Equations of parametric surfaces with base points via syzygies

Suppose S is a parametrized surface in complex projective 3-space P given as the image of φ : P ×P1 → P. The implicitization problem is to compute an implicit equation F = 0 of S using the parametrization φ. An algorithm using syzygies exists for computing F if φ has no base points, i.e. φ is everywhere defined. This work is an extension of this algorithm to the case of a surface with multiple base points of total multiplicity k. We accomplish this in three chapters. In Chapter 2, we develop the concept and properties of Castelnuovo-Mumford regularity in biprojective spaces. In Chapter 3, we give a criterion for regularity in biprojective spaces. These results are applied to the implicitization problem in Chapter 4.