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The surface of lines in a cubic fourfold intersecting a fixed line splitsmotivically into two parts, one of which resembles a K3 surface. We define theanalogue of the Beauville-Voisin class and study the push-forward map to theFano variety of all lines with respect to the natural splitting of theBloch-Beilinson filtration introduced by Mingmin Shen and Charles Vial.

Chow groups of surfaces of lines in cubic fourfolds