Moduli Space Of Stable Maps Research Articles

Hodge Integrals and Hurwitz Numbers via Virtual Localization

We give another proof of Ekedahl, Lando, Shapiro, and Vainshtein's remarkable formula expressing Hurwitz numbers (counting covers of P1 with specified simple branch points, and specified branching over one other point) in terms of Hodge integrals. Our proof uses virtual localization on the moduli space of stable maps. We describe how the proof could be simplified by the proper algebro-geometric definition of a ‘relative space’. Such a space has recently been defined by J. Li.

Regular components of moduli spaces of stable maps

We construct regular components of the moduli space of stable maps from curves of genus g to a product of two projective spaces. These components are generically smooth and have the expected dimension predicted by deformation theory. This result can be seen as a general position theorem for loci in M g consisting of curves carrying exceptional linear series.

ON THE COHOMOLOGY OF $\bar{\mathcal M}_{0,n}({\mathbb P}^1,d)$

Here we investigate rational cohomology of the moduli space of stable maps to the complex projective line with a purely algebro-pgeometric approach. In particular, we prove vanishing theorems for all its odd Betti numbers, and we give an explicit description by generators and relations of its second cohomology group.

A Change of Coordinates on the Large Phase Space¶of Quantum Cohomology

The Gromov–Witten invariants of a smooth, projective variety V, when twisted by the tautological classes on the moduli space of stable maps, give rise to a family of cohomological field theories and endow the base of the family with coordinates. We prove that the potential functions associated to the tautological ψ classes (the large phase space) and the κ classes are related by a change of coordinates which generalizes a change of basis on the ring of symmetric functions. Our result is a generalization of the work of Manin–Zograf who studied the case where V is a point. We utilize this change of variables to derive the topological recursion relations associated to the κ classes from those associated to the ψ classes.

The Characteristic Numbers of Quartic Plane Curves

AbstractThe characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen’s prediction of characteristic numbers of smooth plane curves. A short sketch of a computation of the characteristic numbers of plane cubics is also given as an illustration.

Generating functions for the number of curves on abelian surfaces

Let X be an Abelian surface and C a holomorphic curve in X representing a primitive homology class. The space of genus g curves in the class of C is g dimensional. We count the number of such curves that pass through g generic points and we also count the number of curves in the fixed linear system |C| passing through g-2 generic points. These two numbers, (defined appropriately) only depend on n and g where 2n=C^2+2-2g and not on the particular X or C (n is the number of nodes when a curve is nodal and reduced). Gottsche conjectured that certain quasi-modular forms are the generating functions for the number of curves in a fixed linear system. Our theorem proves his formulas and shows that (a different) modular form also arises in the problem of counting curves without fixing a linear system. We use techniques that were developed in our earlier paper for similar questions on K3 surfaces. The techniques include Gromov-Witten invariants for families and a degeneration to an elliptic fibration. One new feature of the Abelian surface case is the presence of non-trivial Pic^0(X). We show that for any surface S the cycle in the moduli space of stable maps defined by requiring that the image of the map lies in a fixed linear system is homologous to the cycle defined by requiring the image of the map meets b_1 generic loops in S representing the generators of the first integral homology group (mod torsion).

Associativity Relations in Quantum Cohomology

The geometry of moduli spaces of stable maps of genus 0 curves into a complex projective manifold X leads to a system of quadratic equations in the tree-level (genus 0) Gromov-Witten numbers of X. In elementary examples, these equations solve for all such numbers, uniquely and consistently, from starting data. The beautiful paper of Di Francesco and Itzykson [1] presents a number of examples in this context. One of the foundational papers in the area of quantum cohomology, [4], explains this phenomenon, at least in some cases, by proving the first reconstruction theorem. This theorem applies to manifoldsX such thatH∗(X,Q) is generated byH(X). This result gives an effective procedure for solving for genus 0 Gromov-Witten numbers from starting data using the quadratic relations (since this entire paper is concerned only with the genus 0 invariants, we omit explicit mention of genus from now on). In all but the simplest cases there will be more than one way of using the relations to solve for the numbers. In other words, the system of equations is overdetermined. In the same paper the authors ask whether the seemingly redundant equations follow algebraically from the useful ones. Consistency of this overdetermined system of equations, as an algebraic (or combinatorial) statement rather than a geometric statement, has only been noted in the literature in isolated instances, cf. [2]. The equations were predicted on the basis of physical theories and later confirmed by rigorous study of the moduli spaces. The view taken by the physicists is quite useful: the numbers are combined into a generating function and the relations are represented by differential equations. The survey paper [6] documents these early studies. This paper continues in the spirit of these early investigations into the structure of the relations. The main result is a generalization of the first reconstruction theorem. Keeping the hypothesis that the cohomology ring of X is generated by divisors, we show that an initial collection of numbers and relations determines, purely algebraically, the entire system of relations (strong reconstruction) as well as the Gromov-Witten numbers. Examples, in the last section, illustrate the existence of non-geometric solutions. For a manifold X with H = H∗(X,C), the associativity relations can be expressed geometrically by saying that an associated connection on TH is flat, i.e., that the quantum potential function dictates on H the structure of a Frobenius manifold [2, 4]. Near a semisimple solution, the equations for flatness can be proved equivalent (with suitable assumptions) to the well-studied Schlesinger equations, cf. [5]. Strong

On equivariant quantum cohomology

Using the Kontsevich's moduli space of stable maps, we define the equivariant quantum cohomology for generalized flag varieties and make a rigorous computation of quantum cohomology of flag varieties.

Tautological relations for stable maps to a target variety

We define tautological relations for the moduli space of stable maps to a target variety. Using the double ramification cycle formula for target varieties of Janda-Pandharipande-Pixton-Zvonkine, we construct nontrivial tautological relations parallel to Pixton's double ramification cycle relations for the moduli of curves. Examples and applications are discussed.