On parametric surfaces of low degree in P3(C)

Parametrized surfaces of low degrees are very useful in applications, especially in Computer Aided Geometric Design and Geometric Modeling. The precise description of their geometry is not easy in general. Here we study some of the corresponding projective complex surfaces of low implicit degree (i.e., smaller than 12). We show that, generically up to linear changes of coordinates, they are classified by a few number of continuous parameters (called moduli). We present normal forms and provide compact implicit equations for these surfaces and for their singular locus together with a geometric interpretation.