Embeddings of weighted projective spaces

TL;DR

The paper develops a comprehensive framework for understanding when powers of an ample line bundle on weighted projective (toric) spaces yield very ample and projectively normal embeddings by translating the problem to the lattice geometry of rectangular simplices. It introduces and interrelates key invariants ($\mu_{\mathrm{va}}$, $\mu_{\mathrm{norm}}$), LPE criteria, and periodicity phenomena, providing sharp arithmetic criteria and constructive methods for extremal examples, including infinitely many maximally non-normal simplices. A central contribution is the algorithmic construction of large families of maximally non-normal rectangular simplices with distinct prime entries, together with a hypergraph framework that recasts embedding properties as combinatorial conditions. The results extend and connect prior work of Payne, Hering, and Bruns–Gubeladze, yielding new examples and a systematic toolkit to analyze normality and very ampleness in weighted projective settings. Overall, the work advances the understanding of the bridge between toric embeddings, lattice polytopes, and combinatorial models, offering concrete criteria and constructions with potential applications in discrete geometry and integer programming.

Abstract

Let $X$ be a projective toric variety of dimension $n$ and let $L$ be a ample line bundle on $X$. For $k \geq 0$, it is in general difficult to determine whether $L^{\otimes k}$ is very ample and whether it additionally gives a projectively normal embedding. These two properties are equivalent to the very ampleness, respectively normality, of the corresponding polytope. By a result of Ewald-Wessels, both statements are classically known to hold for $k \geq n - 1$. We study embeddings of weighted projective spaces $\mathbb{P}(a_0, \ldots, a_n)$ via their corresponding rectangular simplices $Δ(λ_1, \ldots, λ_n)$. We give multiple criteria (depending on arithmetic properties of the weights $a_i$) to obtain bounds for the power $k$ which are sharp in many cases. We also introduce combinatorial tools that allow us to systematically construct families exhibiting extremal behaviour. These results extend earlier work of Payne, Hering and Bruns-Gubeladze.

Figures (5)

• Figure 1: Implication graph of properties of rectangular simplices.
• Figure 2: The corner polytope $Q$ is shown here in pink.
• Figure 3: Two representations of a hypergraph $G$.
• Figure 4: Some example hypergraphs. Hypergraph $G_2$ corresponds to the complete graph $K_4$, and hypergraph $G_4$ corresponds to the Fano plane.
• Figure 5: The Fano plane.

Theorems & Definitions (96)

• Theorem 1.1
• Theorem 1.2
• Definition 1.3
• Proposition 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 1.7
• Definition 2.1
• Definition 2.2
• Proposition 2.3