CHARACTERIZATION OF

In this paper a new geometric characterization of the $n$th symmetric product of a curve is given. Specifically, we assume that there exists a chain of smooth subvarieties $V_{i}$ of dimension $i$, such that $V_{i}$ is an ample divisor in $V_{i+1}$ and its intersection product with $V_{1}$ is one; that the Albanese dimension of $V_{2}$ is $2$ and the genus of $V_{1}$ 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.