Hyperbolic Secant Varieties of M-Curves

TL;DR

This paper explores the geometry of hyperbolic varieties, focusing on secant varieties of real algebraic curves with many connected components, and establishes new determinantal representations for certain hypersurfaces.

Contribution

It introduces a detailed study of hyperbolic secant varieties of M-curves, including determinantal representations and Ulrich sheaves, advancing understanding of their geometric properties.

Findings

Secant varieties of M-curves have maximal real connected components.

Elliptic normal curves admit definite symmetric determinantal representations.

Existence of symmetric Ulrich sheaves of rank one on these hypersurfaces.

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.