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On the number of points determining a canonical curve. (English) Zbl 0724.14017
The author shows, among other things, that a canonical curve of genus $$g$$ passes through $$g+5$$ points of $${\mathbb{P}}^{g-1}$$. The proof is based on work of Kleppe [via D. Perrin, Mém. Soc. Math. Fr., Nouv. Ser. 28/29, (1987; Zbl 0648.14028)]. One application is to study the singularity given by r general lines through a point in $${\mathbb{A}}^ n$$, with $$n<r<{{n+1}\choose 2}:$$ if $$r-n\neq 8$$, this singularity is smoothable exactly when $$(r-n-2)(n-5)\leq 6\epsilon$$, where $$\epsilon =0$$ or 1 according to the parity of $$r-n$$.
Reviewer: R.Speiser (Provo)

##### MSC:
 14H50 Plane and space curves 14H20 Singularities of curves, local rings 14N10 Enumerative problems (combinatorial problems) in algebraic geometry
##### Keywords:
number of points of curve; canonical curve; genus; singularity