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Plücker conditions on plane rational curves. (English) Zbl 0578.14029
A reduced, irreducible curve, C in \(P^ 2_{{\mathbb{C}}}\) is said to be a Plücker curve if C and its dual curve have only ordinary cusps and simple nodes as singularities. The main result of the paper is the following theorem. Let \(H_ d=P_{{\mathbb{C}}}^{(d+2)(d+1)/2-1}\) be the Hilbert scheme of curves of degree d in \(P^ 2_{{\mathbb{C}}}\), and let \(R_ d\subset H_ d\) be the locally closed subset consisting of reduced irreducible rational curves having only ordinary cusps and nodes as singularities. Then there exists a family \(\{O_ d\}_{d\geq 3}\) of open sets \(O_ d\subset R_ d\) such that: (1) If C is a rational plane curve of degree d, then \(C\in O_ d\) if and only if Č\(\in O_{\check d}\), where Č is the dual curve of C and ď its degree. (2) The family \(\{O_ d\}\) is maximal with respect to the property (1) and such that \(O_ d\) does not contain any non Plücker curve and \(O_ d\) is open in \(R_ d\), \(d\geq 3\).
Reviewer: V.F.Ignatenko

MSC:
14H20 Singularities of curves, local rings
14N05 Projective techniques in algebraic geometry
14C05 Parametrization (Chow and Hilbert schemes)
14H45 Special algebraic curves and curves of low genus
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