On the Arakelov Geometry of Moduli Spaces of Curves

Hain, Richard; Reed, David

doi:10.4310/jdg/1102536200

Cited by 70 publications

(159 citation statements)

References 59 publications

(106 reference statements)

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“…Hain [17] computed ρ * A Θ in terms of tautological classes on M ct g,n , which, via the above observations, implies a formula for the restriction of the double ramification cycle. The result of his computation is: …”

Section: The Double Ramification Cyclementioning

confidence: 99%

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Pixton’s double ramification cycle relations

Clader

Janda

2018

Geom. Topol.

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Section: The Double Ramification Cyclementioning

confidence: 99%

“…In [17], Hain provided such a formula for the restriction of R g,A to the compact-type locus M ct g,n ⊂ M g,n , which parameterizes curves whose dual graph is a tree. His proof relies on an alternative description of the double ramification cycle in terms of the universal Jacobian.…”

Section: Introductionmentioning

confidence: 99%

Pixton’s double ramification cycle relations

Clader

Janda

2018

Geom. Topol.

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“…The construction makes use of the intersection numbers of the given cohomological field theory with the double ramification cycle, the top Chern class of the Hodge bundle and psi-classes on the moduli space of stable Riemann surfaces M g,n . Since the top Chern class of the Hodge bundle vanishes outside of the moduli space of stable curves of compact type, one can use Hain's formula [Hai13] to express the double ramification cycle in computations and, in particular, the consequent polynomiality of the double ramification cycle with respect to ramification numbers.…”

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confidence: 99%

Recursion Relations for Double Ramification Hierarchies

Buryak

Rossi

2015

Commun. Math. Phys.

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Abstract. In this paper we study various properties of the double ramification hierarchy, an integrable hierarchy of hamiltonian PDEs introduced in [Bur15a] using intersection theory of the double ramification cycle in the moduli space of stable curves. In particular, we prove a recursion formula that recovers the full hierarchy starting from just one of the Hamiltonians, the one associated to the first descendant of the unit of a cohomological field theory. Moreover, we introduce analogues of the topological recursion relations and the divisor equation both for the hamiltonian densities and for the string solution of the double ramification hierarchy. This machinery is very efficient and we apply it to various computations for the trivial and Hodge cohomological field theories, and for the r-spin Witten's classes. Moreover we prove the Miura equivalence between the double ramification hierarchy and the Dubrovin-Zhang hierarchy for the Gromov-Witten theory of the complex projective line (extended Toda hierarchy).

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“…Specialising X/S to a hyperelliptic curve of genus g, one obtains the valuation of the hyperelliptic discriminant (up to the factor 3g − 1) as follows from Theorem 1.1. An interesting discussion of the line bundle λ ⊗8g+4 on M g (C) can be found in [13].…”

Section: Weierstrass Points On Arithmetic Surfacesmentioning

confidence: 99%

Local invariants attached to Weierstrass points

Jong

2009

manuscripta math.

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Abstract. Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation d of the hyperelliptic discriminant of X/S, and the valuation δ of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable the invariants d and δ are known to satisfy certain inequalities. We prove an exact formula relating d and δ with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points.

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On the Arakelov Geometry of Moduli Spaces of Curves

Cited by 70 publications

References 59 publications

Pixton’s double ramification cycle relations

Pixton’s double ramification cycle relations

Recursion Relations for Double Ramification Hierarchies

Local invariants attached to Weierstrass points

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