Using the connection between intersection theory on the Deligne-Mumford spaces and the edge scaling of the GUE matrix model (see math.CO/9903176, math.AG/0101147), we express the n-point functions for the intersection numbers as n-dimensional error-function-type integrals and also give a derivation of Witten's KdV equations using the higher Fay identities of Adler, Shiota, and van Moerbeke.
We provide a general method for computing rational Chow rings of moduli of smooth complete intersections. We specialize this result in different ways: to compute the integral Picard group of the associated stack ; to obtain an explicit presentation of rational Chow rings of moduli of smooth complete intersections of codimension two; to prove old and new results on moduli of smooth curves of genus $\leq 5$ and polarized K3 surfaces of degree $\leq 8$.
This is a note on calculating intersection numbers on moduli spaces of curves. A codimension 3 relation among tautological classes on the moduli space of genus 4 curves is given.
We establish the Airy curve case of a conjecture of Gukov and Sułkowski by reducing to Dijkgraaf-Verlinde-Verlinde Virasoro constraints satisfied by the intersection numbers on moduli spaces of algebraic curves.
We give an algebro-geometric derivation of the known intersection theory on the moduli space of stable rank 2 bundles of odd degree over a smooth curve of genus g. We lift the computation from the moduli space to a Quot scheme, where we obtain the intersections by equivariant localization with respect to a natural torus action.
We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.
This paper proves the existence of homeomorphic (diffeomorphic) complex 6-dimensional (7-dim) complete intersections that belong to components of the moduli space of different dimensions. These results are given as a supplement to earlier result on 5-dimensional complete intersections.
We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincare polynomials for sheaves with rank two and three on ruled surfaces.
We explain how to compute top-dimensional intersections of psi-classes on moduli spaces of m-stable curves. On the moduli spaces of Deligne-Mumford stable pointed curves of genus one, these intersection numbers are determined by two recursions, namely the string equation and the dilaton equation. We establish, for each integer m>0, an analogous pair of recursions which determine these intersection numbers on the moduli spaces of m-stable pointed curves of genus one.
We study the moduli space of smooth complete intersections of two quadrics in $\mathbb{P}^n$ by relating it to the geometry of the singular members of the corresponding pencils. Giving an alternative presentation for the moduli space of complete intersections, we compute the Picard group for all $n\geq 3$.
In a recent paper R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. The authors conjectured KdV and Virasoro type equations that completely determine all intersection numbers. In this paper we study these equations in detail. In particular, we prove that the KdV and the Virasoro type equations for the intersection numbers on the moduli space of Riemann surfaces with boundary are equivalent.
We consider a class of tautological top intersection products on the moduli space of stable pairs consisting of semistable vector bundles together with N sections on a smooth complex projective curve C. We show that when N is large, these intersection numbers can equally be computed on the Grothendieck Quot scheme of coherent sheaf quotients of the rank N trivial sheaf on C. The result has applications to the calculation of the intersection theory of the moduli space of semistable bundles on C.
In this paper we consider the topological side of a problem which is the analogue of Sen's S-duality testing conjecture for Hitchin's moduli space of rank 2 stable Higgs bundles of fixed determinant of odd degree over a Riemann surface. We prove that all intersection numbers in the compactly supported cohomology vanish, i.e. "there are no topological L^2 harmonic forms on Hitchin's space". This result generalizes the well known vanishing of the Euler characteristic of the moduli space of rank 2 stable bundles of fixed determinant of odd degree over the given Riemann surface. Our proof shows th…
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. And then we obtain an algorithm expressing intersection numbers $<τ_{n,m}\prod_{i=0}^{r-1}τ_{0,i}^{k_i}>_g$ via correlation functions of primaries.
We introduce conjectures relating the Chow ring of a smooth Artin stack $\mathcal{X}$ to the Chow groups of its possibly singular good moduli space $X$. In particular, we conjecture the existence of an intersection product on a subgroup of the full Chow group $A^*(X)$ coming from strong cycles on $\mathcal{X}$.
Hausel and Rodriguez-Villegas conjectured that the intersection form on the moduli space of stable PGL_n-Higgs bundles on a curve vanishes if the degree is coprime to n. In this note we prove this conjecture. Along the way we show that moduli spaces of stable chains are irreducible for stability parameters larger than the stability condition induced form stability of Higgs bundles.
We present a solution to the W-constraints satisfied by the intersection numbers on the moduli spaces of r-spin curves. We make use of a grading suggested by the selection rule for the correlators determined by the geometry of the moduli space.
We give two recursions for computing top intersections of tautological classes on blowups of moduli spaces of genus-one curves. One of these recursions is analogous to the well-known string equation. As shown in previous papers, these numbers are useful for computing genus-one enumerative invariants of projective spaces and Gromov-Witten invariants of complete intersections.
We show that intersection numbers on the moduli space of stable bundles of coprime rank and degree over a smooth complex curve can be recovered as highest-degree asymptotics in formulas of Vafa-Intriligator type. In particular, we explicitly evaluate all intersection numbers appearing in the Verlinde formula. Our results are in agreement with previous computations of Witten, Jeffrey-Kirwan and Liu. Moreover, we prove the vanishing of certain intersections on a suitable Quot scheme which can be interpreted as giving equations between counts of maps to the Grassmannian.
We characterize transversality, non-transversality properties on the moduli space of genus 0 stable maps to a semipositive symplectic manifold of dimension 4, when GW([point],...,[point]) is enumerative. In particular, we show that the intersection theoretic property depends on the existence of a critical point on a stable map.
We present a series of new results we obtained recently about the intersection numbers of tautological classes on moduli spaces of curves, including a simple formula of the n-point functions for Witten's $τ$ classes, an effective recursion formula to compute higher Weil-Petersson volumes, several new recursion formulae of intersection numbers and our proof of a conjecture of Itzykson and Zuber concerning denominators of intersection numbers. We also present Virasoro and KdV properties of generating functions of general mixed $κ$ and $ψ$ intersections.
Using the celebrated Witten-Kontsevich theorem, we prove a recursive formula of the $n$-point functions for intersection numbers on moduli spaces of curves. It has been used to prove the Faber intersection number conjecture and motivated us to find some conjectural vanishing identities for Gromov-Witten invariants. The latter has been proved recently by X. Liu and R. Pandharipande. We also give a combinatorial interpretation of $n$-point functions in terms of summation over binary trees.
We study the arithmetic of complete intersections in projective space over number fields. Our main results include arithmetic Torelli theorems and versions of the Shafarevich conjecture, as proved for curves and abelian varieties by Faltings. For example, we prove an analogue of the Shafarevich conjecture for cubic and quartic threefolds and intersections of two quadrics.
A bracket is a function that assigns a number to each monomial in variables τ_0, τ_1, ... We show that any bracket satisfying the string and the dilaton relations gives rise to a power series lying in the algebra A generated by the series \sum n^{n-1} q^n/n! and \sum n^n q^n /n! . As a consequence, various series from A appear in the intersection theory of moduli spaces of curves. A connection between the counting of ramified coverings of the sphere and the intersection theory on moduli spaces allows us to prove that some natural generating functions enumerating the ramified coverings lie, yet…
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…
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.
Stabilization of intersection Betti numbers for moduli spaces of one-dimensional sheaves on surfaces
The School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；Shanghai Center for Mathematical Sciences, Fudan University（复旦大学上海数学中心）
We construct a two-level weighted TQFT whose structure coefficents are equivariant intersection numbers on moduli spaces of admissible covers. Such a structure is parallel (and strictly related) to the local Gromov-Witten theory of curves of Bryan-Pandharipande. We compute explicitly the theory using techniques of localization on moduli spaces of admissible covers of a parametrized projective line. The Frobenius Algebras we obtain are one parameter deformations of the class algebra of the symmetric group S_d. In certain special cases we are able to produce explicit closed formulas for such def…
We explicitly compute the diverging factor in the large genus asymptotics of the Weil-Petersson volumes of the moduli spaces of $n$-pointed complex algebraic curves. Modulo a universal multiplicative constant we prove the existence of a complete asymptotic expansion of the Weil-Petersson volumes in the inverse powers of the genus with coefficients that are polynomials in $n$. This is done by analyzing various recursions for the more general intersection numbers of tautological classes, whose large genus asymptotic behavior is also extensively studied.
This paper is to give some concrete examples of the general fibers of the evaluation map of some Kontsevich mapping spaces parametrize low degree rational curves on low degree complete intersection varieties. We prove these examples are Fano or Calabi-Yau vareities of low dimension, e.g, surface and threefold. In usual, the "moduli" fibers are constructed in an abstract way, but here we give a down-to-earth interpretation to some of these fibers from the perspective of projective geometry.
We present certain new properties about the intersection numbers on moduli spaces of curves $\bar{\sM}_{g,n}$, including a simple explicit formula of $n$-point functions and several new identities of intersection numbers. In particular we prove a new identity, which together with a conjectural identity implies the famous Faber's conjecture about relations in $\mathcal R^{g-2}(\sM_g)$. These new identities clarify the mysterious constant in Faber's conjecture and uncover novel combinatorial structures of intersection numbers. We also discuss some numerical properties of Hodge integrals which ha…
We describe algorithms for computing the intersection numbers of divisors and of Chern classes of the Hodge bundle on the moduli spaces of stable pointed curves. We also discuss the implementations and the results obtained. There are several applications. We discuss one in particular: the calculation of the projection in the tautological ring of the moduli space of abelian varieties of the class of the locus of Jacobians.
Let $V$ be a smooth, projective, convex variety. We define tautological $ψ$ and $κ$ classes on the moduli space of stable maps $\M_{0,n}(V)$, give a (graphical) presentation for these classes in terms of boundary strata, derive differential equations for the generating functions of the Gromov-Witten invariants of $V$ twisted by these tautological classes, and prove that these intersection numbers are completely determined by the Gromov-Witten invariants of $V$. This results in families of Frobenius manifold structures on the cohomology ring of $V$ which includes the quantum cohomology as a spe…
We define a new family of open Gromov-Witten type invariants based on intersection theory on the moduli space of pseudoholomorphic curves of arbitrary genus with boundary in a Lagrangian submanifold. We assume the Lagrangian submanifold arises as the fixed points of an anti-symplectic involution and has dimension 2 or 3. In the strongly semi-positive genus 0 case, the new invariants coincide with Welschinger's invariant counts of real pseudoholomorphic curves. Furthermore, we calculate the new invariant for the real quintic threefold in genus 0 and degree 1 to be 30.
A curve on a projective variety is called movable if it belongs to an algebraic family of curves covering the variety. We consider when the cone of movable curves can be characterized without existence statements of covering families by studying the complete intersection cone on a family of blow-ups of complex projective space, including the moduli space of stable six-pointed rational curves, $\bar{M}_{0,6}$, and the permutohedral or Losev-Manin moduli space of four-pointed rational curves. Our main result is that the movable and complete intersection cones coincide for the toric members of th…
The main result of this paper is the statement that the Hodge theoretic Donaldson-Thomas invariant for a quiver with zero potential and a generic stability condition agrees with the compactly supported intersection cohomology of the closure of the stable locus inside the associated coarse moduli space of semistable quiver representations. In fact, we prove an even stronger result relating the Donaldson-Thomas "function" to the intersection complex. The proof of our main result relies on a relative version of the integrality conjecture in Donaldson-Thomas theory. This will be the topic of the s…
We study the ring generated by the Chern classes of tautological line bundles on the moduli space of parabolic bundles of arbitrary rank on a Riemann surface. We show the Poincaré duals to these Chern classes have simple geometric representatives. We use this construction to show that the ring generated by these Chern classes vanishes below the dimension of the moduli space, in analogy with the Newstead-Ramanan conjecture for stable bundles.
We present a topological recursion formula for calculating the intersection numbers defined on the moduli space of open Riemann surfaces. The spectral curve is $x = \frac{1}{2}y^2$, the same as spectral curve used to calculate intersection numbers for closed Riemann surfaces, but the formula itself is a variation of the usual Eynard-Orantin recursion. It looks like the recursion formula used for spectral curves of degree 3, and also includes features present in $β$-deformed models. The recursion formula suggests a conjectural refinement to the generating function that allows for distinguishing…
Recently R. Pandharipande, J. Solomon and R. Tessler initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of the intersection numbers is a specific solution of a system of PDEs, that they called the open KdV equations. In this paper we show that the open KdV equations are closely related to the equations for the wave function of the KdV hierarchy. This allows us to give an explicit formula for the specific solution in terms of Witten's generating series of the intersection numbers on the moduli space of …
This paper studies intersection theory on the compactified moduli space M(n,d) of holomorphic bundles of rank n and degree d over a fixed compact Riemann surface of genus g > 1 where n and d may have common factors. Because of the presence of singularities we work with the intersection cohomology groups defined by Goresky and MacPherson and the ordinary cohomology groups of a certain partial resolution of singularities of M(n,d). Based on our earlier work, we give a precise formula for the intersection cohomology pairing and provide a method to calculate pairings on the partial resolution of s…
We define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the degree of the branch map, which enables us to give a tropical intersection-theoretic definition of tropical triple Hurwitz numbers. We show that our intersection-theoretic definition coincides with the one given by Bertrand, Brugallé and Mikhalkin in the article "Tropical Open Hurwitz numbers" where a Correspondence Theorem for Hurwitz numbers is proved. Thus …
We prove a general formula for the intersection form of two arbitrary monomials in boundary divisors. Furthermore we present a tree basis of the cohomology of $\overline {M}_{0,n}$. With the help of the intersection form we determine the Gram matrix for this basis and give a formula for its inverse. This enables us to calculate the tensor product of the higher order multiplications arising in quantum cohomology and formal Frobenius manifolds. In the context of quantum cohomology this establishes the explicit Künneth formula.
For a split reductive group G over a finite field, we show that the intersection (cohomology) motive of the moduli stack of iterated G-shtukas with bounded modification and level structure is defined independently of the standard conjectures on motivic t-structures on triangulated categories of motives. This is in accordance with general expectations on the independence of l in the Langlands correspondence for function fields.
In [GH], Griffiths and Harris asked whether a projective complex submanifold of codimension two is determined by the moduli of its second fundamental forms. More precisely, given a nonsingular subvariety $X^n \subset {\mathbb P}^{n+2}$, the second fundamental form $II_{X,x}$ at a point $x \in X$ is a pencil of quadrics on $T_x(X)$, defining a rational map $μ^X$ from $X$ to a suitable moduli space of pencils of quadrics on a complex vector space of dimension $n$. The question raised by Griffiths and Harris was whether the image of $μ^X$ determines $X$. We study this question when $X^n \subset {…
The goal of this paper is to study a $p$-adic analog of the joint of the conjectures of André--Oort and André--Pink. More precisely, on a product of ordinary Siegel formal moduli schemes, we study the distribution of points whose components are either CM points or points in Hecke orbits. We use linearity of formal subschemes of the product as the $p$-adic analog of geodesicness over complex numbers. Moreover, we relax the usual incidence relations by using $p$-adic distance. We also study a $p$-adic formal scheme theoretic analog of the Ax--Lindemann theorem.
We prove formulas (found by Witten in 1992 using physical methods) for intersection pairings in the cohomology of the moduli space M(n,d) of stable holomorphic vector bundles of rank n and degree d (assumed coprime) on a Riemann surface of genus g greater than or equal to 2. We also use these formulas for intersection numbers to obtain a proof of the Verlinde formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).
The theory of Q-Cartier divisors on the space of n-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of Q-Divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in P^r is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree d plane curves is explicitly evaluated.
In this paper, we derive the virtual structure constants used in mirror computation of degree k hypersurface in CP^{N-1}, by using localization computation applied to moduli space of polynomial maps from CP^{1} to CP^{N-1} with two marked points. We also apply this technique to non-nef local geometry O(1)+O(-3)->CP^{1} and realize mirror computation without using Birkhoff factorization.
Generalized Hypergeometric Functions for Degree k Hypersurface in CP^{N-1} and Intersection Numbers of Moduli Space of Quasimaps from CP^{1} with Two Marked Points to CP^{N-1}
本文将 CP^{N-1} 中 k 次超曲面的镜像计算所用广义超几何函数实现为带两个标记点的拟映射模空间相交数的生成函数，建立了超曲面镜像对称与拟映射计数之间的具体联系。
Let $C$ be a smooth projective curve of genus $2$. Following a method by O' Grady, we construct a semismall desingularization $\tilde{\mathcal{M}}_{Dol}^G$ of the moduli space $\mathcal{M}_{Dol}^G$ of semistable $G$-Higgs bundles of degree 0 for $G=GL(2,\mathbb{C}), SL(2,\mathbb{C})$. By the decomposition theorem by Beilinson, Bernstein, Deligne one can write the cohomology of $\tilde{\mathcal{M}}_{Dol}^G$ as a direct sum of the intersection cohomology of $\mathcal{M}_{Dol}^G$ plus other summands supported on the singular locus. We use this splitting to compute the intersection cohomology of $…
Goulden, Jackson and Vakil observed a polynomial structure underlying one-part double Hurwitz numbers, which enumerate branched covers of $\mathbb{CP}^1$ with prescribed ramification profile over $\infty$, a unique preimage over 0, and simple branching elsewhere. This led them to conjecture the existence of moduli spaces and tautological classes whose intersection theory produces an analogue of the celebrated ELSV formula for single Hurwitz numbers. In this paper, we present three formulas that express one-part double Hurwitz numbers as intersection numbers on certain moduli spaces. The first …
