Projective hypersurfaces in tropical scheme theory I: the Macaulay ideal

TL;DR

The paper introduces a new construction called the Macaulay tropical ideal, which extends principal ideals to tropical ideals with a universal property and generates non-realizable hypersurface schemes in tropical geometry.

Contribution

The novel contribution is the construction of the Macaulay tropical ideal with a universal property and its application to produce non-realizable hypersurface schemes.

Findings

The Macaulay tropical ideal has a universal property for extending principal ideals.

The construction yields non-realizable degree d hypersurface schemes in projective space for n ≥ 2 and d ≥ 1.

For (d,n) = (1,2), the Macaulay construction matches a previously known non-realizable line construction.

Abstract

A ”tropical ideal” is an ideal in the idempotent semiring of tropical polynomials that is also, degree by degree, a tropical linear space. We introduce a construction based on transversal matroids that canonically extends any principal ideal to a tropical ideal. We call this the Macaulay tropical ideal. It has a universal property: any other extension of the given principal ideal to a tropical ideal with the expected Hilbert function is a weak image of the Macaulay tropical ideal. For each \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n\ge 2$$\end{document}n≥2 and \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$d\ge 1$$\end{document}d≥1, our construction yields a non-realizable degree d hypersurface scheme in \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbb {P}^n$$\end{document}Pn. Maclagan-Rincón produced a non-realizable line in \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\mathbb {P}^n$$\end{document}Pn for each n, and for \documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$(d,n)=(1,2)$$\end{document}(d,n)=(1,2) the two constructions agree. An appendix by Mundinger compares the Macaulay construction with another method for canonically extending ideals to tropical ideals.

Figures