On the Number of Connected Components of T-Hypersurfaces

TL;DR

This paper investigates the maximum number of connected components of T-hypersurfaces in real toric varieties, characterizes when this maximum is achieved, and studies how this number grows with the dilation of the moment polytope.

Contribution

It characterizes the conditions under which T-hypersurfaces attain the upper bound on connected components and shows the existence of such configurations on the standard simplex.

Findings

Upper bound on connected components is not always attainable.

A sharper upper bound is provided for certain triangulations.

Expected number of components grows as order of d^n with dilation.

Abstract

A T-hypersurface is a combinatorial hypersurface of the real locus of a projective toric variety $Y$. It is constructed from a primitive triangulation $K$ of a moment polytope $P$ of $Y$ and a $0$-cochain $\varepsilon$ on $K$ with coefficients in the field with two elements $\mathbb{F}_2$, called a sign distribution. O. Viro showed that when $K$ is convex the T-hypersurface is ambiantly isotopic to a real algebraic hypersurface of $Y$. A. Renaudineau and K. Shaw gave upper bounds on the Betti numbers of T-hypersurfaces in terms of the Hodge numbers of a generic section of the ample line bundle $L$ associated with the moment polytope. In particular, the number of connected components of a T-hypersurface cannot exceed the geometric genus of a generic section of $L$ plus one. In this article we investigate whether this upper bound is attainable. We are able to characterise the couples $(K;\varepsilon)$ leading to T-hypersurfaces realising the Renaudineau-Shaw upper bound on the number of connected components. This theorem generalises B. Haas' theorem for T-curves. In contrast with this results we find that the upper bound is not always attainable on every primitive triangulations. For some of those on which it is not attainable we provide a sharper upper bound. Finally we use our characterisation to show that there always exists a triangulation and a sign distribution on the standard simplex that reach the Renaudineau-Shaw upper bound. We also study the growth of the expected number of connected components of a T-hypersurface as we dilate the moment polytope by $d$ (i.e. we tensorise the line bundle $d$-times with itself) and show that it is always of the order of $d^n$ where $n$ is the dimension of $P$.