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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)
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