Coordinate-wise Powers of Algebraic Varieties

Papri Dey, Paul Görlach, and Nidhi Kaihnsa

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Submission date: 10. Jul. 2018
published in: Beiträge zur Algebra und Geometrie (2020), pp not yet known
DOI number (of the published article): 10.1007/s13366-019-00481-8
Bibtex
Link to arXiv: See the arXiv entry of this preprint.

Abstract:
We introduce and study coordinate-wise powers of subvarieties of ℙⁿ, i.e. varieties arising from raising all points in a given subvariety of ℙⁿ to the r-th power, coordinate by coordinate. This corresponds to studying the image of a subvariety of ℙⁿ under the quotient of ℙⁿ by the action of the finite group ℤ_rⁿ⁺¹. We determine the degree of coordinate-wise powers and study their defining equations, particularly for hypersurfaces and linear spaces. Applying these results, we compute the degree of the variety of orthostochastic matrices and determine iterated dual and reciprocal varieties of power sum hypersurfaces. We also establish a link between coordinate-wise squares of linear spaces and the study of real symmetric matrices with a degenerate eigenspectrum.