Maximal rank and minimal generation of some parametric varieties

In this paper, continuing work of the second author (J. Pure Appl. Algebra 155 (2001) 77) for rational curves, we address the problem of computing the generators of the ideal of an irreducible parametric variety V to the computation of the generators of the ideal of a suitable finite set of points on V. In particular, we consider the case of general parametric surfaces and threefolds and of general parametric surfaces represented by polynomials with base points.