Intersections of loci of admissible covers with tautological classes

Schmitt, Johannes; Zelm, Jason van

doi:10.48550/arxiv.1808.05817

Cited by 5 publications

(7 citation statements)

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“…a pair of a family (X 1 , η 1 ) of multi-scale differentials compatible with an undegeneration of Λ 1 and a family (X 2 , η 2 ) compatible with an undegeneration of Λ 2 . If we forget the differentials, we can construct a family of pointed stable curves (X , z) over some stable graph Π, which is generic as a (Λ 1 , Λ 2 )-stable graph (see [GP03] or [SZ18]). We make Π into a level graph by declaring a vertex v 1 to be on top of v 2 if this holds for either of their images in Λ 1 or in Λ 2 .…”

Section: The Tautological Ringmentioning

confidence: 99%

The Chern classes and the Euler characteristic of the moduli spaces of abelian differentials

Costantini¹,

Möller²,

Zachhuber³

2020

Preprint

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For the moduli spaces of Abelian differentials, the Euler characteristic is one of the most basic intrinsic topological invariants. We give a formula for the Euler characteristic that relies on intersection theory on the smooth compactification by multi-scale differentials. It is a consequence of a formula for the full Chern polynomial of the cotangent bundle of the compactification.The main new technical tools are an Euler sequence for the cotangent bundle of the moduli space of Abelian differentials and computational tools in the Chow ring, such as normal bundles to boundary divisors.

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Section: The Tautological Ringmentioning

confidence: 99%

The Chern classes and the Euler characteristic of the moduli spaces of abelian differentials

Costantini¹,

Möller²,

Zachhuber³

2020

Preprint

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“…By now the package admcycles has been used in a variety of contexts. Its original purpose was computing new examples of admissible cover cycles in [Sv18], e.g. computing the class of the hyperelliptic locus in M 5 and M 6 and the locus of bielliptic cycles in M 4 .…”

Section: Applications Of Admcyclesmentioning

confidence: 99%

“…• admissible cover cycles 1 , such as the fundamental classes of loci of hyperelliptic or bielliptic curves with marked ramification points, as discussed in [Sv18].…”

Section: Introductionmentioning

confidence: 99%

admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves

Delecroix,

Schmitt,

van Zelm

2020

Preprint

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The tautological ring of the moduli space of stable curves has been studied extensively in the last decades. We present a SageMath implementation of many core features of this ring. This includes lists of generators and their products, intersection numbers and verification of tautological relations. Maps between tautological rings induced by functoriality, that is pushforwards and pullbacks under gluing and forgetful maps, are implemented. Furthermore, many interesting cycle classes, such as the double ramification cycles, strata of k-differentials and hyperelliptic or bielliptic cycles are available. In this paper we show how to apply the package, including concrete example computations.be the gluing map associated to Γ. For a class α ∈ H * (M Γ ) given as a product of κ and ψ-classesSuch decorated boundary strata form a generating set (as a Q-vector space) of the tautological ring RH * (M g,n ).Note: Throughout the paper and in the package admcycles, we take the convention of not dividing by the size |Aut(Γ)| of the automorphism group of Γ.

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“…Proof. The proof from [GP03, Proposition 9] of the analogous result for the moduli spaces of stable curves goes through verbatim (see also [Sv18, Section 2] for a more detailed version of the argument).…”

Section: Pullbacks By Gluing Maps and Intersection Productsmentioning

confidence: 99%

Chow rings of stacks of prestable curves I

Bae,

Schmitt

2020

Preprint

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We study the Chow ring of the moduli stack M g,n of prestable curves and define the notion of tautological classes on this stack. We extend formulas for intersection products and functoriality of tautological classes under natural morphisms from the case of the tautological ring of the moduli space M g,n of stable curves. In genus 0, we show that the Chow ring of M 0,n coincides with the tautological ring and give a complete description in terms of (additive) generators and relations. This generalizes earlier results by Keel and Kontsevich-Manin for the spaces of stable curves. Our argument uses the boundary stratification of the moduli stack together with the study of the first higher Chow groups of the strata, in particular providing a new proof of the results of Kontsevich and Manin.6 Note that a priori it is not possible to directly pull back classes in CH * (M g,n ) under the map M g,n (X, β) → M g,n , since this map is in general neither flat nor lci. However, there exists an isomorphismfrom the Chow group of M g,n to its operational Chow group, and operational Chow classes are functorial under arbitrary morphisms. Then, any operational Chow class acts on the Chow group of M g,n (X, β), see Section A.3.

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Intersections of loci of admissible covers with tautological classes

Cited by 5 publications

References 0 publications

The Chern classes and the Euler characteristic of the moduli spaces of abelian differentials

The Chern classes and the Euler characteristic of the moduli spaces of abelian differentials

admcycles -- a Sage package for calculations in the tautological ring of the moduli space of stable curves

Chow rings of stacks of prestable curves I

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