Some intersection numbers of divisors on toroidal compactifications of 𝒜_{ℊ}

Erdenberger, Cord; Grushevsky, Samuel; Hulek, Klaus

doi:10.1090/s1056-3911-09-00512-8

Cited by 4 publications

(12 citation statements)

References 22 publications

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“…The theme of homology (cohomology-) stabilization is indeed a very general one, which has been recently revived through the work of several authors, also in other contexts ( see for instance [131,153,158,203]). …”

Section: Stabilization Results For the Homology Of Moduli Spaces Of Cmentioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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One of the main themes of this long article is the study of projective varieties which are K(H,1)'s, i.e. classifying spaces BH for some discrete group H. After recalling the basic properties of such classifying spaces, an important class of such varieties is introduced, the one of Bagnera-de Franchis varieties, the quotients of an Abelian variety by the free action of a cyclic group. Moduli spaces of Abelian varieties and of algebraic curves enter into the picture as examples of rational K(H,1)'s, through Teichmüller theory. The main trhust of the paper is to show how in the case of K(H,1)'s the study of moduli spaces and deformation classes can be achieved through by now classical results concerning regularity of classifying maps. The Inoue type varieties of Bauer and Catanese are introduced and studied as a key example, and new results are shown. Motivated from this study, the moduli spaces of algebraic varieties, and especially of algebraic curves with a group of automorphisms of a given topological type are studied in detail, following new results by the author, Michael Lönne and Fabio Perroni. Finally, the action of the absolute Galois group on the moduli spaces of such K(H,1) varieties is studied. In the case of surfaces isogenous to a product, it is shown how this yields a faifhtul action on the set of connected components of the moduli space: for each Galois automorphism of order different from 2 there is an Communicated by Efim Zelmanov.The present work took place in the realm of the DFG Forschergruppe 790 "Classification of algebraic surfaces and compact complex manifolds". Part of the article was written when the author was visiting KIAS, Seoul, as KIAS research scholar.

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Section: Stabilization Results For the Homology Of Moduli Spaces Of Cmentioning

confidence: 99%

Topological methods in moduli theory

Catanese

2015

Bull. Math. Sci.

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“…We extend the universal family π : X g → A g to a family over the partial compactification π ′ : X ′ g → A ′ g by globalizing the construction above. We follow the notation of [EGH10], the results and setup of which we now recall. We let X 2 g−1 = X g−1 × A g−1 X g−1 be the fiberwise square, with pr i : X 2 g−1 → X g−1 denoting the projections to the two factors.…”

Section: Notation and Known Resultsmentioning

confidence: 99%

“…The Picard group of the product family X 2 g−1 is generated by the pullback of λ 1 , which we denote by L, by the pullbacks T i = pr * i T of the theta divisors from the two factors, and by the class P of the universal Poincaré bundle, also trivialized along the zero section (see [EGH10]). By abuse of notation, we also use L, T 1 , P and T 2 to denote the pullbacks of these classes to A 1 ( Y ).…”

Section: Notation and Known Resultsmentioning

confidence: 99%

“…Finally, we consider the class of the gluing locus ∆ ∈ A 2 (X ′ g ). Finally, we need to know how these cycles restrict to the bound- [GL08], and [EGH10] we have…”

Section: Notation and Known Resultsmentioning

confidence: 99%

“…Proof. The proof of this formula is a slight extension of the ideas of the proof of [EGH10,Prop. 4.3], where it is shown that ∆| ∆ = P (−P −2T 2 ).…”

Section: Shift-invariant Classesmentioning

confidence: 95%

See 2 more Smart Citations

The zero section of the universal semiabelian variety and the double ramification cycle

Grushevsky¹,

Zakharov²

2014

Duke Math. J.

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We study the Chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties. We describe the subring generated by divisor classes, and compute the class of the partial compactification of the universal zero section, which turns out to lie in this subring. Our formula extends the results for the zero section of the universal uncompactified family.The partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of A g . Our formula provides a first step in a program to understand the Chow groups of A g , especially of the perfect cone compactification, by induction on genus. By restricting to the image of M g under the Torelli map, our results extend the results of Hain on the double ramification cycle, answering Eliashberg's question.

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The class of the locus of intermediate Jacobians of cubic threefolds

Grushevsky

Hulek

2012

Invent. math.

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We study the locus of intermediate Jacobians of cubic threefolds within the moduli space of complex principally polarized abelian fivefolds, and its generalization to arbitrary genus - the locus of abelian varieties with a singular odd two-torsion point on the theta divisor. Assuming that this locus has expected codimension (which we show to be true for genus up to 5), we compute the class of this locus, and of is closure in the perfect cone toroidal compactification, in the Chow, homology, and the tautological ring. We work out the cases of genus up to 5 in detail, obtaining explicit expressions for the classes of the closures of the locus of products of an elliptic curve and a hyperelliptic genus 3 curve, in moduli of principally polarized abelian fourfolds, and of the locus of intermediate Jacobians in genus 5. In the course of our computation we also deal with various intersections of boundary divisors of a level toroidal compactification, which is of independent interest in understanding the cohomology and Chow rings of the moduli spaces.Comment: v2: new section 9 on the geometry of the boundary of the locus of intermediate Jacobians of cubic threefolds. Final version to appear in Invent. Mat

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Some intersection numbers of divisors on toroidal compactifications of 𝒜_{ℊ}

Cited by 4 publications

References 22 publications

Topological methods in moduli theory

Topological methods in moduli theory

The zero section of the universal semiabelian variety and the double ramification cycle

The class of the locus of intermediate Jacobians of cubic threefolds

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