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MiS Preprint

Coordinate-wise Powers of Algebraic Varieties

Papri Dey, Paul Görlach and Nidhi Kaihnsa


We introduce and study coordinate-wise powers of subvarieties of $\mathbb{P}^n$, i.e.\ varieties arising from raising all points in a given subvariety of $\mathbb{P}^n$ to the $r$-th power, coordinate by coordinate. This corresponds to studying the image of a subvariety of $\mathbb{P}^n$ under the quotient of $\mathbb{P}^n$ by the action of the finite group $\mathbb{Z}_r^{n+1}$. 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 coordinatewise squares of linear spaces and the study of real symmetric matrices with a degenerate eigenspectrum.


Related publications

2020 Journal Open Access
Papri Dey, Paul Görlach and Nidhi Kaihnsa

Coordinate-wise powers of algebraic varieties

In: Beiträge zur Algebra und Geometrie, 61 (2020) 3, pp. 473-505