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.


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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