An Additive Basis for the Chow Ring of M<sub>0,2</sub>(P<sup>r</sup>,2)

Cox, Jonathan

doi:10.3842/sigma.2007.085

Cited by 3 publications

(7 citation statements)

References 19 publications

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“…The diagram representation given above for divisors can easily be extended to describe the closures of other degeneration loci. This description is an alternative to the dual graphs used to describe the degeneration loci in [2]. We will use the diagram representation above when referring to their closures of the degeneration loci as opposed to the loci themselves.…”

Section: Generatorsmentioning

confidence: 99%

“…In a previous article ( [2]), we computed the Betti numbers and an additive basis for the Chow ring A * (M 0,2 (P r , 2)) of the moduli space of degree two stable maps from 2-pointed curves to projective space of arbitrary dimension r. These are the first steps in a program to give presentations for these Chow rings. In the present paper, we will complete this program for the special case r = 1 by showing that…”

Section: Introductionmentioning

confidence: 99%

“…We will work over the field C of complex numbers. As in [2], we let n = N ∩ [1, n] be the initial segment consisting of the first n natural numbers. We will also continue to freely make use of the homology isomorphism A * (M 0,n (P r , d)) → H * (M 0,n (P r , d)).…”

Section: Introductionmentioning

confidence: 99%

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A Presentation for the Chow Ring of

Cox

2007

Communications in Algebra

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Section: Generatorsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Presentation for the Chow Ring of

Cox

2007

Communications in Algebra

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“…In fact, the cohomology groups are generated by tautological classes [17], [18]. Betti number calculations in degree 2 can be found in [3]. Betti number calculations in degree 2 can be found in [3].…”

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confidence: 99%

The Betti numbers of \overline{ℳ}_{0,𝓃}(𝓇,𝒹)

Getzler

Pandharipande

2006

J. Algebraic Geom.

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We calculate the Betti numbers of the coarse moduli space of stable maps of genus 0 to projective space, using a generalization of the Legendre transform.Let M 0,n (r, d) be the moduli space of degree d maps from n-pointed, nonsingular, rational curves to P r over C. Kontsevich has introduced a compactification,.) The main result of our paper is a calculation of the Betti numbers of the coarse moduli space M 0,n (r, d).Let K(V, S n ) be the Grothendieck group of quasi-projective varieties over C with action of the symmetric group S n . Denote the class of a variety X in K(V, S n ) by [X]. If Y is an S n -invariant subspace of X, thenLet S be the groupoidWe call a sequence of quasi-projective varieties X = (X (n) | n ≥ 0) over C with actions of S n an S-space. The Grothendieck group of S-spaces decomposes as a productOur main result is a combinatoric formula which relates the classes

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“…Indeed, in the last few years there has been a flurry of research about the cohomological properties of M 0,n (P r , d) after the first steps moved in [2]: it is worth mentioning at least the contributions [1], [3], [20], [21], [22], [16], [17], [18], [19], [9], [4], [5], [6].…”

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confidence: 99%

Towards the cohomology of moduli spaces of higher genus stable maps

Fontanari

2007

Arch. Math.

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Abstract. We prove that the orbifold desingularization of the moduli space of stable maps of genus g = 1 recently constructed by Vakil and Zinger has vanishing rational cohomology groups in odd degree k < 10. 1.Let M g,n (P r , d) be the moduli space of stable maps of degree d from n-pointed curves of genus g to the projective space P r . For a comprehensive introduction to this fashinating geometric object we refer to the classical text [7], which not only contains a careful construction of Kontsevich moduli spaces but also outlines their crucial application to Gromov-Witten theory.For g = 0 it turns out that M 0,n (P r , d) carries a natural structure of smooth orbifold, in particular its rational cohomology is well-defined and well-behaved. Indeed, in the last few years there has been a flurry of research about the cohomological properties of M 0,n (P For higher genus g > 0, instead, M g,n (P r , d) can be arbitrarily singular (see [24]) and it is in general nonreduced with several components of exceptional dimension. As a consequence, a cohomological investigation of the underlying topological spaces has never been addressed and it seems to be completely out of reach. However, at least in the case g = 1, the beautiful construction performed in [25] has recently opened a new frontier to the research in the field. Namely, in [25] it is shown that the closure M 0 1,n (P r , d) of the stratum M 0 1,n (P r , d) corresponding to stable maps with smooth domain allows an orbifold desingularizationwhich can be explicitely described as a sequence of blow-ups along smooth centers. Moreover, the natural (C * ) r+1 -action on M 0 1,n (P r , d)

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An Additive Basis for the Chow Ring of M_0,2(P^r,2)

Cited by 3 publications

References 19 publications

A Presentation for the Chow Ring of

A Presentation for the Chow Ring of

The Betti numbers of \overline{ℳ}_{0,𝓃}(𝓇,𝒹)

Towards the cohomology of moduli spaces of higher genus stable maps

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An Additive Basis for the Chow Ring of M0,2(Pr,2)

Cited by 3 publications

References 19 publications

A Presentation for the Chow Ring of

A Presentation for the Chow Ring of

The Betti numbers of \overline{ℳ}_{0,𝓃}(𝓇,𝒹)

Towards the cohomology of moduli spaces of higher genus stable maps

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Product

Resources

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An Additive Basis for the Chow Ring of M_0,2(P^r,2)