A family of K3 surfaces \({\mathscr {X}}\rightarrow B\) has the Franchetta property if the Chow group of 0-cycles on the generic fiber is cyclic. The generalized Franchetta conjecture proposed by O’Grady asserts that the universal family \({\mathscr {X}}_g\rightarrow {\mathscr {F}}_g\) of polarized K3 of degree \(2g-2\) has the Franchetta property. While this is known only for small g thanks to [7], we prove that for all g there is a hypersurface in \( {\mathscr {F}}_g\) such that the corresponding family has the Franchetta property.

中文翻译：

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关于K3曲面的广义Franchetta猜想的注记

如果通用光纤上的0循环的Chow组是循环的，则K3曲面族（（{\ mathscr {X}} \ rightarrow B \）具有Franchetta属性。O'Grady提出的广义Franchetta猜想断言，偏振度为\（2g-2 \）的极化K3的通用族\（{\ mathscr {X}} _ g \ rightarrow {\ mathcr {F}} _ g \）具有Franchetta属性。尽管由于[7]仅对小g已知，但我们证明了所有g在\（{\ mathscr {F}} _ g \）中都有一个超曲面，因此相应的族具有Franchetta属性。