 limits is given in §\[germlist\], referring to the classification of these curves obtained in §1 of [@MR2002d:14084] and reproduced in §\[appendix\] of this paper.

The proof of Theorem \[main\] (or rather of its more precise form given in Theorem \[mainmain\]) is by an explicit reduction process, and goes along the following lines. The stars mentioned in the statement are obtained by families of translations $\alpha(t)$ (‘germs’) centered at an element $\alpha(0)\not\in{{\mathscr S}}$. To analyze germs centered at points of ${{\mathscr S}}$, we introduce a notion of equivalence of germs (Definition \[equivgermsnew\]), such that equivalent germs lead to the same limit. We then prove that every germ centered at a point of ${{\mathscr S}}$ is essentially equivalent to one with matrix representation $$\begin{pmatrix}
1 & 0 & 0\\
q(t) & t^b & 0\\
r(t) & s(t)t^b & t^c
\end{pmatrix}$$ with $0\le b\le c$ and $q$, $r$, and $s$ polynomials. Here, coordinates are chosen so that the point $p=(1:0:0)$ belongs to ${{\mathscr C}}$. Studying the limits obtained by applying such germs to ${{\mathscr C}}$, we identify five specific types of families (the [*marker germs*]{} listed in §\[germlist\]), reflecting the features of ${{\mathscr C}}$ at $p$ listed in Theorem \[main\], and with the stated kind of limit. We prove that unless the germ is of one of these types, the corresponding limit is already accounted for (for example, it is in the orbit closure of a star of the type mentioned in the statement).

In terms of the graph of the rational map $c$ mentioned above, we prove that every component of the PNC is hit at a general point by the lift in ${{{{\widetilde{{{\mathbb{P}}}}}}}}^8$ of one of the five distinguished types of germs. This yields our set-theoretic description of the PNC. In fact, the lifts intersect the corresponding components transversally, and this will be important in our determination of the multiplicities of the components in [@ghizzII].

The procedure underlying the proof of Theorem \[mainma