 Kim-Lee [@MR2090618]). For example, a singularity analytically isomorphic to $y^2=x^5$ on a [*quartic*]{} leads necessarily to a component of type V (cf. Example \[extwo\]), whereas on a quintic, it leads to either a component of type IV or a component of type V, according to the order of contact with the tangent line. For GIT-stability, see also Remark \[GITrem\].

The enumerative problem considered in [@MR2001h:14068], as well as the question of limits of PGL-translates, makes sense for hypersurfaces of projective space of any dimension. The case of configurations of points in ${{\mathbb{P}}}^1$ is treated in [@MR1244973]. The degree of the orbit closure of a configuration of planes in ${{\mathbb{P}}}^3$ is computed in [@MR2455792]. In general, these problems appear to be very difficult. The techniques used in this paper could in principle be used in arbitrary dimension, but the case-by-case analysis (which is already challenging for curves in ${{\mathbb{P}}}^2$) would likely be unmanageable in higher dimension. By contrast, the techniques developed in [@ghizzII] should be directly applicable: once ‘marker germs’ have been determined, computing the multiplicities of the corresponding components of the PNC should be straightforward, using the techniques of [@ghizzII].

[^1] We also thank the referee of this paper and [@ghizzII], for the careful reading of both papers and for comments that led to their improvement.

Set-theoretic description of the PNC {#prelim}
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Limits of translates {#rough}
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We work over ${{\mathbb{C}}}$. We choose homogeneous coordinates $(x:y:z)$ in ${{\mathbb{P}}}^2$, and identify ${\text{\rm PGL}}(3)$ with the open set of nonsingular matrices in the space ${{\mathbb{P}}}^8$ parametrizing $3\times 3$ matrices. We consider the right action of ${\text{\rm PGL}}(3)$ on the space ${{\mathbb{P}}}^N={{\mathbb{P}}}H^0({{\mathbb{P}}}^2, \mathcal O(d))$ of degree-$d$ plane curves; if $F(x,y,z)=0$ is an equation for a plane curve $