Limits of PGL(3)-translates of plane curves, I

TL;DR

This paper classifies all possible limits of families of translates of a fixed complex plane curve by analyzing the projective normal cone of a related rational map, aiding in understanding orbit closures under PGL(3).

Contribution

It provides a set-theoretic description of the projective normal cone of the base scheme for the rational map associated with plane curves, advancing the understanding of curve orbit limits.

Findings

Classified all limits of PGL(3)-translates of plane curves.

Described the projective normal cone set-theoretically.

Laid groundwork for computing orbit closure degrees.

Abstract

We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the $P^8$ of 3x3 matrices to the $P^N$ of plane curves of degree $d$. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, performed in our earlier paper "Linear orbits of arbitrary plane curves".