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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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