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.
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.
