Algebraic geometry 43: Proper maps

Algebraic geometry 43: Proper maps

by Richard E Borcherds

Formal & Physical Sciences Mathematics PBMGeometryPBMWAlgebraic geometry
🎙 Richard E Borcherds 👥 82K 📅 June 16, 2020 ⏱ 25 min 👁 2K 📄 lecture 🧭 2026-08-17
Available in: English (current) Français

Keywords

proper mapuniversally closedcomplete varietyprojective spaceblow-up

Summary

This lecture on algebraic geometry, part of a series by Richard Borcherds, focuses on proper maps and their significance. It begins by revisiting the resultant of two homogeneous polynomials, showing that the image of the intersection of two hypersurfaces under projection is closed, which is a key property. The lecture then discusses the concept of compactness in algebraic geometry, noting that the Zariski topology makes all varieties compact, so a different notion is needed. Proper maps are introduced as the correct analog, defined as universally closed morphisms. The main goal is to prove that projective varieties are complete, meaning the map to a point is proper. The proof uses the resultant to show that the projection from P^1 × A^m to A^m is closed, and then employs a blow-up to handle the general case for P^n. The blow-up of P^n at a point is shown to be a P^1-bundle over P^{n-1}, which allows an inductive proof. The lecture concludes by illustrating the technique of using blow-ups to turn rational maps into regular maps, with the example of a ruled surface.

180 words

Critical Evaluation

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous introduction to proper maps, a fundamental concept in algebraic geometry. The argumentation is solid, building from concrete examples with resultants to abstract definitions and proofs. The use of the resultant to demonstrate closedness of projections is particularly illuminating, and the step-by-step proof that projective varieties are complete is well-structured. The lecturer also highlights common pitfalls, such as the image of a polynomial map not being closed, which enhances understanding. The explanation of the blow-up technique is insightful, showing how it resolves rational maps and enables induction.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on standard algebraic geometry, referencing Hartshorne’s textbook. The mathematical rigor is high, with precise definitions and proofs. The title accurately reflects the content, as the lecture is indeed about proper maps. No external sources are cited in the video, but the reliance on established theory ensures reliability. The lecture is part of a series, which provides context and continuity.

171 words

Title / Content Match

The title accurately reflects the content, as the lecture focuses on proper maps and their role in algebraic geometry.

Quality & Reliability

9/10

The lecture is given by a renowned mathematician, Richard Borcherds, and is part of a well-structured course on algebraic geometry. The content is rigorous, based on standard references like Hartshorne, and the proofs are carefully presented.

Key Moments

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

Cited Sources

  • Algebraic geometry — The course is based on chapter I of Hartshorne's textbook.

Concurring Sources

  • Algebraic geometry — The lecture follows the standard treatment in Hartshorne's textbook.

Contribution & Novelties

The lecture provides a clear and rigorous exposition of proper maps, a key concept in algebraic geometry. It connects the notion of properness to the intuitive idea of compactness, and demonstrates the importance of the resultant in proving closedness of projections. The use of blow-ups to resolve rational maps is a powerful technique that is well illustrated. This lecture is particularly valuable for students learning algebraic geometry, as it bridges abstract definitions with concrete examples.

Pour aller plus loin :

  • Proper morphism — Wikipedia article on proper morphisms, which are the algebraic geometry analog of proper maps.
  • Complete variety — Wikipedia article on complete varieties, which are the analog of compact spaces.
  • Blowing up — Wikipedia article on blow-ups, a technique used in the lecture.
  • Resultant — Wikipedia article on resultants, which are used in the proof.

137 words

Radar Profile

The radar profile shows high scores across all dimensions, indicating a lecture that is both informative and rigorous, with a strong technical level. The balance between quantity and quality of information is excellent, making it a reliable resource for advanced students.

Reliability 9/10