In this paper we give two explicit relations among 1-cycles modulo rational equivalence on a smooth cubic hypersurface 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...

Given a curve of genus 3 with an unramified double cover, we give an explicit description of the associated Prym-variety. We also describe how an unramified double cover of a non-hyperelliptic genus 3 curve can be embedded in the jacobian of a curve of genus 2 over its field of definition and how this can be used to do Chabautyand Brauer-Manin type calculations for curves of genus 5 with an unr...

We prove that families of Prym varieties abelian Galois covers P1 in Ag do not give rise to high dimensional Shimura subvareties. This contributes the classification such totally geodesic subvarieties locus and extends results E. Colombo et al. which deal with 1-parameter families. Our method is based on analyzing Monodromy family.