## 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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### References:

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