  We classify the orbits of nets of conics under the action of the projective
linear group and we determine the specializations of these orbits, using
geometric and algebraic methods. We study related geometric questions, as the
parametrization of planar cubics. We show that Artinian algebras of Hilbert
function H=(1,3,3,0) determined by nets, can be smoothed - deformed to a direct
sum of fields; and that algebras of Hilbert function H=(1,r,2,0), determined by
pencils of quadrics, can also be smoothed.
  In a Historical Appendix A we describe related work prior to 1977. In an
Update Appendix B we survey some developments since 1977 concerning nets of
conics, related geometry, and deformations of Artinian algebras of small
length.
