Classification of degree two curves in the symmetric square with positive self-intersection

We give a precise classification of the pairs (C, B) with C a smooth curve of genus g and B C⁽²⁾ a curve of degree two and posit?"""""""""""""""""""""""""""""""""""""""""""""""""""""""""""""e self-intersection. We prove that there are no such pairs if g < p_a(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 × C.