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The variety of infinitely near points of order $$n$$ to points of the plane. (Variété des points infiniment voisins d’ordre $$n$$ de points du plan.) (French. Abridged English version) Zbl 0815.14002
For each integer $$n \geq 1$$, a variety $$S_ n$$ is defined, which parametrizes the infinitely near points of order $$n$$, to points of the projective plane $$P$$: one has $$S_ 1 = P$$, $$S_{n+1} = \text{Proj}_{S_ n} E_ n$$, with $$E_ n$$ a locally free $${\mathcal O}_{S_ n}$$-module of rank two. A divisor $$Y_ n$$ of $$S_ n$$ and an embedding of $$S_ n - Y_ n$$ in the Hilbert scheme $$\text{Hilb}_ nP$$ are described, $$S_ 4$$ is studied more closely and some results of Halphen for the number of points of a plane curve satisfying a given differential equation are interpreted.

##### MSC:
 14B10 Infinitesimal methods in algebraic geometry 14C05 Parametrization (Chow and Hilbert schemes)
##### Keywords:
infinitely near points; Hilbert scheme