Stable pairs on curves and surfaces

During the last years there has been growing interest in vector bundles with additional structures, e.g. parabolic and level structures. This paper results from an attempt to construct quasi-projective moduli spaces for framed bundles, i.e. bundles together with an isomorphism to a fixed bundle on a divisor as introduced in [Do], [L1] and [Lü]. More generally one can ask for bundles with a homomorphism to a fixed sheaf E0. We use techniques of geometric invariant theory to construct projective moduli spaces. This leads to natural stability conditions. In contrast to the pure bundle case an extra parameter appears in the definition of stability.