Characterization of curves in $C^{(2)}$

TL;DR

This paper characterizes irreducible curves in the symmetric square of a curve, providing conditions for when a curve admits a degree one morphism to the symmetric square with specific properties.

Contribution

It establishes a precise criterion linking irreducible curves in $C^{(2)}$ to morphisms from other curves, revealing the structure of such curves through degree considerations.

Findings

Characterization of irreducible curves in $C^{(2)}$

Existence of a smooth curve $D$ with specific morphisms

Conditions for degree one morphisms to $C^{(2)}$

Abstract

In this paper we characterize the irreducible curves lying in $C^{(2)}$. We prove that a curve $B$ has a degree one morphism to $C^{(2)}$ with image a curve of degree $d$ with irreducible preimage in $C\times C$ if and only if there exists an irreducible smooth curve $D$ and morphisms from $D$ to $C$ and $B$ of degrees $d$ and $2$ respectively forming a diagram which does not reduce.