Degree two curves in $C^{(2)}$

TL;DR

This paper classifies degree two curves with positive self-intersection in the symmetric square of a smooth curve, establishing non-existence in certain genus ranges and providing examples with specific properties.

Contribution

It provides a detailed classification of degree two curves in $C^{(2)}$, including non-existence results and explicit examples with particular genus and intersection properties.

Findings

No such pairs exist if $g < p_a( ilde{B}) < 2g-1$

Analyzes singularities and self-intersection of degree two curves in $C^{(2)}$

Constructs examples with arithmetic genus in the Brill-Noether range on $C imes C$

Abstract

In this paper we give a precise classification of the pairs $(C,\widetilde{B})$ with $C$ a smooth curve of genus $g$ and $\widetilde{B}\subset C^{(2)}$ a curve of degree two and positive self-intersection. We prove that there are no such pairs if $g < p_a(\widetilde{B})< 2g-1$. We study the singularities and self-intersection of any degree two curve in $C^{(2)}$. Moreover, we give examples of curves with arithmetic genus in the Brill-Noether range and positive self-intersection on $C\times C$.