Abstract
We study the smoothness question for some families of real and complex varieties with cyclic or dihedral symmetry. This question is related to deep properties of the Vandermonde matrix on the roots of unity, also known as the Discrete Fourier Transform matrix. We present some partial results on these questions.
Para Antonio Campillo, en sus 65 años.
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Notes
- 1.
An abstract version of this construction was studied in [3], giving results about the topology of the quotient surfaces under the cyclic group.
- 2.
- 3.
In the case with equations of different degrees, it is not clear how smoothness could be expressed in terms of the matrix V n.
- 4.
Compare with the linear case d = 1 in which it is sufficient for smoothness that one of the minors is non-zero.
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Acknowledgements
Conversations with Enrique Artal, Pablo Barrera, Shirley Bromberg, Peter Bürgisser, Sylvain Cappell, Marc Chaperon, Antonio Costa, Genaro de la Vega, Javier Elizondo, Matthias Franz, Ignacio Luengo, Mike Shub, Denis Sullivan, Yuri Tschinkel, Luis Verde, Alberto Verjovsky, Felipe Zaldívar and Adrián Zepeda have been very helpful.
Special mention is deserved by Matthias Franz who has dedicated a great effort to the Vandermonde minors question. Long discussions with him have clarified many aspects of the theory (and of the present paper) and the proof of Theorem 2 is due to him. Much joint work is still in progress and will certainly be the object of a joint future paper more closely related to his interests and point of view.
This work was partially supported by a Papiit-UNAM grant IN111415.
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López de Medrano, S. (2018). Smoothness in Some Varieties with Dihedral Symmetry and the DFT Matrix. In: Greuel, GM., Narváez Macarro, L., Xambó-Descamps, S. (eds) Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics. Springer, Cham. https://doi.org/10.1007/978-3-319-96827-8_16
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