Algebraic geometry 47: Resolution of curve singularities

Item: Algebraic geometry 47: Resolution of curve singularities
Rating: 5
Author: PostSingularity Institute

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMGeometry PBMWAlgebraic geometry

🎙 Richard E Borcherds 👥 82K 📅 June 19, 2020 ⏱ 21 min 👁 3K 📄 lecture 🧭 2026-08-17

Available in: English (current) Français

Keywords

resolution of singularitiesblowupplane curvesmultiplicityNewton polygonPuiseux seriescharacteristic zero

Summary

This lecture is part of an online algebraic geometry course based on Hartshorne’s book. The main goal is to show that singularities of plane curves over a field of characteristic zero can be resolved by repeatedly blowing up the singular points. The lecturer begins by explaining how a function field of transcendence degree one corresponds to a projective curve, possibly singular, and that resolving singularities yields a nonsingular projective curve. The method, attributed to Isaac Newton, involves an algorithm: at each step, blow up the singular point and repeat until the curve becomes nonsingular. The key idea is to analyze the lowest-degree homogeneous part of the defining polynomial, which determines the multiplicity and the directions of the singularity. Blowing up separates these directions, reducing multiplicity unless all tangent directions coincide. In that case, a change of variables can be made to simplify the polynomial, and the process is analyzed using a Newton polygon. The lecturer shows that the process terminates unless the polynomial has multiple factors, which is excluded for irreducible curves. The proof relies on the characteristic being zero, as the coefficient of a certain term vanishes in positive characteristic. The lecture concludes with a mention of Puiseux series, which are related to Newton’s original approach.

207 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous explanation of a fundamental result in algebraic geometry. The argumentation is solid, building from the problem of constructing nonsingular curves from function fields to the resolution of singularities via blowups. The use of examples, such as the curve y^2 + x^9 + y^5 x^3, helps illustrate the process. The lecturer carefully explains the Newton polygon method and the termination argument, highlighting the role of characteristic zero. The presentation is logical and well-paced, making complex ideas accessible to advanced students.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on standard material from Hartshorne’s ‘Algebraic Geometry’, a widely respected textbook. The proof sketch is mathematically accurate, with appropriate caveats about characteristic zero and multiple factors. The title accurately reflects the content. The lecturer is a Fields medalist, adding to the credibility. No external sources are cited in the description, but the reliance on Hartshorne is explicit.

162 words

Title / Content Match

The title accurately reflects the content, which focuses on resolving singularities of algebraic curves.

Quality & Reliability

9/10

The lecture is given by a renowned mathematician, Richard Borcherds, and presents a rigorous proof sketch of resolution of curve singularities via blowups, based on Hartshorne's textbook. The content is mathematically sound and well-structured, with clear explanations and examples. The proof relies on standard algebraic geometry techniques and is presented with appropriate caveats about characteristic zero.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Cited Sources

Algebraic Geometry (book) — The course is based on Chapter I of this textbook.

Concurring Sources

Hartshorne's Algebraic Geometry — The lecture follows the content of this textbook.

Contribution & Novelties

This lecture provides a clear and rigorous exposition of the resolution of curve singularities via blowups, a classical result. The originality lies in the pedagogical approach, using the Newton polygon to visualize the effect of blowups on monomials and explaining the termination argument in detail. The connection to Puiseux series and Newton’s original method adds historical context.

Pour aller plus loin :

Resolution of singularities — Overview of the general problem and its history.
Blowing up — Definition and properties of blowups in algebraic geometry.
Newton polygon — Tool used to analyze polynomial equations.
Puiseux series — Series expansions for algebraic functions, introduced by Newton.

105 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a lecture that is both information-dense and technically rigorous. The balance between quantity and quality of information is strong, with a high level of technical depth suitable for an advanced audience.

Reliability 9/10