A perfect obstruction theory for moduli of coherent systems

March, 2018

Let C be a curve of genus g. A coherent system on C is a pair (E,V), where E is a finite rank vector bundle on C and V is a linear subspace of the space of global sections of E. The type of a coherent system (E,V) is a triple (n,d,k), where n is the rank of E, d is the degree of E and k is the dimension of V. The notion of stability for a coherent system (E,V) differs from the stability of the bundle E and depends on the choice of a real parameter α. The moduli space of α-stable coherent systems of type (n,d,k) has an expected dimension β=β(n,d,k) which depends on the genus of the curve C and on the type of the coherent systems. We construct a perfect obstruction theory for the moduli spaces of α-stable coherent systems which has rank equal to the expected dimension β. In our construction we do not fix one curve, but we work on families of Gorenstein projective curves.