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A Harnack theorem in space. (Un théorème de Harnack dans l’espace.) (French) Zbl 0829.14027
Let $$C$$ be a smooth algebraic curve in $$\mathbb{P}^n$$ of degree $$d$$. Then it is known that the genus of $$C$$ does not exceed the Castelnuovo bound $$C(d,n)$$: $C(d,n) = ({1\over 2}) k(k - 1) (n - 1) + \varepsilon k; \quad k = \bigl[ (d - 1) / (n - 1) \bigr],\;d - 1 = m(n - 1) + \varepsilon.$ Besides, for a smooth algebraic curve $$C$$ defined over $$\mathbb{R}$$ of genus $$g$$, the space $$C(\mathbb{R})$$ of real points has at most $$g + 1$$ connected components (Harnack’s inequality). Therefore, for a smooth real algebraic curve $$C$$ in $$\mathbb{P}^n$$ of degree $$d$$, $$C(\mathbb{R})$$ has at most $$C(d,n) + 1$$ connected components.
In the paper it is shown that, for any integer $$c$$ with $$0 \leq c \leq C(d,n)$$ and for any positive integers, $$d,n$$, there exists a smooth real algebraic curve $$C$$ in $$\mathbb{P}^n$$ of degree $$d$$ with $$c + 1$$ connected components. The method of the proof is simlyfying the isolated double points of certain rational curves defined over $$\mathbb{R}$$.

##### MSC:
 14P25 Topology of real algebraic varieties 14F45 Topological properties in algebraic geometry 14H50 Plane and space curves 14N10 Enumerative problems (combinatorial problems) in algebraic geometry