Nets of conics and associated Artinian algebras of length 7

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