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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
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