Parabolic Coverings I:The Case of Curves

Here we define the notion of a parabolic curve (X, Q), parabolic coverings and parabolic fundamental group both topological and algebraic. We prove some analogies with the etale coverings and classical fundamental groups. Then we define parabolic genus g(X, Q) guided by the Hurwitz theorem for ramified coverings and show that g(X, Q)-1 gets multiplied by the degree under a parabolic covering. Then we prove a projection formula for parabolic vector bundles under parabolic coverings, using which we obtain a cohomological criterion for semistability for parabolic bundles. We also obtain a criterion for a given map to factor through a parabolic covering in terms of parabolic bundles.