Summary of Algebraic Geometry Applications

Parameterization & Singularities & Genus

Computing polynomial and rational parameterization maps, for 
algebraic curves, algebraic surfaces, trimmed patches of low degree
surfaces, and algebraic varieties, in general.

Computing necessary and sufficient criterion (Genus, plurigenera) 
for the rationality of algebraic curves and surfaces.

Efficient polynomial and rational approximations of arbitrary 
algebraic curves and surfaces

Robust display of arbitrary rational parametric surfaces, and trimmed
surface patches

Local resolution of topology of gradient vector fields of high index.

Combinatorics of Splines and Mesh Complexes

The dimension of algebraic curve and surface splines in their implicit bases, 
with maximal continuity allowable in their basis functions

The derivation of differential and integral properties of patch 
complexes built from low degree algebraic surfaces