Nontautological bielliptic cycles

Zelm, Jason van

doi:10.2140/pjm.2018.294.495

Cited by 15 publications

(24 citation statements)

References 11 publications

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“…It is therefore natural to define an extension of the tautological ring obtained by adding all such cycles. As shown in [27,47], this eventually gives strictly more cycle classes.…”

Section: Discussionmentioning

confidence: 80%

“…The class of any locus of admissible covers of genus 0 curves with fixed degree d and fixed ramification profile is tautological (see [18]). However, not all classes coming from spaces of admissible covers are tautological (see for example [27,47]). In fact such classes are the prime example of explicit nontautological algebraic classes.…”

Section: Motivation: Admissible Cover Cyclesmentioning

confidence: 99%

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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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“…It is therefore natural to define an extension of the tautological ring obtained by adding all such cycles. As shown in [27,47], this eventually gives strictly more cycle classes.…”

Section: Discussionmentioning

confidence: 80%

Section: Motivation: Admissible Cover Cyclesmentioning

confidence: 99%

Intersections of loci of admissible covers with tautological classes

Schmitt

Zelm

2020

Sel. Math. New Ser.

Self Cite

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“…(iii) Cycles (φ g/h,d ) * (1) are known in some cases to be non-tautological, see, for example, [18,34]. We defer a more detailed discussion of this phenomenon to the next section.…”

Section: Hurwitz Loci On Moduli Spaces Of Curvesmentioning

confidence: 99%

“…In the classical situation, the answer is k = g − 1 due to Looijenga and Faber, see [15,26]. However, when one allows H-tautological classes, this becomes false when g = 12 due to van Zelm's result that the bielliptic locus on M 12 is non-tautological, see [34,Theorem 2].…”

Section: Further Directionsmentioning

confidence: 99%

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The $${\mathcal {H}}$$-tautological ring

Lian

2021

Sel. Math. New Ser.

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We extend the theory of tautological classes on moduli spaces of stable curves to the more general setting of moduli spaces of admissible Galois covers of curves, introducing the so-called $${\mathcal {H}}$$ H -tautological ring. The main new feature is the existence of restriction-corestriction morphisms remembering intermediate quotients of Galois covers, which are a rich source of new classes. In particular, our new framework includes classes of Harris–Mumford admissible covers on moduli spaces of curves, which are known in some (and speculatively many more) examples to lie outside the usual tautological ring. We give additive generators for the $${\mathcal {H}}$$ H -tautological ring and show that their intersections may be algorithmically computed, building on work of Schmitt-van Zelm. As an application, we give a method for computing integrals of Harris-Mumford loci against tautological classes of complementary dimension, recovering and giving a mild generalization of a recent quasi-modularity result of the author for covers of elliptic curves.

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Non-tautological Hurwitz cycles

Lian

2021

Math. Z.

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We show that various loci of stable curves of sufficiently large genus admitting degree d covers of positive genus curves define non-tautological algebraic cycles on $${\overline{{\mathcal {M}}}}_{g,N}$$ M ¯ g , N , assuming the non-vanishing of the d-th Fourier coefficient of a certain modular form. Our results build on those of Graber-Pandharipande and van Zelm for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by Schmitt-van Zelm and the author.

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Nontautological bielliptic cycles

Cited by 15 publications

References 11 publications

Intersections of loci of admissible covers with tautological classes

Intersections of loci of admissible covers with tautological classes

The $${\mathcal {H}}$$-tautological ring

Non-tautological Hurwitz cycles

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