,and a new proof of the ELSV formula

In this paper we give a new proof of the ELSV formula. First, we refine an argument of Okounkov and Pandharipande in order to prove (quasi-)polynomiality of Hurwitz numbers without using the ELSV formula (the only way to do that before used the ELSV formula). Then, using this polynomiality we give a new proof of the Bouchard–Mariño conjecture. After that, using the correspondence between the Givental group action and the topological recursion coming from matrix models, we prove the equivalence of the Bouchard–Mariño conjecture and the ELSV formula (it is a refinement of an argument by Eynard).     

Chamber behavior of double Hurwitz numbers in genus 0

We study double Hurwitz numbers in genus zero counting the number of covers CP¹→CP¹ 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.     

Generators for the tautological algebra of the moduli space of curves

In this paper, we prove that the tautological algebra in cohomology of the moduli space M_g of smooth projective curves of genus g is generated by the first [g/3] Mumford-Morita-Miller classes. This solves a part of Faber's conjecture (Moduli of Curves and Abelian Varieties Vieweg, Braunschweig, 1999) concerning the structure of the tautological algebra affirmatively. More precisely, for any k we express the kth Mumford-Morita-Miller class e_k as an explicit polynomial in the lower classes for all genera g=3k−1,3k−2,…,2.     

Degree formulae for offset curves

In this paper, we present three different formulae for computing the degree of the offset of a real irreducible affine plane curve C given implicitly, and we see how these formulae particularize to the case of rational curves. The first formula is based on an auxiliary curve, called S, that is defined depending on a non-empty Zariski open subset of R². The second formula is based on the resultant of the defining polynomial of C, and the polynomial defining generically S. The third formula expresses the offset degree by means of the degree of C and the multiplicity of intersection of C and the hodograph H to C, at their intersection points.     

Hurwitz theory and the double ramification cycle

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 results by many groups of people on the subject, but, perhaps more importantly, collect a number of conjectures and problems which are still open.     

Genus Hurwitz numbers: Recursions, formulas, and graph-theoretic interpretations

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 and . (Hurwitz numbers essentially count irreducible genus 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, Jackson and Vainshtein, and extend results of Hurwitz and many others.

     

Moduli spaces of metric graphs of genus 1 with marks on vertices

In this paper we study homotopy type of certain moduli spaces of metric graphs. More precisely, we show that the spaces , which parametrize the isometry classes of metric graphs of genus 1 with n marks on vertices are homotopy equivalent to the spaces TM1,n, which are the moduli spaces of tropical curves of genus 1 with n marked points. Our proof proceeds by providing a sequence of explicit homotopies, with key role played by the so-called scanning homotopy. We conjecture that our result generalizes to the case of arbitrary genus.     

Polynomial Hurwitz Numbers and Intersections on \overline M _{0,k}

We express Hurwitz numbers of polynomials of arbitrary topological type in terms of intersection numbers on the moduli space of curves of genus zero with marked points.     

An algebro-geometric proof of Witten's conjecture

We present a new proof of Witten's conjecture. The proof is based on the analysis of the relationship between intersection indices on moduli spaces of complex curves and Hurwitz numbers enumerating ramified coverings of the -sphere.

     

Kontsevich's formula and the WDVV equations in tropical geometry

Using Gromov-Witten theory the numbers of complex plane rational curves of degree d through 3d−1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.     

Two results on homogeneous Hessian nilpotent polynomials

Let z=(z₁,…,z_n) and , the Laplace operator. A formal power series P(z) is said to be Hessian Nilpotent (HN) if its Hessian matrix is nilpotent. In recent developments in [M. de Bondt, A. van den Essen, A reduction of the Jacobian conjecture to the symmetric case, Proc. Amer. Math. Soc. 133 (8) (2005) 2201-2205. [MR2138860]; G. Meng, Legendre transform, Hessian conjecture and tree formula, Appl. Math. Lett. 19 (6) (2006) 503-510. [MR2170971]. See also math-ph/0308035; W. Zhao, Hessian nilpotent polynomials and the Jacobian conjecture, Trans. Amer. Math. Soc. 359 (2007) 249-274. [MR2247890]. See also math.CV/0409534], the Jacobian conjecture has been reduced to the following so-called vanishing conjecture (VC) of HN polynomials: for any homogeneous HN polynomialP(z) (of degreed=4), we haveΔ^mP^m+1(z)=0for anym?0. In this paper, we first show that the VC holds for any homogeneous HN polynomial P(z) provided that the projective subvarieties Z_P and Z_σ2 of CPⁿ⁻¹ determined by the principal ideals generated by P(z) and , respectively, intersect only at regular points of Z_P. Consequently, the Jacobian conjecture holds for the symmetric polynomial maps F=z−∇P with P(z) HN if F has no non-zero fixed point w∈Cⁿ with . Secondly, we show that the VC holds for a HN formal power series P(z) if and only if, for any polynomial f(z), Δ^m(f(z)P(z)^m)=0 when m?0.     

Bounds on the number of numerical semigroups of a given genus

Lower and upper bounds are given for the number n_g of numerical semigroups of genus g. The lower bound is the first known lower bound while the upper bound significantly improves the only known bound given by the Catalan numbers. In a previous work the sequence n_g is conjectured to behave asymptotically as the Fibonacci numbers. The lower bound proved in this work is related to the Fibonacci numbers and so the result seems to be in the direction to prove the conjecture. The method used is based on an accurate analysis of the tree of numerical semigroups and of the number of descendants of the descendants of each node depending on the number of descendants of the node itself.     

The Moduli Space of Curves, Double Hurwitz Numbers, and Faber��s Intersection Number Conjecture

We define the dimension 2g − 1 Faber-Hurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of \mathbbP₁{{\mathbb{P}_1}} with given ramification over ∞ 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 Y{\Psi} -classes. As genus 0 double Hurwitz numbers with at most 3 parts over ∞ are well understood, we obtain Faber’s Intersection Number Conjecture for up to 3 parts, and an approach to the Conjecture in general (bypassing the Virasoro Conjecture). We also recover other geometric results in a unified manner, including Looijenga’s theorem, the socle theorem for curves with rational tails, and the hyperelliptic locus in terms of κ _g–2.     

Moduli of stable sheaves on a smooth quadric and a Brill-Noether locus

We prove that the moduli space of stable sheaves of rank 2 with the Chern classes c₁=O_Q(1,1) and c₂=2 on a smooth quadric Q in P₃ is isomorphic to P₃. Using this identification, we give a new proof that a Brill-Noether locus, defined as the closure of the stable bundles with at least three linearly independent sections, on a non-hyperelliptic curve of genus 4, is isomorphic to the Donagi-Izadi cubic threefold.     

Hurwitz spaces and liftings to the Valentiner group

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, a 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 found by Bogomolov and Kulikov.     

On Hurwitz numbers and Hodge integrals

In this paper we find an explicit formula for the number of topologically different ramified coverings C → CP¹ (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.