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The data singular and the data isotropic loci for affine cones. (English) Zbl 1365.14068

For a given real algebraic variety, its Euclidean Distance Degree (ED degree) is roughly speaking the number of critical points of a generic point for the standard Euclidean distance function in the ambient space. This number is constant provided that we considered all points real and complex, and hence some extra care must be taken when doing this, as there are some special exceptional data points for which this number is smaller than the ED degree which arise from the complex formulation of the problem.
For instance, a critical point may wander off into the singular locus of the variety. Points of this nature as said to be in the ED data singular locus. Another anomaly happens when the critical point becomes isotropic with respect to the Euclidean inner product. These points are said to be in the ED data isotropic locus.
In the paper under review, connections between these two special loci and the dual variety are presented in the case where \(X\) is an affine cone. To be more precise, denoting with \(X^*\) the standard dual variety of the complex variety \(X\subset\mathbb C^n,\,DS(X)\) (resp. \(DI(X)\)) the set of all ED data singular (resp. isotropic) locus of \(X,\) and \(Q=\{\sum_{i=1}^nx_i^2=0\},\) the main results of this paper are:
Theorem. If \(X\) is an irreducible affine cone that is not a linear space, then \(X^*\subset DS(X)\subset X^*+\mathrm{Sing}(X). \)
Theorem. If \(X\) is an irreducible affine cone that is not a linear space, then \(X^*\subset DI(X)\subset X^*+(Q\cap X).\)
Examples coming from applications which show that these contentions are sharp but can also be strict are provided at the end of the paper.

MSC:

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
55R80 Discriminantal varieties and configuration spaces in algebraic topology

Software:

Macaulay2
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References:

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