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Nodal curves in \(\mathbb{P}^3(\mathbb{C})\). (English) Zbl 0607.14019

Here we prove the following result: For all integers \((d,g)\) such that there is a smooth connected curve in \({\mathbb{P}}^ 3\) of genus \(g\) and degree \(d,\) and every t with \(0\leq t\leq g\), there is an irreducible curve \(C\subset {\mathbb{P}}^ 3\) of arithmetic genus \(g,\) degree \(d,\) with exactly \(t\) nodes as only singularities. - The proof uses Severi-Wahl-Tannenbaum’s theory of nodal curves on rational surfaces, and the proof by Gruson-Peskine of Halphen’s conjecture about the possible \((d,g)\) for smooth curves in \({\mathbb{P}}^ 3\).

MSC:

14H20 Singularities of curves, local rings
14N05 Projective techniques in algebraic geometry
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