On relations among 1-cycles on cubic hypersurfaces

TL;DR

This paper establishes explicit relations among 1-cycles on smooth cubic hypersurfaces, demonstrating how these relations can be used to understand the structure of the Chow group and the intermediate Jacobian.

Contribution

It provides explicit relations among 1-cycles on cubic hypersurfaces and applies these to reprove known results and describe the intermediate Jacobian via Prym-Tjurin varieties.

Findings

CH_1(X) is generated by lines for high-dimensional cubics

Intermediate Jacobian is isomorphic to a Prym-Tjurin variety

Explicit relations among 1-cycles involving secant lines

Abstract

In this paper we give two explicit relations among 1-cycles modulo rational equivalence on a smooth cubic hypersurfaces $X$. Such a relation is given in terms of a (pair of) curve(s) and its secant lines. As the first application, we reprove Paranjape's theorem that $\mathrm{CH}_1(X)$ is always generated by lines and that it is isomorphic to $\Z$ if the dimension of $X$ is at least 5. Another application is to the intermediate jacobian of a cubic threefold $X$. To be more precise, we show that the intermediate jacobian of $X$ is naturally isomorphic to the Prym-Tjurin variety constructed from the curve parameterizing all lines meeting a given curve on $X$. The incidence correspondences play an important role in this study. We also give a description of the Abel-Jacobi map for 1-cycles in this setting.