The pullback of a theta divisor to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\overline{\mathcal {M}}_{g,n}}$\end{document}

Müller, Fabian

doi:10.1002/mana.201200072

Cited by 10 publications

(3 citation statements)

References 7 publications

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“…Hain's formula answers Eliashberg's question. Analogue results are proven by M üller [25]. Grushevsky and Zakharov [13], [14] give a formula for the pull-back of the theta divisor to the spaces M ct g,n classifying pointed curves of compact type and the space M g,n of stable pointed curves.…”

Section: The Degree G + 1 Relationmentioning

confidence: 57%

Tautological classes on the moduli space of hyperelliptic curves with rational tails

Tavakol

2018

Journal of Pure and Applied Algebra

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We study tautological classes on the moduli space of stable n-pointed hyperelliptic curves of genus g with rational tails. Our result gives a complete description of tautological relations. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology. The connection with recent conjectures by Pixton is also discussed. CONTENTS TAUTOLOGICAL CLASSES ON THE MODULI SPACE H rt g,n 3 Everything mentioned above concerns tautological classes in Chow. The Gorenstein property of R * (H rt g,n ) implies the same results in cohomology. This shows that there is no difference between Chow and cohomology as long as we restrict to tautological classes. Using a result of Petersen and Tommasi, which was our motivation for this project, we prove the following:Corollary 0.2. The cycle class map induces an isomorphism between the tautological ring of the moduli space H rt g,n in Chow and monodromy invariant classes in cohomology.At the end we discuss the connection between the relations on the space H rt g,n and Pixton's relations on M g,n . Conventions 0.3. We consider algebraic cycles modulo rational equivalence. All Chow groups are taken with Q-coefficients.

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Section: The Degree G + 1 Relationmentioning

confidence: 57%

Tautological classes on the moduli space of hyperelliptic curves with rational tails

Tavakol

2018

Journal of Pure and Applied Algebra

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“…Recent seminal work exposes the fundamental algebraic attributes of these spaces [Mc, KZ, Mö, EMM, EMa, Fil]. Furthermore, the condition of the existence of a holomorphic or meromorphic one-form of fixed signature has been used previously both explicitly and under many guises to obtain divisor classes [Cu,D,C,CC1,L,Mü,GZ,F2,FV] and in lower genus, higher codimension cycles [CC2,CT,Bl] in M g and M g,n . In contrast, conditions from k-differentials for k ≥ 2 have remained an untapped source for effective cycles and questions of the relation of these strata to the birational geometry of M g,n have remained largely unexamined.…”

Section: Introductionmentioning

confidence: 99%

$k$-differentials on curves and rigid cycles in moduli space

Mullane

2021

Doc. Math.

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For g ≥ 2, j = 1, . . . , g and n ≥ g + j we exhibit infinitely many new rigid and extremal effective codimension j cycles in M g,n , the Deligne-Mumford compactification of the moduli of npointed curves of genus g. The extremal cycles constructed correspond to the strata of quadratic differentials and projections of these strata under forgetful morphisms. We further show the same holds for k-differentials with k ≥ 3 if the strata are irreducible. We compute the class of the divisors in the case of quadratic differentials which contain the first known examples of effective divisors on M g,n with negative ψ i coefficients.

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“…Remark 1.2. Apart from M g,n itself, other examples of algebro-geometrically interesting varieties that contain a Teichmüller disk are the theta-null divisor (see Müller [Mül13] and Grushevsky-Zakharov [GZ14]), the anti-ramification locus (see Farkas-Verra [FV13]), and the Weierstrass divisor (see Cukierman [Cuk89]). More examples are listed in Mullane [Mul17] and the Kodaira dimensions of many such loci are computed in Gendron [Gen15].…”

Section: Introductionmentioning

confidence: 99%

GL2ℝ–invariant measures in marked strata : generic marked points, Earle–Kra for strata and illumination

Apisa

2020

Geom. Topol.

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We classify GL(2, R) invariant point markings over components of strata of Abelian differentials. Such point markings exist only when the component is hyperelliptic and arise from marking Weierstrass points or two points exchanged by the hyperelliptic involution. We show that these point markings can be used to determine the holomorphic sections of the universal curve restricted to orbifold covers of subvarieties of the moduli space of Riemann surfaces that contain a Teichmüller disk. The finite blocking problem is also solved for translation surfaces with dense GL(2, R) orbit.

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The pullback of a theta divisor to \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}${\overline{\mathcal {M}}_{g,n}}$\end{document}

Cited by 10 publications

References 7 publications

Tautological classes on the moduli space of hyperelliptic curves with rational tails

Tautological classes on the moduli space of hyperelliptic curves with rational tails

$k$-differentials on curves and rigid cycles in moduli space

GL2ℝ–invariant measures in marked strata : generic marked points, Earle–Kra for strata and illumination

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