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

In this paper we give a precise classification of the pairs $(C,\tilde{B})$ with $C$ a smooth curve of genus $g$ and $\tilde{B}\subset C^{(2)}$ 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 $C\times C$.