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We show that prime Fano threefolds of genus 8 have a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, a certain tautological subring of the Chow ring of powers of injects into cohomology.
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Robert Laterveer, Algebraic cycles and Fano threefolds of genus 8. Port. Math. 78 (2021), no. 3, pp. 255–280DOI 10.4171/PM/2069