In this paper we describe an algorithm for implicitizing rational hypersurfaces in case there exists at most a finite number of base points. It is based on a technique exposed in Busé and Jouanolou [2002], where implicit equations are obtained as determinants of certain graded parts of a so-called approximation complex. We detail and improve this method by providing an in-depth study of the cohomology of such a complex. In both particular cases of interest of curve and surface implicitization we also yield explicit algorithms which only involves linear algebra routines.

Abstract. 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 proof 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.Introduction. In this article we prove bounds for the Castelnuovo-Mumford regularity of projective schemes in terms of the degrees of defining equations, very much in the spirit of Bertram, Ein, and Lazarsfeld ([BEL]). Our methods are based on liaison theory. They also lead to results in positive characteristic and provide information on defining ideals, even if these are not saturated or unmixed.The following gives a flavor of one of our main results (Theorem 4.7(a)).

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