Existence of coherent systems
  A coherent system of type (r,d,k) on a curve C is a pair (E,V) where E is a vector bundle of rank r and degree d and V is a space of sections of E of dimension k. There is a condition of stability on coherent systems that depends on a parameter \alpha and allows to construct moduli spaces for such pairs. This paper shows non-emptiness of these moduli spaces when k>r and some mild conditions on the degree and genus are satisfied. 
Montserrat Teixidor i Bigas