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Generatrices of rational curves. (English) Zbl 1006.51016
Let $$c_0,\ldots,c_g$$ be projectively related rational curves of degree $$\leq d$$ in the real projective $$n$$-space. The generatrix of $$c_0,\ldots,c_g$$ is the one-parametric set of subspaces $$U(s)$$ spanned by corresponding points of $$c_0,\ldots,c_g$$. Assume that $$U(s)$$ is of generic dimension $$g$$, and denote by $$\delta$$ the degree of $$G$$, by $$\nu_i$$ the (finite) number of subspaces $$U(s)$$ of dimension $$g-i$$, and by $$\omega$$ the dimension of the variety of all rational curves of degree $$\leq d$$ that can be used to generate $$G$$.
The author proves: $$\delta+\omega=dg+d+g$$, $$\omega-\Sigma i\nu_i=g$$, and $$\delta+\Sigma i\nu_i=d(g+1)$$.
The proof uses the geometry of rational parameterized representations which was developed by the author recently.

##### MSC:
 51N15 Projective analytic geometry 53A17 Differential geometric aspects in kinematics 51N35 Questions of classical algebraic geometry 14N05 Projective techniques in algebraic geometry
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