Asymptotics of discrete Okounkov bodies and thresholds

TL;DR

This work introduces and analyzes discrete Okounkov bodies as a robust framework for the asymptotics of algebraic-geometric thresholds in the big cone. It develops volume quantiles, joint k–m limits via Grassmannians, and a discrete-to-continuum theory (rooftop Okounkov bodies and idealized constructs) to unify the asymptotics of stability thresholds δ_k and global log canonical thresholds α_k. The authors prove that δ_τ exists for τ ∈ [0,1], with δ_0 = α and δ_1 = δ, and establish precise convergence rates when the optimizing valuation is divisorial, notably α_k = α + O(k^{−1/n}) and δ_k = δ + O(k^{−1}) in corresponding cases. A high-dimensional generalization of Weierstrass gaps emerges from the gaps between Δ_k and Δ, linking classical gap theory to modern threshold analysis through discrete Okounkov bodies and tail-distribution techniques. The results rely on 1-parameter families of Okounkov bodies, detailed lattice-point estimates, and a comprehensive valuation-theoretic framework, with implications for K-stability and canonical metrics in the big class setting.

Abstract

This article initiates the study of discrete Okounkov bodies and higher-dimensional Weierstrass gap phenomena, with applications to asymptotic analysis of stability and global log canonical thresholds.

Figures (9)

• Figure 1: A 1-parameter family of Okounkov bodies and a discrete Okounkov body associated to a divisorial valuation with $\#(\Delta_k)=d_k=4$ and $\#(\Delta\cap {\mathbb Z}^n/k)=D_k=8$.
• Figure 2: A possible Okounkov body $\Delta$ of some big line bundle over a surface with respect to some flag $Y_\bullet:Y_0\supset Y_1\supset Y_2$. The dots are the $k$-th lattice points $\Delta\cap{\mathbb Z}^2/k$, of which the solid dots are the ones corresponding to sections in $R_k$. Let $v=\mathrm{ord}_{Y_1}$. Then there are no solid dots whose first coordinate exceeds $\max_{\Delta_k}\!p_1$ (or $S_{k,1}(v)$ in the notation of Definition \ref{['SkmDef']}; $\max_{\Delta_k}\!p_1={\mathcal{S}}_0(v)$ in the notation of Definition \ref{['S0Def']}).
• Figure 3: The discrete Okounkov bodies $\Delta_k\subset\Delta\cap{\mathbb Z}/k\subset\Delta=[0,4]$ of ${\mathcal{O}}_C(K_C)$ over a smooth quartic plane curve $C$, with the flag chosen to be $C\supset\{p\}$, for $k\in\{1,2\}$. There are six cases: $p$ is not a 2-Weierstrass point, $p$ a flex point, $p$ a hyperflex point, and $p$ a $s$-sextactic point for $s\in\{1,2,3\}$, respectively Ver, AS,Cayley.
• Figure 4: The discrete Okounkov bodies $\Delta_k\subset\Delta\cap{\mathbb Z}/k\subset\Delta=[0,1]$ of ${\mathcal{O}}_C(p)$ over a smooth quartic plane curve $C$, with the flag chosen to be $C\supset\{p\}$, for $k\in\{1,\ldots,5\}$. There are three cases: $p$ is a non-Weierstrass point, i.e., its gap sequence is $1,2,3$; $p$ is a flex point, for which the gap sequence is $1,2,4$; $p$ is a hyperflex point, for which the gap sequence is $1,2,5$Ver.
• Figure 5: The Okounkov body of ${\mathcal{O}}_X(1,1)$ over $X={\mathbb P}^1\times{\mathbb P}^1$, with the flag chosen to be $X\supset C\supset \{p\}$, where $C$ is a smooth $(2,1)$-curve. The discrete Okounkov body $\Delta_k\subset\Delta\cap{\mathbb Z}^2/k$ is drawn in three cases: $k=1$, $k=2$ with $p$ not a ramification point, and $k=2$ with $p$ a ramification point, respectively.

Theorems & Definitions (130)

• Theorem 1.2
• Theorem 1.4
• Theorem 1.5
• Definition 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 1.10
• Theorem 1.11
• Definition 1.12
• Definition 1.13