[Paper Review] Irrational Toric Varieties

Ata Firat Pi̇r|arXiv (Cornell University)|Sep 13, 2018

Algebraic Geometry and Number Theory6 references3 citations

TL;DR

This paper develops a theory of irrational toric varieties associated to arbitrary fans in ℝⁿ, constructing them as (ℝ>⁰)ⁿ-equivariant cell complexes dual to the fan. The key contribution is showing that such a variety is homeomorphic to the polytope it defines when the fan is the normal fan of a polytope, and proving that the space of Hausdorff limits of translates of an irrational toric variety is homeomorphic to the secondary polytope of its exponent set.

ABSTRACT

Classical toric varieties are among the simplest objects in algebraic geometry. They arise in an elementary fashion as varieties parametrized by monomials whose exponents are a finite subset $\mathcal{A}$ of $\mathbb{Z}^n$. They may also be constructed from a rational fan $Σ$ in $\mathbb{R}^n$. The combinatorics of the set $\mathcal{A}$ or fan $Σ$ control the geometry of the associated toric variety. These toric varieties have an action of an algebraic torus with a dense orbit. Applications of algebraic geometry in geometric modeling and algebraic statistics have long studied the nonnegative real part of a toric variety as the main object, where the set $\mathcal{A}$ may be an arbitrary set in $\mathbb{R}^n$. These are called irrational affine toric varieties. This theory has been limited by the lack of a construction of an irrational toric variety from an arbitrary fan in $\mathbb{R}^n$. We construct a theory of irrational toric varieties associated to arbitrary fans. These are $(\mathbb{R}_>)^n$-equivariant cell complexes dual to the fan. Such an irrational toric variety is projective (may be embedded in a simplex) if and only if its fan is the normal fan of a polytope, and in that case, the toric variety is homeomorphic to that polytope. We use irrational toric varieties to show that the space of Hausdorff limits of translates an irrational toric variety associated to a finite subset $\mathcal{A}$ of $\mathbb{R}^n$ is homeomorphic to the secondary polytope of $\mathcal{A}$.

Motivation & Objective

• To extend classical toric geometry to irrational settings where exponent sets are in ℝⁿ rather than ℤⁿ.
• To resolve the lack of a fan-based construction for irrational toric varieties, which previously hindered the study of Hausdorff limits.
• To establish a homeomorphism between the space of Hausdorff limits of translates of an irrational toric variety and the secondary polytope of its exponent configuration.
• To develop a functorial, duality-based construction of irrational toric varieties as cell complexes with torus action.
• To enable applications in geometric modeling and algebraic statistics by providing a topological and combinatorial framework for non-rational configurations.

Proposed method

• Construct irrational toric varieties as (ℝ>⁰)ⁿ-equivariant cell complexes, with cells dual to cones in a fan Σ ⊂ ℝⁿ.
• Define the cell structure so that the poset of cell closures is anti-isomorphic to the face lattice of the fan Σ.
• Use the fan Σ to define a topological space XΣ that generalizes classical toric varieties, even when Σ is not rational.
• Prove that XΣ is compact if and only if Σ is a complete fan, extending a classical result.
• Show that when Σ is the normal fan of a polytope P, then XΣ is homeomorphic to P, generalizing the projective case.
• Establish a homeomorphism between the space of Hausdorff limits of translates of XΣ and the secondary polytope of the exponent set A via the secondary fan Σ(A).

Experimental results

Research questions

• RQ1 Can a theory of irrational toric varieties be constructed from arbitrary fans in ℝⁿ, even when the fan is not rational?
• RQ2 Is there a homeomorphism between the space of Hausdorff limits of translates of an irrational toric variety and the secondary polytope of its exponent set?
• RQ3 How does the irrational toric variety XΣ relate to the classical toric variety YΣ when Σ is rational?
• RQ4 Can the irrational toric variety XΣ be recovered from its topological and equivariant structure, ensuring the construction is functorial?
• RQ5 What is the role of the secondary fan in classifying degenerations of irrational toric varieties?

Key findings

• The irrational toric variety XΣ associated to a fan Σ ⊂ ℝⁿ is a (ℝ>⁰)ⁿ-equivariant cell complex whose cell structure is dual to the face lattice of Σ.
• When Σ is rational, XΣ is homeomorphic to the nonnegative real part of the classical toric variety YΣ.
• XΣ is compact if and only if Σ is a complete fan, extending a classical result to the irrational setting.
• If Σ is the normal fan of a polytope P, then XΣ is homeomorphic to P, establishing a topological realization of the polytope.
• The space of Hausdorff limits of translates of XΣ is homeomorphic to the secondary polytope of the exponent set A, resolving a gap in prior work.
• The construction is functorial: maps of fans induce equivariant maps of irrational toric varieties, and the fan can be reconstructed from the variety.