Hyperbolic secant varieties of M-curves
-
Mario Kummer
and
Rainer Sinn
Abstract
We relate the geometry of curves to the notion of hyperbolicity in real algebraic geometry. A hyperbolic variety is a real algebraic variety that (in particular) admits a real fibered morphism to a projective space whose dimension is equal to the dimension of the variety. We study hyperbolic varieties with a special interest in the case of hypersurfaces that admit a real algebraic ruling. The central part of the paper is concerned with secant varieties of real algebraic curves where the real locus has the maximal number of connected components, which is determined by the genus of the curve. For elliptic normal curves, we further obtain definite symmetric determinantal representations for the hyperbolic secant hypersurfaces, which implies the existence of symmetric Ulrich sheaves of rank one on these hypersurfaces. We also use this to derive better bounds on the size of semidefinite representations for convex hulls of real algebraic curves of genus 1.
Funding statement: The authors have been supported by the Deutsche Foschungsgemeinschaft under Grants No. 421473641 and No. 426054364.
Acknowledgements
Part of this work was done while the authors were visiting the Simons Institute for the Theory of Computing. We would like to thank Kristian Ranestad for helpful discussions.
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Articles in the same Issue
- Frontmatter
- Artin–Mazur heights and Yobuko heights of proper log smooth schemes of Cartier type, and Hodge–Witt decompositions and Chow groups of quasi-F-split threefolds
- Kazhdan–Lusztig conjecture via zastava spaces
- Monodromic model for Khovanov–Rozansky homology
- Hyperbolic secant varieties of M-curves
- On the sharp lower bounds of modular invariants and fractional Dehn twist coefficients
- Degeneration of curves on some polarized toric surfaces
- Noether–Severi inequality and equality for irregular threefolds of general type
- Erratum to Table of Contents (J. reine angew. Math. 785 (2022), i–iv)
Articles in the same Issue
- Frontmatter
- Artin–Mazur heights and Yobuko heights of proper log smooth schemes of Cartier type, and Hodge–Witt decompositions and Chow groups of quasi-F-split threefolds
- Kazhdan–Lusztig conjecture via zastava spaces
- Monodromic model for Khovanov–Rozansky homology
- Hyperbolic secant varieties of M-curves
- On the sharp lower bounds of modular invariants and fractional Dehn twist coefficients
- Degeneration of curves on some polarized toric surfaces
- Noether–Severi inequality and equality for irregular threefolds of general type
- Erratum to Table of Contents (J. reine angew. Math. 785 (2022), i–iv)
