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Simple constructions of algebraic curves with nodes. (English) Zbl 0783.14013
The author gives a proof of the following results: There exists an integral nondegenerate (i.e. lying in no hyperplane) curve of degre $$d \geq n$$ in $$\mathbb{P}^ n$$, with $$\delta$$ real nodes and no other singular points, for all $$\delta$$ from 0 to the Castelnuovo’s bound. Over the field of complex numbers, this result was proved by Severi in the case $$n=2$$; the case $$n \geq 3$$ was proved by A. Tannenbaum [Math. Ann. 240, 213-221 (1979; Zbl 0385.14008) and Compos. Math. 41, 107-126 (1980; Zbl 0399.14018)], by using deformation theory. In the paper under review the author uses a method based on the simplification of nodes of certain Lissajous’s curves, following a technique described in a previous paper [C. R. Acad. Sci., Paris, Sér. I 315, No. 5, 561-565 (1992; see the preceding review)].
Reviewer: C.Cumino (Torino)

##### MSC:
 14H20 Singularities of curves, local rings 14H50 Plane and space curves
##### Keywords:
real nodes; Castelnuovo’s bound; Lissajous’s curves
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##### References:
 [1] R. Benedetti , J.-J. Risler : Real algebraic and semi-algebraic sets . Actualités Mathématiques (1990). · Zbl 0694.14006 [2] D. Pecker : Courbes gauches ayant beaucoup de points multiples réels , to appear. · Zbl 0783.14012 [3] E.I. Shustin : Real Plane Algebraic Curves with Many Singularities . Preprint Samara State University, 1991. · Zbl 0786.14035 [4] A. Tannenbaum : Families of algebraic curves with nodes , Compositio Mathematica 41 (1980), 107-126. · Zbl 0399.14018 [5] A. Tannenbaum : On the geometric genera of projective curves , Math. Ann. 240(3) (1979), 213-221. · Zbl 0385.14008
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