[Paper Review] Compactifications of subvarieties of tori

Jenia Tevelev|ArXiv.org|Dec 16, 2004

Geometric and Algebraic Topology14 references4 citations

TL;DR

This paper introduces a new method for constructing compactifications of subvarieties of algebraic tori using polyhedral structures on non-archimedean amoebas, resulting in tropical compactifications with desirable geometric properties such as combinatorial normal crossings and Cohen-Macaulay singularities. The key contribution is a constructive, combinatorial framework that generalizes Kapranov's visible contour and provides a log canonical model for complements of hyperplane arrangements.

ABSTRACT

We study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non-archimedean amoebas. These compactifications have many nice properties, for example any k boundary divisors intersect in codimension k. We consider some examples including $M_{0,n}\subset\bar M_{0,n}$ (and more generally log canonical models of complements of hyperplane arrangements) and compact quotients of Grassmannians by a maximal torus.

Motivation & Objective

• To develop a systematic method for constructing well-behaved compactifications of subvarieties in algebraic tori.
• To establish conditions under which such compactifications have desirable geometric properties, such as combinatorial normal crossings and Cohen-Macaulay singularities.
• To generalize Kapranov's visible contour construction to higher-codimension subvarieties and hyperplane arrangement complements.
• To relate tropical compactifications to existing quotient constructions like GIT and Hilbert quotients.
• To show that the log canonical line bundle of the compactification is very ample and globally generated.

Proposed method

• Uses the closure of a subvariety $X$ in a smooth toric variety $\mathbb{P}$ to define a tropical compactification via the multiplication map $\Psi: T \times \overline{X} \to \mathbb{P}$, requiring faithful flatness.
• Applies the Hilbert scheme of $\mathbb{P}$ to construct a Groebner toric variety $\mathbb{P}_{\text{Gr}}$ as the normalization of the $T$-orbit closure of $[\overline{X}]$.
• Defines the visible contour $\overline{X}_{\text{vc}}$ as the closure of $X/T_X$ in $\mathbb{P}_{\text{Gr}}$, characterized by $\overline{X}_{\text{vc}} = \{[Y] \in \mathbb{P}_{\text{Gr}} \mid e \in Y\}$, where $e$ is the identity.
• Employs non-archimedean amoebas and Lafforgue's transversality argument to ensure the existence of such compactifications.
• Uses matroid decompositions of the hypersimplex $\Delta(r,n)$ induced by weight functions $\omega$ to analyze the initial ideals and cross-ratios.
• Relies on the fan $\mathcal{F}_{\text{Laf}}$ associated with Lafforgue's transversality to compare coarsenings of matroid decompositions and initial ideals.

Experimental results

Research questions

• RQ1 Can any subvariety of a torus admit a tropical compactification with smooth toric ambient space and combinatorial normal crossings?
• RQ2 What conditions ensure that a tropical compactification has a smooth multiplication map and toroidal singularities?
• RQ3 How does the visible contour construction generalize Kapranov's original definition for Grassmannians?
• RQ4 Can the log canonical line bundle of the compactification be globally generated and very ample?
• RQ5 How do cross-ratios on the compactification reflect the combinatorial structure of the initial ideal?

Key findings

• Every subvariety $X$ of a torus admits a tropical compactification $\overline{X}$ in a smooth toric variety $\mathbb{P}$, with boundary divisors intersecting in codimension $k$ for $k$ divisors.
• If $X$ is schön, then any tropical compactification $\overline{X} \subset \mathbb{P}$ has a smooth multiplication map, is regularly embedded, and has toroidal singularities.
• The log canonical line bundle of $\overline{X}_{\text{vc}}$ is globally generated and equal to the determinant of the normal bundle.
• For complements of essential, connected hyperplane arrangements, $\overline{X}_{\text{vc}}$ is a log canonical model and the log canonical bundle is very ample.
• The initial ideal $\mathcal{I}_\omega$ determines whether any cross-ratio on the compactification is $0$, $1$, $\infty$, or none of the above.
• The interior of any cone in the tropical fan $\mathcal{F}_{\text{trop}}$ lies in the interior of a cone in the Lafforgue fan $\mathcal{F}_{\text{Laf}}$, ensuring proper coarsening of matroid decompositions.