On O'Grady's Generalized Franchetta Conjecture

Abstract

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.

The Generalized Franchetta Conjecture for Some Hyperkähler Fourfolds

“…This statement is very similar to the "generalized Franchetta conjecture" for K3 surfaces, which was formulated by O'Grady [32, Section 5] and proven in certain cases by Pavic-Shen-Yin [33].…”

“…for some non-zero integer d. We observe that (11) implies that the restriction δ 00 | S b is of degree 0 for all b ∈ B. But then, reasoning exactly as in [33], we must have that δ 00 | S b is rationally trivial. (More in detail: by construction, one has…”

“…The proof of theorems 3.1 and 3.2 is based on Voisin's method of "spread" of algebraic cycles [43], [44], [45], [46], combined with results of Pavic-Shen-Yin on the generalized Franchetta conjecture for K3 surfaces [33].…”

Tautological classes on moduli spaces of hyper-Kähler manifolds

“…(2) The second Chern class c 2 (T S ) equals 24c S ∈ CH 0 (S). Conjecture 2 has been confirmed when g ≤ 12 (see [46]) but remains widely open for large g. Let T π be the relative tangent bundle of π. According to [58,Theorem 10.19], Conjecture 2 is equivalent to the following Conjecture 3.…”

Lagrangian Subvarieties in the Chow Ring of Some Hyperkähler Varieties

“…2 NB: after the present paper was written, the paper [15] appeared, which explores closely related questions. Both the present paper and [15] are inspired by [39].…”