Results 1  10
of
62
Relative virtual localization and vanishing of tautological classes on moduli spaces of curves
 Duke Math. J
"... ABSTRACT. We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As ..."
Abstract

Cited by 68 (6 self)
 Add to MetaCart
(Show Context)
ABSTRACT. We prove a localization formula for the moduli space of stable relative maps. As an application, we prove that all codimension i tautological classes on the moduli space of stable pointed curves vanish away from strata corresponding to curves with at least i − g + 1 genus 0 components. As consequences, we prove and generalize various conjectures and theorems about various moduli spaces of curves (due to Getzler, Ionel, Faber, Looijenga, Pandharipande, Diaz, and others). This theorem appears to be the geometric content behind these results; the rest is straightforward graph combinatorics. The theorem also suggests the importance of the stratification of the moduli space by number of rational components. CONTENTS
Algebraic cobordism revisited
"... Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provid ..."
Abstract

Cited by 52 (7 self)
 Add to MetaCart
(Show Context)
Abstract. We define a cobordism theory in algebraic geometry based on normal crossing degenerations with double point singularities. The main result is the equivalence of double point cobordism to the theory of algebraic cobordism previously defined by Levine and Morel. Double point cobordism provides a simple, geometric presentation of algebraic cobordism theory. As a corollary, the Lazard ring given by products of projective spaces rationally generates all nonsingular projective varieties modulo double point degenerations. Double point degenerations arise naturally in relative DonaldsonThomas theory. We use double point cobordism to prove all the degree 0 conjectures in DonaldsonThomas theory: absolute, relative, and equivariant. 0.1. Overview. A first idea for defining cobordism in algebraic geometry is to impose the relation
On a proof of a conjecture of MarinoVafa on Hodge integrals
"... Abstract. We prove a remarkable formula for Hodge integrals conjectured by Mariño and Vafa [23] based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms. 1. ..."
Abstract

Cited by 46 (15 self)
 Add to MetaCart
Abstract. We prove a remarkable formula for Hodge integrals conjectured by Mariño and Vafa [23] based on large N duality, using functorial virtual localization on certain moduli spaces of relative stable morphisms. 1.
Tautological relations and the rspin Witten conjecture
"... In [23, 24], Y.P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.P. Lee conjectured that the two sets of relations coincide and proved ..."
Abstract

Cited by 43 (11 self)
 Add to MetaCart
(Show Context)
In [23, 24], Y.P. Lee introduced a notion of universal relation for formal Gromov–Witten potentials. Universal relations are connected to tautological relations in the cohomology ring of the moduli space Mg,n of stable curves. Y.P. Lee conjectured that the two sets of relations coincide and proved the inclusion (tautological relations) ⊂ (universal relations) modulo certain results announced by C. Teleman. He also proposed an algorithm that, conjecturally, computes all universal/tautological relations. Here we give a geometric interpretation of Y.P. Lee’s algorithm. This leads to a much simpler proof of the fact that every tautological relation gives rise to a universal relation. We also show that Y.P. Lee’s algorithm computes the tautological relations correctly if and only if the Gorenstein conjecture on the tautological cohomology ring of Mg,n is true. These results are first steps in the task of establishing an equivalence between formal and geometric Gromov–Witten theories. In particular, it implies that in any semisimple 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.
Virasoro constraints for target curves
, 2003
"... We prove generalized Virasoro constraints for the relative GromovWitten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. Descendents of the odd cohomology are then controlled by monodr ..."
Abstract

Cited by 38 (9 self)
 Add to MetaCart
(Show Context)
We prove generalized Virasoro constraints for the relative GromovWitten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. Descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations. As an outcome of our results, the relative theories of target curves are
The moduli space of curves and GromovWitten theory
, 2006
"... The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology r ..."
Abstract

Cited by 26 (4 self)
 Add to MetaCart
The goal of this article is to motivate and describe how GromovWitten theory can and has provided tools to understand the moduli space of curves. For example, ideas and methods from GromovWitten theory have led to both conjectures and theorems showing that the tautological part of the cohomology ring has a remarkable and profound structure. As an illustration, we describe a new approach to Faber’s intersection number conjecture via branched covers of the projective line (work with I.P. Goulden and D.M. Jackson, based on work with T. Graber). En route we review the work of a large number of mathematicians.
The moduli space of curves, double Hurwitz numbers, and Faber’s intersection number conjecture
, 2006
"... We define the dimension 2g − 1 FaberHurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of P 1 with given ramification over ∞ and sufficiently many fixed ramification points elsewhere. Degeneration of the target and judicious localizatio ..."
Abstract

Cited by 17 (2 self)
 Add to MetaCart
(Show Context)
We define the dimension 2g − 1 FaberHurwitz Chow/homology classes on the moduli space of curves, parametrizing curves expressible as branched covers of 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 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 ψclasses. 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 κg−2.
A short proof of the λgConjecture without GromovWitten theory: Hurwitz theory and the moduli of curves
"... Abstract. We give a short and direct proof of the λgConjecture. The approach is through the EkedahlLandoShapiroVainshtein theorem, which establishes the “polynomiality ” of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of GromovWitten theory. We brief ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
Abstract. We give a short and direct proof of the λgConjecture. The approach is through the EkedahlLandoShapiroVainshtein theorem, which establishes the “polynomiality ” of Hurwitz numbers, from which we pick off the lowest degree terms. The proof is independent of GromovWitten theory. We briefly describe the philosophy behind our general approach to intersection numbers and how it may be extended to other intersection number conjectures.