2011/02/13 by Mingmin Shen, Shen, Mingmin
Computer Science · Mathematics · #14C25 #14N25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1102.2550
openalex publication_date 2011/02/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 CH1(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.