Algebraic cycles on cubic hypersurfaces and Fano scheme of lines

In this dissertation, I study algebraic and geometric structures linking the cubic hypersurfaces and the associated Fano variety of lines in terms of algebraic cycles. The first part is the cylinder homomorphism. The family of lines over the Fano variety of a cubic hypersurface defines the cylinder homomorphism. On the cohomology group, the cylinder homomorphism is fundamental to study other objects such as Hodge structures and intermediate Jacobians. Shimada showed that this map is always surjective. I prove that the cylinder homomorphism is universally surjective on the Chow groups, which generalizes the results for low dimensional cycles on the cubic hypersurface by Mingmin and René. The universally surjectivity means that after any field extension, the associated cylinder homomorphism remains surjective. Moreover, I applied the result to conclude the integral Hodge and Tate conjecture for one-cycles on the variety of lines. The second part investigates certain relations between the Chow motives of cubic hypersurfaces and the Fano variety of lines. The theory of motives is a program initiated by Grothendieck that aims to unify vast good cohomology theories of algebraic varieties. Using a group law-like structure on a smooth cubic hypersurface established by Galkin-Shinder, I prove a formula of decomposition of the diagonal of the Hilbert square of the cubic hypersurface. The formula refines a result of Laterveer, which shows the Chow motive of the Fano variety of lines is essentially controlled by the Chow motive of the cubic hypersurface