Schemes 45: Blowing up schemes

Value of the Information & Strength of the Argument

The lecture provides a clear and rigorous introduction to blowing up in scheme theory. It builds on previously established concepts (Proj, sheaves of ideals) and carefully explains the motivation and technical details. The argumentation is solid, with explicit checks and examples. The distinction between the inverse image ideal sheaf and the pullback is well explained, addressing a common source of confusion. The applications to resolution of singularities and elimination of indeterminacy are presented with sufficient context, though the resolution of singularities is only briefly mentioned.

Scientific Rigor, Source Quality, Title Accuracy

The lecture is based on a standard reference (Hartshorne’s ‘Algebraic Geometry’), and the lecturer is a well-known mathematician. The content is mathematically rigorous, with definitions and proofs sketched appropriately for a lecture. The title accurately reflects the content. No external sources are cited beyond the textbook, but the lecture is self-contained and relies on established mathematical knowledge.

The title accurately reflects the content, which focuses on the blowing up operation in scheme theory.

8/10

Lecture by a renowned mathematician, based on a standard textbook (Hartshorne), with rigorous definitions and proofs sketched. The content is mathematically sound and well-structured.

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

• 00:00Introduction and recall of blowing up a point in affine space.
• 02:00General construction of blowing up along a sheaf of ideals using Proj.
• 05:00Verification that the general construction matches the classical example.
• 08:00Discussion of the effect of blowing up: making the ideal sheaf invertible.
• 10:00Explanation of the difference between inverse image ideal sheaf and pullback.
• 12:00Affine example illustrating the difference between the two operations.
• 14:00Sketch of the proof that the blow-up makes the ideal invertible.
• 16:00Application to resolution of singularities (Hironaka's theorem).
• 18:00Application to eliminating points of indeterminacy of rational maps.
• 20:00General method for resolving indeterminacy using blowing up.

• Algebraic Geometry — Based on Chapter II of Hartshorne's textbook.

• Algebraic Geometry — The lecture follows the treatment in Hartshorne's textbook, which is a standard reference.