Nets of Conics and associated Artinian algebras of length 7

TL;DR

This paper classifies nets of conics under projective transformations, explores their geometric and algebraic properties, and demonstrates how associated Artinian algebras can be smoothed or deformed, advancing understanding of their structure and classification.

Contribution

It provides a comprehensive classification of nets of conics, analyzes their orbit structure, and shows how related Artinian algebras can be smoothed or deformed, extending prior work in algebraic geometry.

Findings

Classification of nets of conics under group action

Artinian algebras of specific Hilbert functions can be smoothed

Connections between nets of conics and deformations of Artinian algebras

Abstract

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.