Algebraic cycles and Fano threefolds of genus 8

Abstract

We show that prime Fano threefolds Y of genus 10 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 Y injects into cohomology.

Some Motivic Properties of Gushel-Mukai Sixfolds

“…In dimension two, a smooth quartic in P 3 has an MCK decomposition, but a very general surface of degree ≥ 7 in P 3 should not have one [15,Proposition 3.4]. For a discussion in greater detail, and further examples of varieties with an MCK decomposition, the reader may check [51, Section 8], as well as [55], [52], [16], [29], [30], [31], [32], [33], [34], [35], [36], [37], [15], [38], [39], [40] We say that Y → B has the Franchetta property if Y → B has the Franchetta property in codimension j for all j.…”

Motives and the Pfaffian-Grassmannian equivalence

“…Remark 4.3. In the set-up of Corollary 4.2, the varieties X and Y are actually birational [47], and the isomorphism of motives can be readily obtained by exploiting the specific form of the birationality [34]. However, the proof given here does not rely on the birationality.…”

Algebraic cycles and intersections of three quadrics

“…As for surfaces: a smooth quartic in P 3 has an MCK decomposition, but a very general surface of degree ≥ 7 in P 3 should not have an MCK decomposition [15,Proposition 3.4]. For more detailed discussion, and examples of varieties with an MCK decomposition, we refer to [40,Section 8], as well as [46], [41], [16], [24], [25], [26], [27], [15], [28], [29], [30].…”