algebraic geometry 35 More on blow ups

Value of the Information & Strength of the Argument

The lecture provides valuable insights into the geometric and algebraic aspects of blow-ups. The argumentation is rigorous and well-structured, building on previous lectures and standard references. The examples are carefully chosen to illustrate key concepts, such as the non-orientability of the blow-up of the real plane and the resolution of a rational map. The discussion of blowing up along ideals and the potential introduction of singularities is particularly instructive.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on Hartshorne’s ‘Algebraic Geometry’, a standard and authoritative reference. The content is presented with mathematical rigor, and the explanations are clear. The title accurately reflects the content, which is a continuation of the discussion on blow-ups. The lecture does not cite external sources beyond the textbook, but the mathematical arguments are self-contained and reliable.

The title accurately reflects the content, which continues the discussion of blow-ups with examples and generalizations.

9/10

Lecture by a renowned mathematician, based on a standard textbook (Hartshorne), with rigorous mathematical content and clear explanations. The video is part of a structured course, indicating careful preparation.

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

• 00:00Introduction and overview of the lecture.
• 00:30Example: blowing up a point in the real affine plane.
• 02:00Discussion of the topology of the blow-up: it is a Möbius band.
• 05:00Example: rational map from P^1 × P^1 to P^2.
• 08:00Blowing up the undefined point to obtain a regular map.
• 10:00Comparison of blow-ups: P^1 × P^1 blown up at one point equals P^2 blown up at two points.
• 12:00Generalization: blowing up along subvarieties and ideals.
• 15:00Blowing up along a quasi-coherent sheaf of graded algebras.
• 18:00Hironaka's theorem on resolution of singularities.
• 20:00Example of blowing up along the ideal (x^2, y^2) leading to the Whitney umbrella singularity.

• Algebraic Geometry (book) — The course is based on Chapter I of Hartshorne's textbook.

• Algebraic Geometry (book) — The lecture follows the content of Hartshorne's textbook, which is a standard reference.