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

TL;DR

This paper advances the understanding of the projective normal cone associated with complex plane curves, providing a cycle-level description crucial for computing the degree of PGL(3)-orbit closures, which are significant in enumerative geometry.

Contribution

It determines the PNC as a cycle for plane curves, building on previous set-theoretic descriptions, and facilitates the calculation of orbit closure degrees.

Findings

PNC determined as a cycle for plane curves

Explicit description of limits of PGL(3)-translates

Foundation for computing orbit closure degrees

Abstract

Every complex plane curve C determines a subscheme S of the $P^8$ of 3x3 matrices, whose projective normal cone (PNC) captures subtle invariants of C. In "Limits of PGL(3)-translates of plane curves, I" we obtain a set-theoretic description of the PNC and thereby we determine all possible limits of families of plane curves whose general element is isomorphic to C. The main result of this article is the determination of the PNC as a cycle; this is an essential ingredient in our computation in "Linear orbits of arbitrary plane curves" of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, an invariant of natural enumerative significance.