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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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