Motives of smooth families and cycles on threefolds

Abstract

Let X --> S be a smooth projective family of surfaces over a smooth curve S such that the generic fiber is a surface with Weil H^2 spanned by divisors and trivial H^1. We prove that if the relative motive of X/S is finite-dimensional the Chow group CH^2(X) with coefficients in Q is generated by a multisection and vertical cycles, i.e. one-dimensional cycles lying in closed fibers of the map X --> S. If S is the projective line P^1 then CH^2(X) is a direct sum of n+1 copies of Q where n<=b_2 and b_2 is the second Betti number of the generic fiber. Vertical generators in CH^2(X) can be concretely expressed in terms of spreads of algebraic generators of the above H^2. We also show where such families are naturally arising from by spreading out surfaces over C.