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The geometry of rational parameterized representations. (English) Zbl 1083.14049
The author starts with the fact that a point in $$\mathbb{P}^n(\mathbb{R})$$ can be given by a (special) vector $${\mathbf x}\in\mathbb{R}^{n+1}$$. To a set of such vectors $$({\mathbf x}_i)$$ $$(i= 0,\dots, n)$$, he attaches the polynomial $$X(s)=\sum^d_0 s^i{\mathbf x}_i$$. An appropriate definition is given to extend the base space to $$\mathbb{R}\cup\infty$$. Then he studies the algebraic variety generated by projective transformations of $$\mathbb{P}^n(\mathbb{R}\cup\infty)$$ and real fractional linear transformations $$t(s)= (as+\beta)/(\gamma s+\delta)$$ acting as $$X(s)\to(\gamma s+\delta)^d X(t(s))$$. The projective space of all rational parametrized such functions is $$\mathbb{P}^{dn+ d+ n}$$, isomorphic to a matrix space investigated by W. Rath [Abh. Math. Semin. Univ. Hamb. 63, 177–196 (1993; Zbl 0791.53014)]. He determines the subvarieties of defect at least $$i$$ and related kernel varieties (shown all to be algebraic) and shows that the group generated by the transformations indicated is the full automorphism group of the variety of all 1-kernels. A last section describes the setting in which the theory is useful in geometric research.
##### MSC:
 14J50 Automorphisms of surfaces and higher-dimensional varieties 51A05 General theory of linear incidence geometry and projective geometries 14Q05 Computational aspects of algebraic curves
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