Math 216: Foundations of algebraic geometry 2007-08

The course webpage for the 2011-12 version of the course is here, along with more recent lecture notes. Go there first.

All notes in four big files

Here are the notes: Fall, winter, spring, problem sets. These notes are quite rough, and roughly reflect what I did in class. I have reordered and improved them for the 2009-10 class, but these 2007-08 files (with all remaining errors and infelicities) will stay frozen. I didn't get around to typing up notes for some of the later classes, but you can find some of that discussion in the 2005-06 course notes.

References: The posted lecture notes will be rough, so I recommend having another source you like, for example Mumford's Red Book of Varieties and Schemes (the original edition is better, as Springer introduced errors into the second edition by retyping it), and Hartshorne's Algebraic Geometry. Both books are on reserve at the library. Mumford (2nd ed) may be availble online (with a Stanford account) from Springer (thanks Justin!). Hartshorne should be available at the bookstore. For background on commutative algebra, I'd suggest consulting Eisenbud's Commutative Algebra with a view toward algebraic geometry or Atiyah and MacDonald's Commutative Algebra. For background on abstract nonsense, Weibel's Introduction to Homological Algebra is good to have handy. Justin Walker also points out that Freyd's Abelian Categories is available online (free and legally) here.

Homework: Unlike most advanced graduate courses, there will be homework. It is important --- this material is very dense, and the only way to understand it is to grapple with it at close range.

Notes: Notes for the classes in ps and pdf formats will be posted here. Caution: All of these notes are quite rough, and just approximate transcriptions of my lecture notes. I encourage you to take notes yourselves, and not just rely on these. However, if you feel like pointing out improvements, I would appreciate it, as these notes are crudely extracted from a larger set of notes that I hope to make available eventually. Note that I give the dates of the last important update. Unimportant updates that do not change anything substantive will not be flagged.

Baiju Bhatt has pointed out Brian Osserman's very helpful cheatsheets that might help you keep track of the myriad definitions. He has one for properties of schemes and one for properties of morphisms of schemes.

Fall quarter

Winter quarter quadric surface

Spring quarter

  • Class 41 (Tues. Apr. 1): maps to projective schemes extend over smooth codimension one points (the "curve to projective extension theorem"); and the left-over topic of normalization.
  • Class 42 (Thurs. Apr. 3): a bunch of equivalent categories, each roughly of "curves over k"; the degree of a map of curves.
  • Class 43 (Tues. Apr. 8): Towards "fun with curves". Black boxes: Serre duality, Riemann-Hurwitz. A criterion for a map to be a closed immersion.
  • Class 44 (Thurs. Apr. 10): some initial crucial observations about curves; curves of genus 0, curves of genus 2.
  • Class 45 (Tues. Apr. 15): hyperelliptic curves (and the conclusion of curves of genus 2); curves of genus 3. The 28 bitangents, and the 27 lines on a cubic surface (not included in these notes).
  • Class 46 (Thurs. Apr. 17): curves of genus at least 3; more glimpses of moduli; elliptic curves (start): line bundles of degree 0 (K=O), degree 1 (points form a group) (notes included in class 47).
  • Class 47 (Tues. Apr. 22): elliptic curves continued: line bundles of degree 2 (the j-invariant), degree 3 (Weierstrass normal form; the group law).
  • Class 48 (Thurs. Apr. 24): elliptic curves continued: Poncelet's theorem; fun counterexamples using elliptic curves.
  • Class 49 (Tues. Apr. 29): the Picard group has "dimension" g; the "moduli space of curves" as "dimension" 3g-3.
  • Class 50 (Thurs. May 1): differentials: motiation and game plan, the affine case (two of three definitions), affine versions of the relative cotangent sequence and the conormal exact sequence.
  • No class May 6 (Bea Yormark memorial service).
  • Class 51 (Thurs. May 8): differentials continued: a general definition, and many examples, including the Euler exact sequence.
  • Class 52 (Tues. May 13): differentials and varieties over algebraically closed fields: Riemann-Hurwitz formula, Bertini's theorem, the conormal exact sequence for nonsingular varieties.
  • Class 53 (Thurs. May 15): flatness, algebraic definition and easy facts, .
  • Class 54 (Tues. May 20): flat pullback commutes with higher pushforwards, symmetry of Tor, an ideal-theoretic criterion for flatness, for coherent modules over Noetherian local rings flat = free, flatness over nonsingular curves.
  • Class 55 (Thurs. May 22): dimensions behave well for flat morphisms, flatness implies constant euler characteristic and consequences thereof; degree of a line bundle over a flat family of curves is locally constant, a nonprojective proper surface.
  • Class 56 (Tues. May 27): cohomology and base change theorems, semicontinuity theorem, Grauert's theorem.
  • Classes 57 and 58 (Thurs. May 29 and Tues. June 3): smooth, etale, unramified: definitions and easier consequences, harder facts, generic smoothness in characteristic 0, Kleiman-Bertini theorem and applications, introduction to the Schubert calculus.

    Many thanks to many people, including especially Justin Walker, for improving the class notes immeasurably!

    Creative Commons License All content on this website (including course notes) is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
    Back to my home page.