[Paper Review] Algebro-geometric characterization of Cayley polytopes

Atsushi Ito|arXiv (Cornell University)|Feb 29, 2012

Advanced Combinatorial Mathematics10 references3 citations

TL;DR

This paper provides an algebro-geometric characterization of Cayley polytopes via the geometry of associated toric varieties, showing that a lattice polytope $P$ is a Cayley polytope of length $r+1$ if and only if its associated polarized toric variety $(X_P, L_P)$ is covered by $r$-planes. A key result establishes that for length 2, this condition is equivalent to the Seshadri constant $\varepsilon(X_P, L_P; 1) = 1$, offering a numerical criterion for lattice width one polytopes.

ABSTRACT

In this paper, we give an algebro-geometric characterization of Cayley polytopes. As a special case, we also characterize lattice polytopes with lattice width one by using Seshadri constants.

Motivation & Objective

• To provide an algebro-geometric characterization of Cayley polytopes using the geometry of associated polarized toric varieties.
• To establish a criterion for a lattice polytope to be a Cayley polytope of length $r+1$ without requiring $X_P$ to be $\mathbb{Q}$-factorial or $L_P$ to be very ample.
• To characterize lattice polytopes of lattice width one using Seshadri constants, linking combinatorics to positivity in algebraic geometry.

Proposed method

• Use the toric variety $(X_P, L_P)$ associated with a lattice polytope $P$ to translate combinatorial properties into geometric ones.
• Define and employ the notion of a variety being covered by $r$-planes, i.e., containing an $r$-dimensional projective space through every general point.
• Apply Seshadri constants $\varepsilon(X, L; 1)$ as invariants measuring positivity of line bundles to characterize the case $r=1$ (lines).
• Use blow-ups and numerical positivity criteria (Kleiman's criterion) to analyze the behavior of line bundles and their strict transforms under birational maps.
• Leverage the fact that $\varepsilon(X_P, L_P; 1) = 1$ implies the existence of rational curves of degree one through very general points.
• Prove equivalence between $\varepsilon(X_P, L_P; 1) = 1$, line coverings, and Cayley structure of length 2 via reduction to base-point-free linear systems and curve class analysis.

Experimental results

Research questions

• RQ1 When is a lattice polytope $P$ a Cayley polytope of length $r+1$ in terms of the geometry of its associated toric variety $(X_P, L_P)$?
• RQ2 Can the condition of being covered by $r$-planes characterize Cayley polytopes without assumptions on singularities or ampleness?
• RQ3 What is the precise algebro-geometric meaning of a lattice polytope having lattice width one?
• RQ4 How can Seshadri constants be used to detect Cayley structures in toric varieties?
• RQ5 Is there a numerical invariant that characterizes lattice polytopes of width one via positivity of line bundles?

Key findings

• A lattice polytope $P \subset \mathbb{R}^n$ is a Cayley polytope of length $r+1$ if and only if its associated polarized toric variety $(X_P, L_P)$ is covered by $r$-planes.
• For the case $r=1$, the Seshadri constant $\varepsilon(X_P, L_P; 1) = 1$ holds if and only if $P$ is a Cayley polytope of length 2.
• The equivalence between $\varepsilon(X_P, L_P; 1) = 1$ and the existence of lines through every very general point holds without requiring $L_P$ to be very ample or $X_P$ to be $\mathbb{Q}$-factorial.
• The Seshadri constant $\varepsilon(X_P, L_P; 1)$ is exactly 1 for all lattice polytopes of lattice width one.
• The result refines earlier work on dual defects by showing that positive dual defect implies Cayley structure, now with a precise geometric and numerical characterization.
• The proof technique relies on blow-ups and numerical positivity, using Kleiman’s criterion to detect nef but not ample divisors and to construct rational curves of degree one.