Weighted projective spaces and a generalization of Eves’ theorem. (English) Zbl 1296.51034

Let \(\mathcal S\) be a finite set of signed lengths of directed segments in the projectively extended (real) Euclidean plane. The theorem of Howard H. Eves (1911–2004) is applicable to \({\mathcal S}\) and yields ratios (that are elements of the real projective line \({\mathbb R}P^1\)) being projectively invariant; a special case is the cross ratio. Avoiding any notion of distance, the author develops a generalization of Eves’ theorem (gEt) in terms of linear algebra and projective geometry. In the gEt \({\mathbb R}P^1\) is replaced with a weighted projective space \({\mathbb K}P(p)\) where \({\mathbb K}\) is a commutative field and \(p=(p_0,\dots,p_n)\) (“weight vector”). Configurations to which gEt applies are described by a weighting, coloring, and indexing scheme. To each configuration \({\mathcal S}\) of points of \({\mathbb K}P^D\) that satisfies a certain condition depending on the weight vector \(p\) an element \(E_p\in{\mathbb K}P(p)\) is assigned and \(E_p\) is invariant under “morphisms” of the configuration which generalize projective transformations.
Moreover, the author introduces the new notion of reconstructibility of a weighted projective space; “the two main results are that complex weighted projective spaces are all reconstructible, and that some real weighted projective spaces are not.”


51N15 Projective analytic geometry
51A20 Configuration theorems in linear incidence geometry
14E05 Rational and birational maps
14N05 Projective techniques in algebraic geometry
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