On the variety of nets of quadrics defining twisted cubics

Ellingsrud, Geir; Piene, Ragni; Strømme, Stein Arild

doi:10.1007/bfb0078179

Cited by 34 publications

(78 citation statements)

References 4 publications

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“…Ellingsrud-Piene-Strømme moduli space of twisted cubic curves, constructed as the GIT quotient of C 2 ⊗ C 3 ⊗ C 4 by the action of GL 2 × GL 3 (see [3]) V 16 rank 3 tautological vector bundle on G(2, 3, 4) V 16 rank 2 tautological vector bundle on G(2, 3, 4) V 18 rank 2 vector bundle on OG(3, 9) corresponding to a spin representation For all g listed above, Mukai showed that a general K3 surface over F g is given as the zero locus of a general global section of U g (the cases g ≤ 5 are classical).…”

Section: Mukai Models and The Basic Settingmentioning

confidence: 99%

“…The variety G 16 = G(2, 3, 4) is realized as a GIT quotient of C 2 ⊗ C 3 ⊗ C 4 by the obvious action of GL 2 × GL 3 on the first two factors. As described in [3] (see also [11]), there are two tautological vector bundles V 16 and V 16 of rank 3 and 2 respectively, as well as a morphism…”

Section: Polarized K3 Surfaces As Nonunique Complete Intersectionsmentioning

confidence: 99%

“…Further, it was shown in [3,Proposition 2] that the Chow ring CH * (G (2,3,4)) is generated by the Chern classes of V 16 , V 16 . To prove Theorem 0.2 we have to take care of the second Chern classes of both tautological bundles.…”

mentioning

confidence: 99%

“…Let i : S ֒→ G(2, 3, 4) be the embedding of a K3 surface S in G (2,3,4) as in the Mukai model. By the same reasoning as in Section 2, we have the following relation in CH 0 (S):…”

mentioning

confidence: 99%

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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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Section: Mukai Models and The Basic Settingmentioning

confidence: 99%

Section: Polarized K3 Surfaces As Nonunique Complete Intersectionsmentioning

confidence: 99%

mentioning

confidence: 99%

“…Let i : S ֒→ G(2, 3, 4) be the embedding of a K3 surface S in G (2,3,4) as in the Mukai model. By the same reasoning as in Section 2, we have the following relation in CH 0 (S):…”

mentioning

confidence: 99%

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On O'Grady's Generalized Franchetta Conjecture

Pavic

Shen

Yin

2016

Int Math Res Notices

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“…We define the variety H to be the irreducible component of Hilb 3t+1 (P(A)) containing rational normal cubics in P(A), as constructed in [EPS87]. The open subset H c consisting of points which are Cohen-Macaulay embeds in G(C 3 , S 2 A) by means of the vector bundle U * H whose fiber over [Γ] ∈ H c is Tor R 1 (R/J Γ,P 3 , C) 2 C 3 .…”

Section: Nets Of Dual Quadrics and 3-instanton Bundles On Pmentioning

confidence: 99%