Abstract
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We define an analogue of the Arnold surface for odd
degree flexible curves, and we use it to double branch cover
-flexible embeddings,
where
-flexible
is a condition to be added to the classical notion of a flexible curve. This allows us to
obtain a Viro–Zvonilov-type inequality: an upper bound on the number of nonempty
ovals of a curve of odd degree. We investigate our method for flexible curves in a
quadric to derive a similar bound in two cases. We also digress around a possible
definition of nonorientable flexible curves, for which our method still works and a
similar inequality holds.
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Keywords
algebraic curves, flexible curves, Hilbert's 16th problem,
double branched covers, nonorientable surfaces
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Mathematical Subject Classification
Primary: 14P25, 57M12
Secondary: 14H45, 57S25
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Milestones
Received: 15 May 2023
Revised: 15 February 2024
Accepted: 9 March 2024
Published: 16 April 2024
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| © 2024 MSP (Mathematical Sciences
Publishers). Distributed under the Creative Commons
Attribution License 4.0 (CC BY). |
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