The Hodge-Deligne polynomials of some moduli spaces of coherent systems

A coherent system on a smooth projective curve C consists of a pair (E,V) where E is a vector bundle on C (of rank n and degree d) and V is a subspace (of dimension k) of $H^0(C,E)$. For each triple (n,d,k) there is a family of moduli spaces of coherent systems, depending on a real positive parameter $\alpha$. It is known that these moduli spaces change only if we pass through a finite set of critical values, so we have a finite number of distinct moduli spaces labeled according to the corresponding interval in the real line. The final moduli space is in general very simple to study, while not so much is known about the intermediate moduli spaces and the first one (which has strong relations with the Brill-Noether locus $B(n,d,k)$). In particular, an interesting open problem is that of computing the Hodge-Deligne polynomials of such moduli spaces. In the present work we get some explicit results in the cases (n=2,k=1) and (n=3,k=1), together with some general techniques that in principle could be used to tackle also more complicated cases. We give also some partial results on the cases (n=4,k=1) and (n=2,k=2).