Segre invariant and a stratification of the moduli space of coherent systems

The aim of this paper is to generalize the m-Segre invariant for vector bundles to coherent systems. Let X be a non-singular irreducible complex projective curve of genus g >= 0 and G(alpha; n,d,k) be the moduli space of alpha -stable coherent systems of type (n,d,k) on X. For any pair of integers (m,t) with 0 < m < n, 0 <= t <= k we define the (m,t)-Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the (m,t)-Segre invariant induces a stratification of the moduli space G(alpha; n,d,k) into locally closed subvarieties G(alpha; n,d,k; m,t; s) according to the value s of the function. We determine an above bound for the (m,t)-Segre invariant and compute a bound for the dimension of the different strata G(alpha; n,d,k; m,t; s). Moreover, we give some conditions under which the different strata are nonempty. To prove the above results, we introduce the notion of coherent systems of subtype (a).