The Picard groups of the moduli spaces of curves

Arbarello, Enrico; Cornalba, Maurizio

doi:10.1016/0040-9383(87)90056-5

Cited by 147 publications

(190 citation statements)

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“…To compute the Kodaira dimension for n ≥ 11, we first need to express the canonical divisor K M1,n in terms of generators of the rational Picard group of M 1,n . We briefly recall such generators and some of their relations: for more details the reader is referred, for instance, to [2].…”

Section: Proposition 2 Fix G and N Non-negative Integers N > 2 − 2gmentioning

confidence: 99%

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Moduli of curves and spin structures via algebraic geometry

Bini

Fontanari

2006

Trans. Amer. Math. Soc.

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Abstract. Here we investigate some birational properties of two collections of moduli spaces, namely moduli spaces of (pointed) stable curves and of (pointed) spin curves. In particular, we focus on vanishings of Hodge numbers of type (p, 0) and on computations of Kodaira dimension. Our methods are purely algebro-geometric and rely on an induction argument on the number of marked points and the genus of the curves.

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Section: Proposition 2 Fix G and N Non-negative Integers N > 2 − 2gmentioning

confidence: 99%

“…If λ 1 denotes the first Chern class of the Hodge bundle on M 1,1 , then we have λ = π * (λ 1 ) (see for instance [2], (6)). Moreover, since λ 1 is ample on …”

Section: Theorem 3 We Havementioning

confidence: 99%

Moduli of curves and spin structures via algebraic geometry

Bini

Fontanari

2006

Trans. Amer. Math. Soc.

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“…For g ≥ 3, let F 0 g ⊂ F g be the open dense subset parametrizing polarized K3 surfaces with trivial automorphism groups, which carries a universal family X → F 0 g . Motivated by Franchetta's conjecture on the moduli spaces of curves (see [1]), O'Grady asked the following question in [12], referred to as the generalized Franchetta conjecture.Question 0.1 (Generalized Franchetta conjecture). Given a class α ∈ CH 2 (X) and a closed point s ∈ F 0 g , is it true that α| Xs ∈ Zc Xs ?…”

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confidence: 99%

On O'Grady's Generalized Franchetta Conjecture

Pavic

Shen

Yin

2016

Int Math Res Notices

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Abstract. We study relative zero cycles on the universal polarized K3 surface X → Fg of degree 2g − 2. It was asked by O'Grady if the restriction of any class in CH 2 (X) to a closed fiber Xs is a multiple of the Beauville-Voisin canonical class c Xs ∈ CH 0 (Xs). Using Mukai models, we give an affirmative answer to this question for g ≤ 10 and g = 12, 13, 16, 18, 20. 0. Introduction Throughout, we work over the complex numbers. Let S be a projective K3 surface. In [2], Beauville and Voisin studied the Chow ring CH * (S) of S. They showed that there is a canonical class c S ∈ CH 0 (S) represented by a point on a rational curve in S, which satisfies the following properties:(i) The intersection of two divisor classes on S always lies in Zc S ⊂ CH 0 (S).(ii) The second Chern class c 2 (T S ) equals 24c S ∈ CH 0 (S). This result is rather surprising since the Chow group CH 0 (S) is infinite-dimensional by Mumford's theorem [7].Let F g denote the moduli space of (primitively) polarized K3 surfaces of degree 2g − 2. For g ≥ 3, let F 0 g ⊂ F g be the open dense subset parametrizing polarized K3 surfaces with trivial automorphism groups, which carries a universal family X → F 0 g . Motivated by Franchetta's conjecture on the moduli spaces of curves (see [1]), O'Grady asked the following question in [12], referred to as the generalized Franchetta conjecture.Question 0.1 (Generalized Franchetta conjecture). Given a class α ∈ CH 2 (X) and a closed point s ∈ F 0 g , is it true that α| Xs ∈ Zc Xs ? The goal of this paper is to give an affirmative answer to Question 0.1 for a list of small values of g. By the work of Mukai [8,9,10,11], for these g a general polarized K3 surface can be realized in a variety with "small" Chow groups as a complete intersection with respect to a vector bundle.Theorem 0.2. The generalized Franchetta conjecture holds for g ≤ 10 and g = 12, 13, 16, 18, 20.The paper is organized as follows. In Section 1 we review Mukai's constructions and make some comments about Question 0.1. In Section 2 we prove Theorem 0.2 for all cases except g = 13, 16. Two independent proofs are presented, one using Voisin's result [17], the other via a direct calculation. The cases g = 13, 16 have a different flavor and are treated in Section 3. Acknowledgement. We are grateful to Rahul Pandharipande for his constant support and his enthusiasm in this project. We also thank Kieran O'Grady for his careful reading of a preliminary version of this paper.

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“…The next proposition, which is a slight extension of Proposition 2, contains all the geometric information which we will need to compute the Deligne products in (1). We use the same notations as above, but also denote by π = π N −m,m the forgetful map from M g,N to M g,N −m for some m ≥ 0 and use ξ 1 , .…”

Section: Proposition 2 ([6])mentioning

confidence: 99%

“…The Picard group of M g,N is known to be free of rank N + 1 [4] and has a Z-basis given by the Mumford class λ (the line bundle whose fiber at C is detH 0 (C, K C ) ⊗ detH 1 (C, K C ) −1 ) and the "tautological line bundles" i := P * i (K N ), where K N is the relative canonical line bundle (relative dualizing sheaf) of π [1,5]. The i carry metrics in such a way that their first Chern forms give the Kähler metrics on M g,N defined by Takhtajan-Zograf [9,10] in terms of Eisenstein series associated to punctured Riemann surfaces [11,12].…”

Section: Introductionmentioning

confidence: 99%