Hyperbolic Secant Varieties of M-Curves

Research output: Contribution to journalResearch articleContributedpeer-review

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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.

Details

Original languageEnglish
Pages (from-to)125-162
Number of pages38
Journal Journal für die reine und angewandte Mathematik : the world's oldest mathematical periodical
Volume2022
Issue number787
Publication statusPublished - 2022
Peer-reviewedYes

External IDs

Scopus 85129275546
WOS 000783649500001
Mendeley 469739a6-bf22-3140-ba9e-eb82e7e9e1cc

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