Generalized Boundary Strata Classes

Pixton, Aaron

doi:10.1007/978-3-319-94881-2_9

Cited by 11 publications

(13 citation statements)

References 6 publications

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“…When X is a point, the relations constructed here specialize to the double ramification cycle relations for M g,n conjectured by Pixtion [28] and proven by Clader and Janda in [7]. In fact, our proof follows the strategy of [7].…”

Section: 2mentioning

confidence: 75%

Tautological relations for stable maps to a target variety

Bae

2020

Arkiv För Matematik

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Section: 2mentioning

confidence: 75%

Tautological relations for stable maps to a target variety

Bae

2020

Arkiv För Matematik

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“…Pixton's formula's opened new directions in the subject: new formulas for Hodge classes [35, Section 3], new relations in the tautological ring of

{\bar{scriptM}}_{g, n}

[17], new connections to the loci of meromorphic differentials [24, Appendix], and connections to new integrable hierarchies [9]. 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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“…In [52], Pixton has defined a subalgebra of the tautological ring R * (M g,n ) spanned by generalized boundary strata classes: tautological classes [Γ] associated to prestable graphs Γ of genus g with n legs. If Γ is a semistable graph (every genus 0 vertex is incident to at least two legs or half-edges), then Pixton's definition takes a simple form.…”

Section: Pixton's Generalized Boundary Strata Classesmentioning

confidence: 99%

The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves

Molcho¹,

Pandharipande²,

Schmitt³

2021

Preprint

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We bound from below the complexity of the top Chern class λg of the Hodge bundle in the Chow ring of the moduli space of curves: no formulas for λg in terms of classes of degrees 1 and 2 can exist. As a consequence of the Torelli map, the 0-section over the second Voronoi compactification of the moduli of principally polarized abelian varieties also can not be expressed in terms of classes of degree 1 and 2. Along the way, we establish new cases of Pixton's conjecture for tautological relations.In the log Chow ring of the moduli space of curves, however, we prove λg lies in the subalgebra generated by logarithmic boundary divisors. The proof is effective and uses Pixton's double ramification cycle formula together with a foundational study of the tautological ring defined by a normal crossings divisor. The results open the door to the search for simpler formulas for λg on the moduli of curves after log blow-ups. Contents 1 Introduction 2 λ g in the Chow ring 3 The log Chow ring 4 Relationship with logarithmic geometry 5 The divisor subalgebra of log Chow 6 Pixton's formula for λ g ∈ CH ⋆ (M g ) 7 The bChow ring A The fourth cohomology group of M g B Computations in admcycles

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Generalized Boundary Strata Classes

Cited by 11 publications

References 6 publications

Tautological relations for stable maps to a target variety

Tautological relations for stable maps to a target variety

Double ramification cycles with target varieties

The Hodge bundle, the universal 0-section, and the log Chow ring of the moduli space of curves

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