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

Abstract

Abstract We give a precise classification of the pairs (C, B͠) with C a smooth curve of genus g and B͠ ⊂ C (2) a curve of degree two and positive self-intersection. We prove that there are no such pairs if g < pa (B͠) < 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.