[Paper Review] Desingularization in Computational Applications and Experiments

Anne Frühbis-Krüger|arXiv (Cornell University)|Jan 16, 2013

Polynomial and algebraic computation12 references3 citations

TL;DR

This paper presents practical computational methods for applying resolution of singularities in algebraic geometry, focusing on extracting invariants like the intersection form, dual graph, discrepancies, log-canonical threshold, and topological zeta-function from algorithmically computed resolution data. It demonstrates these techniques using Singular software and illustrates their relevance to Bernstein-Sato polynomials and experimental desingularization in positive characteristic.

ABSTRACT

After briefly recalling some computational aspects of blowing up and of representation of resolution data common to a wide range of desingularization algorithms (in the general case as well as in special cases like surfaces or binomial varieties), we shall proceed to computational applications of resolution of singularities in singularity theory and algebraic geometry, also touching on relations to algebraic statistics and machine learning. Namely, we explain how to compute the intersection form and dual graph of resolution for surfaces, how to determine discrepancies, the log-canoncial threshold and the topological Zeta-function on the basis of desingularization data. We shall also briefly see how resolution data comes into play for Bernstein-Sato polynomials, and we mention some settings in which desingularization algorithms can be used for computational experiments. The latter is simply an invitation to the readers to think themselves about experiments using existing software, whenever it seems suitable for their own work.

Motivation & Objective

• To bridge theoretical resolution of singularities with practical computational applications in algebraic geometry and singularity theory.
• To demonstrate how algorithmic resolution data—generated via blowing up—can be used to compute key invariants such as the dual graph and intersection form.
• To explore the role of resolution data in computing discrepancies, log-canonical thresholds, and topological zeta-functions.
• To suggest experimental uses of desingularization algorithms in challenging settings, such as positive characteristic and Bernstein-Sato polynomial analysis.
• To provide accessible, code-illustrated examples using the Singular software system to enable reproducible research and tool familiarity.

Proposed method

• Utilizes algorithmic desingularization via blow-ups, particularly a variant of Villamayor’s algorithm implemented in Singular.
• Represents resolution data across multiple charts, with practical focus on identifying and gluing common points rather than purely theoretical gluing.
• Applies embedded resolution to compute discrepancies and log-canonical thresholds, requiring detailed tracking of exceptional divisors and multiplicities.
• Employs resolution data to compute the topological zeta-function by analyzing the structure of the exceptional divisor and its components.
• Uses the known form of the Bernstein-Sato polynomial for monomial functions after principalization to infer roots from exceptional multiplicities.
• Proposes experimental use of desingularization algorithms in positive characteristic, especially for testing structural failures in Hironaka-type approaches.

Experimental results

Research questions

• RQ1 How can resolution data from algorithmic desingularization be systematically used to compute the dual graph and intersection form of surface singularities?
• RQ2 What resolution data is required to compute discrepancies and the log-canonical threshold, and how is this data structured in embedded resolution?
• RQ3 In what ways can resolution data inform the computation of the topological zeta-function and the Bernstein-Sato polynomial?
• RQ4 What experimental insights can be gained from applying desingularization algorithms in positive characteristic, particularly regarding the failure of maximal contact and order increase?
• RQ5 How might known resolution data help predict or understand the denominators of roots of the Bernstein-Sato polynomial?

Key findings

• The dual graph and intersection form of surface singularities can be computed directly from resolution data using only the basic structure of exceptional divisors and their intersections.
• Discrepancies and the log-canonical threshold require embedded resolution data, including the strict transform and multiplicities of exceptional divisors.
• The topological zeta-function is computable from the resolution data by analyzing the configuration of the exceptional divisor and its components.
• The roots of the Bernstein-Sato polynomial for a monomial function after principalization are rational numbers of the form $-k/r_i$, where $r_i$ are the exceptional multiplicities.
• The failure of maximal contact and the increase of order in auxiliary ideals in positive characteristic are key obstacles to Hironaka-style desingularization, as identified by Hauser.
• Algorithmic desingularization in positive characteristic can be used experimentally to search for counterexamples or structural patterns, particularly in dimension two using modified implementations.