Moduli Space Of Stable Curves Research Articles

Integration on moduli spaces of stable curves through localization

We introduce a new method of calculating intersections on M ¯ g , n , using localization of equivariant cohomology. As an application, we give a proof of Mirzakhani's recursion relation for calculating intersections of mixed ψ and κ 1 classes.

Tautological relations and the $r$-spin Witten conjecture

In a series of two preprints, Y.-P. Lee studied relations satisfied by all formal Gromov-Witten potentials, as defined by A. Givental. He called them "universal relations" and studied their connection with tautological relations in the cohomology ring of moduli spaces of stable curves. Building on Y.-P. Lee's work, we give a simple proof of the fact that every tautological relation gives rise to a universal relation (which was also proved by Y.-P. Lee modulo certain results announced by C. Teleman). In particular, this implies that in any semi-simple Gromov-Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov-Witten potentials coincide. As the most important application, we show that our results suffice to deduce the statement of a 1991 Witten conjecture on r-spin structures from the results obtained by Givental for the corresponding formal Gromov-Witten potential. The conjecture in question states that certain intersection numbers on the moduli space of r-spin structures can be arranged into a power series that satisfies the r-KdV (or r-th higher Gelfand-Dikii) hierarchy of partial differential equations.

GIT constructions of moduli spaces of stable curves and maps

This largely expository paper first gives an introduction to Hilbert stability and its use in Gieseker's GIT construction of $\overline{M}_g$. Then I review recent work in this area--variants for unpointed curves that arise in Hassett's log minimal model program, starting with Schubert's moduli space of pseudostable curves, and constructions for weighted pointed stable curves and for pointed stable maps due to Swinarski and to Baldwin and Swinarski respectively. The focus is on the steps at which new ideas are needed. Finally, I list open problems in the area, particularly some arising in the log minimal model program that seem inaccessible to current techniques.

Positive divisors on quotients of [formula omitted] and the Mori cone of [formula omitted

We prove that if m ≥ n − 3 then every S m -invariant F -nef divisor on the moduli space of stable n -pointed curves of genus zero is linearly equivalent to an effective combination of boundary divisors. As an application, we determine the Mori cone of the moduli spaces of stable curves of small genus with few marked points.

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 3 d − 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.

Compactification of the moduli space of hyperplane arrangements

Consider the moduli space M 0 M^0 of arrangements of n n hyperplanes in general position in projective ( r − 1 ) (r-1) -space. When r = 2 r=2 the space has a compactification given by the moduli space of stable curves of genus 0 0 with n n marked points. In higher dimensions, the analogue of the moduli space of stable curves is the moduli space of stable pairs: pairs ( S , B ) (S,B) consisting of a variety S S (possibly reducible) and a divisor B = B 1 + ⋯ + B n B=B_1+\dots +B_n , satisfying various additional conditions. We identify the closure of M 0 M^0 in the moduli space of stable pairs as Kapranov’s Hilbert quotient compactification of M 0 M^0 , and give an explicit description of the pairs at the boundary. We also construct additional irreducible components of the moduli space of stable pairs.

Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves

We consider the moduli space \mathcal H_{g,n} of n -pointed smooth hyperelliptic curves of genus g . In order to get cohomological information we wish to make \mathbb S_n -equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfil. Thus, if we can make \mathbb S_n -equivariant counts of \mathcal H_{g,n} for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for \mathcal H_{g,n} for all g and for n \leq 7 . These results are applied to the moduli spaces of stable curves of genus 2 with up to 7 points, and this gives us the \mathbb S_n -equivariant Galois (resp. Hodge) structure of their \ell -adic (resp. Betti) cohomology.

Stable degenerations of symmetric squares of curves

The stable (in the sense of the relative minimal model program) degenerations of symmetric squares of smooth curves of genus g>2 are computed. This information is used to prove that the component of the moduli space of stable surfaces parameterizing such surfaces is isomorphic to the moduli space of stable curves of genus g.

Pointed admissible G-covers and G-equivariant cohomological field theories

For any finite group G we define the moduli space of pointed admissible G-covers and the concept of a G-equivariant cohomological field theory (G-CohFT), which, when G is the trivial group, reduces to the moduli space of stable curves and a cohomological field theory (CohFT), respectively. We prove that taking the ‘quotient’ by G reduces a G-CohFT to a CohFT. We also prove that a G-CohFT contains a G-Frobenius algebra, a G-equivariant generalization of a Frobenius algebra, and that the ‘quotient’ by G agrees with the obvious Frobenius algebra structure on the space of G-invariants, after rescaling the metric. We then introduce the moduli space of G-stable maps into a smooth, projective variety X with G action. Gromov–Witten-like invariants of these spaces provide the primary source of examples of G-CohFTs. Finally, we explain how these constructions generalize (and unify) the Chen–Ruan orbifold Gromov–Witten invariants of of Fantechi and Göttsche.

Moduli of Stable Surfaces

This article is a survey of basic facts about the moduli spaces of stable surfaces. These spaces are projective compactifications of the moduli spaces of minimal surfaces of general type. They share few of the nice features of the moduli spaces of stable curves. Some of the main pathologies are presented with some historical remarks and some more recent results on the boundary of these spaces.

Combinatorial Properties of Stable Spin Curves#

The geometry of the moduli space of stable spin curves is studied, with emphasis on its combinatorial properties. In this context, the standard graph-theoretic framework is not just a book-keeping device: some purely combinatorial results are proved, having moduli- theoretic applications. In particular, certain strata of the moduli space of stable curves are characterized by a (finite) set of integers, measuring the non-reducedness of the scheme of spin curves, and definable in purely graph-theoretical terms. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

Matrix Integrals and Feynman Diagrams in the Kontsevich Model

We review some relations occurring between the combinatorial intersection theory on the moduli spaces of stable curves and the asymptotic behavior of the ’t Hooft-Kontsevich matrix integrals. In particular, we give an alternative proof of the Witten-Di Francesco-ItzyksonZuber theorem —which expresses derivatives of the partition function of intersection numbers as matrix integrals— using techniques based on diagrammatic calculus and combinatorial relations among intersection numbers. These techniques extend to a more general interaction potential.

Curves of genus 2 and Desargues configurations

A Desargues configuration is the configuration of 10 points and 10 lines of the classical theorem of Desargues in the complex projective plane. For a precise definition see Section 2. The Greek mathematician C. Stephanos showed in 1883 (see [10]) that one can associate to every Desargues configuration a curve of genus 2 in a canonical way. Moreover he proved that the induced map from the moduli space of Desargues configurations to the moduli space of curves of genus 2 is birational. Our main motivation for writing this paper was to understand the last result. Stephanos needed about a hundred pages of classical invariant theory to prove it. We apply instead a simple argument of Schubert calculus to prove a slightly more precise version of his result. Let MD denote the (coarse) moduli space of Desargues configurations. It is a threedimensional quasiprojective variety. On the other hand, let M 6 denote the moduli space of stable binary sextics. There is a canonical isomorphism H2nD1 GM 6 (see [1]). Here H2 denotes the moduli space of stable curves of genus 2 in the sense of Deligne–Mumford and D1 the boundary divisor parametrizing two curves of genus 1 intersecting transversely in one point. So instead of curves of genus 2, we may speak of binary sextics. The main result of the paper consists of the following two statements:

Nef Divisors in Codimension One on the Moduli Space of Stable Curves

Let Mg be the moduli space of smooth curves of genus g [ges ] 3, and M¯g the Deligne-Mumford compactification in terms of stable curves. Let M¯g[1] be an open set of M¯g consisting of stable curves of genus g with one node at most. In this paper, we determine the necessary and sufficient condition to guarantee that a -divisor D on M¯g is nef over M¯g[1], that is, (D · C) [ges ] 0 for all irreducible curves C on M¯g with C ∩ M¯g[1] ≠ [empty ].

Higher cycles on the moduli space of stable curves

We construct a number of analytic cycles on the moduli space of stable curves by using three moduli spaces: the moduli space of tori with one marked point, that of spheres with four marked points, and that of tori with two marked points. We then prove the linear independence of the cycles in the rational homology groups in order to improve Wolpert's estimates for even degree Betti numbers of the moduli space of stable curves.

Stable log surfaces and limits of quartic plane curves

Let C⊂ℙ2 be a smooth plane curve of degree d≥ 4. We regard pairs (ℙ}2,C) as stable log surfaces, higher-dimensional analogs to pointed stable curves. Using the log minimal model program, Kollar, Shepherd-Barron, and Alexeev have constructed projective moduli spaces for stable log surfaces. Unfortunately, few explicit examples of these moduli spaces are known. The purpose of this paper is to give a concrete description of these spaces for plane curves of small degree. In particular, we show that the moduli space of stable log surfaces corresponding to quartic plane curves coincides with the moduli space of stable curves of genus three.

Lefschetz fibrations and the Hodge bundle

Integral symplectic 4-manifolds may be described in terms of Lefschetz fibrations. In this note we give a formula for the signature of any Lefschetz fibration in terms of the second cohomology of the moduli space of stable curves. As a consequence we see that the sphere in moduli space defined by any (not necessarily holomorphic) Lefschetz fibration has positive "symplectic volume"; it evaluates positively with the Kahler class. Some other applications of the signature formula and some more general results for genus two fibrations are discussed.

The Semi-Classical Approximation for Modular Operads

We study the contribution of one-loop graphs (the semi-classical expansion) problem in the setting of modular operads. As an application, we calculate the Betti numbers of the Deligne-Mumford-Knudsen moduli spaces of stable curves of genus 1 with n marked smooth points.

ON THE SLOPES OF THE MODULI SPACES OF CURVES

Let Mg be the moduli space of smooth curves of genus g and Mg the moduli space of stable curves of genus g. Then Mg = MgU A, where A = YJg£ ] A*> Ao is the closure of the locus of genus g — 1 curves with one double point, and A* (i ^ 0) is the the closure of the locus of the stable curves of type (i,g — i). Let A be the class of the Hodge line bundle on A4S, let Si be the class of Aj if i ^ 1, and let Si be half of the class of Ai . Let S = So H h 4 [11], and on the moduli functor, the canonical class is 13A — 2S, the only difference is the coefficient of Si (see [11, 9] for the details). This class is called effective if n(aX — bS) is effective for sufficiently large and sufficiently divisible n. Harris and Morrison [10] define the slope of the moduli space as

Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves

Let f : X → Y f : X \to Y be a surjective and projective morphism of smooth quasi-projective varieties over an algebraically closed field of characteristic zero with dim ⁡ f = 1 \dim f = 1 . Let E E be a vector bundle of rank r r on X X . In this paper, we would like to show that if X y X_y is smooth and E y E_y is semistable for some y ∈ Y y \in Y , then f ∗ ( 2 r c 2 ( E ) − ( r − 1 ) c 1 ( E ) 2 ) f_*\left ( 2rc_2(E) - (r-1)c_1(E)^2 \right ) is weakly positive at y y . We apply this result to obtain the following description of the cone of weakly positive Q \mathbb {Q} -Cartier divisors on the moduli space of stable curves. Let M ¯ g \overline {\mathcal {M}}_g (resp. M g \mathcal {M}_g ) be the moduli space of stable (resp. smooth) curves of genus g ≥ 2 g \geq 2 . Let λ \lambda be the Hodge class, and let the δ i \delta _i ’s ( i = 0 , … , [ g / 2 ] i = 0, \ldots , [g/2] ) be the boundary classes. Then, a Q \mathbb {Q} -Cartier divisor x λ + ∑ i = 0 [ g / 2 ] y i δ i x \lambda + \sum _{i=0}^{[g/2]} y_i \delta _i on M ¯ g \overline {\mathcal {M}}_g is weakly positive over M g \mathcal {M}_g if and only if x ≥ 0 x \geq 0 , g x + ( 8 g + 4 ) y 0 ≥ 0 g x + (8g + 4) y_0 \geq 0 , and i ( g − i ) x + ( 2 g + 1 ) y i ≥ 0 i(g-i) x + (2g+1) y_i \geq 0 for all 1 ≤ i ≤ [ g / 2 ] 1 \leq i \leq [g/2] .