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

Getzler, Ezra; Pandharipande, Rahul

doi:10.1090/s1056-3911-06-00425-5

Cited by 17 publications

(16 citation statements)

References 17 publications

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“…A G-variety is a variety with an action of a group G. An S-space X is a sequence of 𝑆 𝑛 -varieties X (𝑛) for 𝑛 ≥ 0. Getzler-Pandharipande define a Grothendieck ring of S-spaces [14]. We briefly generalize this formalism to the case of (𝑆 𝑚 × 𝑆 𝑛 )-varieties.…”

Section: The Grothendieck Ring Of S 2 -Spacesmentioning

confidence: 99%

Equivariant Hodge polynomials of heavy/light moduli spaces

Kannan,

Serpente,

Yun

2024

Forum of Mathematics, Sigma

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Let $\overline {\mathcal {M}}_{g, m|n}$ denote Hassett’s moduli space of weighted pointed stable curves of genus g for the heavy/light weight data $$\begin{align*}\left(1^{(m)}, 1/n^{(n)}\right),\end{align*}$$ and let $\mathcal {M}_{g, m|n} \subset \overline {\mathcal {M}}_{g, m|n}$ be the locus parameterizing smooth, not necessarily distinctly marked curves. We give a change-of-variables formula which computes the generating function for $(S_m\times S_n)$ -equivariant Hodge–Deligne polynomials of these spaces in terms of the generating functions for $S_{n}$ -equivariant Hodge–Deligne polynomials of $\overline {\mathcal {M}}_{g,n}$ and $\mathcal {M}_{g,n}$ .

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Section: The Grothendieck Ring Of S 2 -Spacesmentioning

confidence: 99%

Equivariant Hodge polynomials of heavy/light moduli spaces

Kannan,

Serpente,

Yun

2024

Forum of Mathematics, Sigma

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“…This section owes much to Getzler and Pandharipande, who provide the framework for computing the Betti numbers of all the spaces M 0,n (P r , d) in [9]. However, we will take the definitions and basic results from other sources, and prove their theorem in the special case that we need.…”

Section: Proposition 1 (Homology Isomorphism)mentioning

confidence: 99%

“…However, we will take the definitions and basic results from other sources, and prove their theorem in the special case that we need. We will compute a formula for the Poincaré polynomials of the moduli spaces M 0,2 (P r , 2) using what are called Serre polynomials in [9] and Serre characteristics in [10]. (These polynomials are also known as virtual Poincaré polynomials or E-polynomials.)…”

Section: Proposition 1 (Homology Isomorphism)mentioning

confidence: 99%

“…We accomplish this in Section 2 by using the equivariant Serre polynomial method of Getzler and Pandharipande. I owe much gratitude to Getzler and Pandharipande for supplying a copy of their unpublished preprint [9], which provided the inspiration for Section 2. In Section 3, we use the presentations of the Chow rings for the moduli spaces M 0,0 (P r , d) from [21] together with excision to determine a generating set for the Chow ring A * (M 0,2 (P r , 2)).…”

Section: Introductionmentioning

confidence: 99%

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

Cox¹

2007

SIGMA

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Abstract. We begin a study of the intersection theory of the moduli spaces of degree two stable maps from two-pointed rational curves to arbitrary-dimensional projective space. First we compute the Betti numbers of these spaces using Serre polynomial and equivariant Serre polynomial methods developed by E. Getzler and R. Pandharipande. Then, via the excision sequence, we compute an additive basis for their Chow rings in terms of Chow rings of nonlinear Grassmannians, which have been described by Pandharipande. The ring structure of one of these Chow rings is addressed in a sequel to this paper.

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

Cited by 17 publications

References 17 publications

Equivariant Hodge polynomials of heavy/light moduli spaces

Equivariant Hodge polynomials of heavy/light moduli spaces

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

Towards the cohomology of moduli spaces of higher genus stable maps

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

Cited by 17 publications

References 17 publications

Equivariant Hodge polynomials of heavy/light moduli spaces

Equivariant Hodge polynomials of heavy/light moduli spaces

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

Towards the cohomology of moduli spaces of higher genus stable maps

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Product

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