Liaison and Castelnuovo-Mumford regularity

In this article we establish bounds for the Castelnuovo-Mumford regularity of projective schemes in terms of the degrees of their defining equations. The main new ingredient in our proofs is to show that generic residual intersections of complete intersection rational singularities again have rational singularities. When applied to the theory of residual intersections this circle of ideas also sheds new light on some known classes of free resolutions of residual ideals.