Characterization of $C^{(n)}$

TL;DR

This paper provides a new geometric characterization of the nth symmetric product of a curve using a chain of smooth subvarieties with specific intersection properties, linking algebraic and geometric features.

Contribution

It introduces a novel geometric criterion involving a chain of subvarieties that characterizes the symmetric product of a curve.

Findings

The variety is isomorphic to the symmetric product of a curve under the given conditions.

The intersection properties and Albanese dimension are key to the characterization.

The genus of the first subvariety equals the irregularity of the variety.

Abstract

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.