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Universal generation of the cylinder homomorphism of cubic hypersurfaces

Research paper by Renjie Lyu

Indexed on: 29 Oct '18Published on: 29 Oct '18Published in: arXiv - Mathematics - Algebraic Geometry



Abstract

In this article, we prove that the Chow group of algebraic cycles of a smooth cubic hypersurface $X$ over an arbitrary field $k$ is generated, via the natural cylinder homomorphism, by the algebraic cycles of its Fano variety of lines $F(X)$, under an assumption on the $1$-cycles of $X/k$. As an application, if $X/\mathbb{C}$ is a smooth complex cubic fourfold, our result provides a proof of the integral Hodge conjecture for curve classes on the polarized hyper-K\"ahler variety $F(X)$. In addition, when $k$ is finitely generated over its prime subfield with $\text{char}(k)\neq 2, 3$, we use our conclusion to prove the integral analog of the Tate conjecture for $1$-cycles on $F(X)$ for a smooth cubic fourfold $X/k$.