Characterization of C(n)

Abstract

In this paper a new geometric characterization of the th symmetric product of a curve is given. Specifically, we assume that there exists a chain of smooth subvarieties Vi of dimension i, such that Vi is an ample divisor in Vi+1 and its intersection product with V1 is one; that the Albanese dimension of V2 is 2 and the genus of is equal to the irregularity of the variety. We prove that in this case the variety is isomorphic to the symmetric product of a curve.