Classification of real hyperplane singularities by real log canonical thresholds

TL;DR

This work develops an algebraic framework to study the real log canonical threshold $\lambda_\mathbb{R}$ and its multiplicity $m_\mathbb{R}$ for real hyperplane arrangements, linking them to the complex invariants $\lambda_\mathbb{C}$ and $m_\mathbb{C}$ and to Watanabe's singular learning theory. It derives explicit combinatorial formulas using building sets and wonderful compactifications, proving $\lambda_\mathbb{R}=\lambda_\mathbb{C}$ and $m_\mathbb{R}=m_\mathbb{C}$ under natural real-point conditions, and providing efficient computational tools via a SageMath implementation. The paper applies these results to central and non-central real hyperplane arrangements, detailing how the rlct governs the asymptotics of high-dimensional volume integrals and the stochastic complexity in singular models. These contributions connect birational geometry with statistical learning, offering practical methods to evaluate rlct and its multiplicity and enabling broader analysis of singular models and their asymptotics. The inclusion of explicit algorithms and examples enhances applicability to geometry, combinatorics, and statistical asymptotics.

Abstract

The log canonical threshold (lct) is a fundamental invariant in birational geometry, essential for understanding the complexity of singularities in algebraic varieties. Its real counterpart, the real log canonical threshold (rlct), also known as the learning coefficient, has become increasingly relevant in statistics and machine learning, where it plays a critical role in model selection and error estimation for singular statistical models. In this paper, we investigate the rlct and its multiplicity for real (not necessarily reduced) hyperplane arrangements. We derive explicit combinatorial formulas for these invariants, generalizing earlier results that were limited to specific examples. Moreover, we provide a general algebraic theory for real log canonical thresholds, and present a SageMath implementation for efficiently computing the rlct and its multiplicity in the case or real hyperplane arrangements. Applications to examples are given, illustrating how the formulas can also be used to analyze the asymptotic behavior of high-dimensional volume integrals.

Figures (3)

• Figure 1: Comparison of normal crossing configurations.
• Figure 2: The area of the set of all $(x,y) \in [-1,1]^2$ with $|x^3y - xy^3| \leq \varepsilon$ goes to $0$ like $C \varepsilon^{1/2}$, for some constant $C$, because $\mathop{\mathrm{rlct}}\nolimits(f) = (1/2, 1)$, see Example \ref{['example:recover-tubes']}(d).
• Figure 3: The surface defined by $|xy^2z^2 (x+y+z)| = \varepsilon$ in a compact neighbourhood of the origin in $\mathbb{R}^3$, for small $\varepsilon >0$. It consists of sixteen connected components. As $\mathop{\mathrm{rlct}}\nolimits(xy^2z^2 (x+y+z)) = (1/2,3)$, see Example \ref{['example:four-planes']}, the area between these components asymptotically equals $C\varepsilon^{1/2}(-\ln \varepsilon)^2$, for some constant $C$.

Theorems & Definitions (27)

• Definition 2.1
• proof
• proof
• Theorem 3.1
• proof
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Lemma 3.5
• proof