Classifying generic algebras: The local case

In this paper we complete the classification up through dimension seven of generic algebras, i.e., algebras or families whose orbits are dense in a reduced irreducible component of the structure constant scheme parametrizing n dimensional algebras. We also parametrize the irreducible components of families which are candidates to be quasi-generic. We adduce an example to show why the eight dimensional local algebras seem to lie outside the realm of the current techniques.