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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
##### Keywords:
plane rational curve; Plücker curve; dual curve; Hilbert scheme
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