Non-existence of torically maximal hypersurfaces

TL;DR

This paper proves that hyperplanes are the only torically maximal hypersurfaces in higher-dimensional projective spaces, extending the understanding of real algebraic curves to hypersurfaces.

Contribution

It establishes the non-existence of non-hyperplane torically maximal hypersurfaces in higher dimensions, a significant extension of known results for curves.

Findings

Hyperplanes are the only torically maximal hypersurfaces in higher dimensions.

Non-existence of other torically maximal hypersurfaces beyond hyperplanes.

Extends the classification of torically maximal curves to higher-dimensional hypersurfaces.

Abstract

Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gau{\ss} map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.