A calculus for the moduli space of
                    curves

Pandharipande, Rahul

doi:10.1090/pspum/097.1/01682

Cited by 25 publications

(16 citation statements)

References 83 publications

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“…For n = 1, 2, 3 we can compute the class of the bielliptic locus in terms of decorated stratum classes and thus the corresponding intersection numbers with λ n+1 λ n−1 . The results agree with the coefficients in F(u) predicted by the formula (37) above. This provides a further nontrivial check for our computations.…”

Section: Remarksupporting

confidence: 88%

Intersections of loci of admissible covers with tautological classes

Schmitt

Zelm

2020

Sel. Math. New Ser.

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For a finite group G, let $$\overline{\mathcal {H}}_{g,G,\xi }$$ H ¯ g , G , ξ be the stack of admissible G-covers $$C\rightarrow D$$ C → D of stable curves with ramification data $$\xi $$ ξ , $$g(C)=g$$ g ( C ) = g and $$g(D)=g'$$ g ( D ) = g ′ . There are source and target morphisms $$\phi :\overline{\mathcal {H}}_{g,G,\xi }\rightarrow \overline{\mathcal {M}}_{g,r}$$ ϕ : H ¯ g , G , ξ → M ¯ g , r and $$\delta :\overline{\mathcal {H}}_{g,G,\xi }\rightarrow \overline{\mathcal {M}}_{g',b}$$ δ : H ¯ g , G , ξ → M ¯ g ′ , b , remembering the curves C and D together with the ramification or branch points of the cover respectively. In this paper we study admissible cover cycles, i.e. cycles of the form $$\phi _* [\overline{\mathcal {H}}_{g,G,\xi }]$$ ϕ ∗ [ H ¯ g , G , ξ ] . Examples include the fundamental classes of the loci of hyperelliptic or bielliptic curves C with marked ramification points. The two main results of this paper are as follows: firstly, for the gluing morphism $$\xi _A:\overline{\mathcal {M}}_A\rightarrow \overline{\mathcal {M}}_{g,r}$$ ξ A : M ¯ A → M ¯ g , r associated to a stable graph A we give a combinatorial formula for the pullback $$\xi ^*_A \phi _*[\overline{\mathcal {H}}_{g,G,\xi }]$$ ξ A ∗ ϕ ∗ [ H ¯ g , G , ξ ] in terms of spaces of admissible G-covers and $$\psi $$ ψ classes. This allows us to describe the intersection of the cycles $$\phi _*[\overline{\mathcal {H}}_{g,G,\xi }]$$ ϕ ∗ [ H ¯ g , G , ξ ] with tautological classes. Secondly, the pull–push $$\delta _*\phi ^*$$ δ ∗ ϕ ∗ sends tautological classes to tautological classes and we give an explicit combinatorial description of this map. We show how to use the pullbacks to algorithmically compute tautological expressions for cycles of the form $$\phi _* [\overline{\mathcal {H}}_{g,G,\xi }]$$ ϕ ∗ [ H ¯ g , G , ξ ] . In particular, we compute the classes "Equation missing" and "Equation missing" of the hyperelliptic loci in $$\overline{\mathcal {M}}_5$$ M ¯ 5 and $$\overline{\mathcal {M}}_6$$ M ¯ 6 and the class "Equation missing" of the bielliptic locus in $$\overline{\mathcal {M}}_4$$ M ¯ 4 .

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Section: Remarksupporting

confidence: 88%

Intersections of loci of admissible covers with tautological classes

Schmitt

Zelm

2020

Sel. Math. New Ser.

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“…For a sampling of the subsequent study and applications, see [6–8, 10, 12, 16, 23, 31, 32, 45, 56, 57]. We refer the reader to [35, Section 0] and [50, Section 5] for more leisurely introductions to the subject.…”

Section: Introductionmentioning

confidence: 99%

Double ramification cycles with target varieties

Janda

Pandharipande

Pixton

et al. 2020

Journal of Topology

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Let X be a nonsingular projective algebraic variety over C, and let M g,n,β (X) be the moduli space of stable maps f : (C, x 1 ,. .. , x n) → X from genus g, n-pointed curves C to X of degree β. Let S be a line bundle on X. Let A = (a 1 ,. .. , a n) be a vector of integers which satisfy n i=1 a i = β c 1 (S). Consider the following condition: the line bundle f * S has a meromorphic section with zeroes and poles exactly at the marked points x i with orders prescribed by the integers a i. In other words, we require f * S (− n i=1 a i x i) to be the trivial line bundle on C. A compactification of the space of maps based upon the above condition is given by the moduli space of stable maps to rubber over X and is denoted by M ∼ g,A,β (X, S). The moduli space carries a virtual fundamental class [M ∼ g,A,β (X, S)] vir ∈ A * M ∼ g,A,β (X, S) in Gromov-Witten theory. The main result of the paper is an explicit formula (in tautological classes) for the push-forward via the forgetful morphism of [M ∼ g,A,β (X, S)] vir to M g,n,β (X). In case X is a point, the result here specializes to Pixton's formula for the double ramification cycle proven in [28]. Several applications of the new formula are given. 1 Contents 0 Introduction 2 1 Curves with an rth root 13 2 GRR for the universal line bundle 27 3 Localization analysis 36 4 Applications 51 We define the set G g,n,β (X) of X-valued stable graphs as follows. A graph Γ ∈ G g,n,β (X) consists of the data

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“…See[23] for a survey of the Faber-Zagier relations and related topics.3 A proof (unpublished) was found by Faber and Zagier in 2002. The result is also derived in [26, Section 3].…”

mentioning

confidence: 99%

Tautological relations via 𝑟-spin structures

Pandharipande¹,

Pixton²,

Zvonkine³

2019

J. Algebraic Geom.

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Relations among tautological classes on M g,n are obtained via the study of Witten's r-spin theory for higher r. In order to calculate the quantum product, a new formula relating the r-spin correlators in genus 0 to the representation theory of sl 2 (C) is proven. The Givental-Teleman classification of CohFTs is used at two special semisimple points of the associated Frobenius manifold. At the first semisimple point, the R-matrix is exactly solved in terms of hypergeometric series. As a result, an explicit formula for Witten's r-spin class is obtained (along with tautological relations in higher degrees). As an application, the r = 4 relations are used to bound the Betti numbers of R * (M g ). At the second semisimple point, the form of the R-matrix implies a polynomiality property in r of Witten's r-spin class.In the Appendix (with F. Janda), a conjecture relating the r = 0 limit of Witten's r-spin class to the class of the moduli space of holomorphic differentials is presented. 0 Introduction 0.1 Overview Let M g,n be the moduli space of stable genus g curves with n markings. Letbe the subring of tautological classes in cohomology 1 . The subringsare defined together as the smallest system of Q-subalgebras closed under push-forward via all boundary and forgetful maps, see [6,7,12]. There has been substantial progress in the understanding of RH * (M g,n ) since the study began in the 1980s [21]. The subject took a new turn in 2012 with the family of relations conjectured in [25]. We refer the reader to [23] for a survey of recent developments.Witten's r-spin class defines a Cohomological Field Theory (CohFT) for each integer r ≥ 2. Witten's 2-spin theory concerns only the fundamental classes of the moduli spaces of curves (and leads to no new geometry). In our previous paper [24], we used Witten's 3-spin theory to construct a family of relations among tautological classes of M g,n equivalent (in cohomology) to the relations proposed in [25]. Our goal here is to extend our study of tautological relations to Witten's r-spin theory for all r ≥ 3.Taking [24] as a starting point, Janda has completed a formal study of tautological relations obtained from CohFTs. Two results of Janda are directly relevant here:(i) The relations for r = 3 are valid in Chow [14,16].(ii) The relations for r ≥ 4 are implied by the relations for r = 3 [14,15]. By (i) and (ii) together, all of the r-spin relations that we find will be valid in Chow. However, since our methods here are cohomological, we will use the language of cohomology throughout the paper.

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A calculus for the moduli space of curves

Cited by 25 publications

References 83 publications

Intersections of loci of admissible covers with tautological classes

Intersections of loci of admissible covers with tautological classes

Double ramification cycles with target varieties

Tautological relations via 𝑟-spin structures

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