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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.”

MSC:

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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