Generatrices of rational curves. (English) Zbl 1006.51016

Schröcker, Hans-Peter

J. Geom. 73, No. 1-2, 134-147 (2002).

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.

Reviewer: Rolf Riesinger (Wien)

MSC:

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

Keywords:

projectively related rational curves; geometry of rational parameterized representations; kernel variety; projective kinematics