Math 216: Foundations of algebraic geometry

Spring quarter: Tuesday and Thursday 9:35-10:50 in 381-T

There are several types of courses that can go under the name of "introduction to algebraic geometry": complex geometry; the theory of varieties; a non-rigorous examples-based course; algebraic geometry for number theorists (perhaps focusing on elliptic curves); and more. There is a place for each of these courses. This course will deal with schemes, and will attempt to be faster and more complete and rigorous than most, but with enough examples and calculations to help develop intuition for the machinery. Such a course is normally a "second course" in algebraic geometry, and in an ideal world, people would learn this material over many years. This is not an ideal world. To make things worse, I am experimenting with the material, and trying to see if a non-traditional presentation will make it possible to help people learn this material better, so this year's course is only a second approximation. (See here for a first approximation.)

In short, this not a course to take casually. But if you have the interest and time and energy, I will do my best to make this rewarding.

References: I hope to type up each day's notes fairly soon after the lecture (usually within a week). You should also take notes yourself, and not count on these. The posted lecture notes will be rough, so I would 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.

The notes from the class two years ago are available 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

Class 1 (Mon. Sept. 24): about this course; what is algebraic geometry; motivation and program; a bit of category theory.
Class 2 (Wed. Sept. 26): Yoneda's lemma (which we'll never use); limits and colimits; adjoints. (Revised Sept. 28 because definitions of full and faithful were swapped. Thanks Nathan! And again Oct. 1 to remove a reference to Yoneda.)
- Problem set 1 (due Fri. Oct. 5). Problem 2 patched Oct. 3.
Class 3 (Mon. Oct. 1): abelian categories, kernels, cokernels, images, exactness, homology. Introduction to sheaves, presheaves, sheaves, examples and motivation. Updated Oct. 6 to fix typo in definition of "monic".
Class 4 (Wed. Oct. 3): morphisms of (pre)sheaves; properties determined at the level of stalks; sheaves of abelian groups, and O_X modules, form abelian categories. Mild update Nov. 1.
- Problem set 2 (due Fri. Oct. 12).
Class 5 (Mon. Oct. 8): inverse image sheaf; sheaves on a base; toward schemes; the underlying set of an affine scheme. Mild update Nov. 1.
Class 6 (Wed. Oct. 10): more examples of underlying sets of affine schemes; the Zariski topology; topological definitions. Mild update Nov. 1.
- Problem set 3 (due Fri. Oct. 19).
Classes 7 and 8 (Mon. Oct. 15 and Wed. Oct. 17): decomposition into irreducible components; the function I(.); distinguished open sets; the structure sheaf, and the definition of a scheme. Updated Nov. 15.
- Problem set 4 (due Fri. Oct. 27).
Classes 9 and 10 (Mon. Oct. 22 and Wed. Oct. 24): topological properties; reducedness and integrality; affine-local properties; normality and factoriality; associated points and fuzzy pictures take 1. Updated Nov. 13 to add quasiseparated schemes.
- Problem set 5 (due Fri. Nov. 2).
Classes 11 and 12 (Mon. Oct. 29 and Wed. Oct. 31): associated points continued; introduction to morphisms of schemes; morphisms of ringed spaces; from locally ringed spaces to morphisms of schemes; some types of morphisms. Updated Nov. 8 (some of the discussion has been moved to the class 13 notes).
- Problem set 6 (due Fri. Nov. 9). Updated Nov. 9.
Class 13 (Mon. Nov. 5): some types of morphisms (quasicompact and quasiseparated, open immersion, affine, finite, closed immersion, locally closed immersion); construction related to "smallest closed subschemes" (scheme-theoretic image, scheme-theoretic closure, induced reduced subscheme, and the reduction of a scheme); more finiteness conditions on morphisms ((locally) of finite type, quasifinite, (locally) of finite presentation). Updated Nov. 13 to include quasiseparated morphisms. Small update Nov. 15.
- Problem set 7 (due Fri. Nov. 16).
Class 14 (Wed. Nov. 7): Projective varieties, the Proj construction, examples, maps of graded rings and maps of projective schemes, important exercises. Updated Nov. 15.
Classes 15 and 16 (Mon. Nov. 12 and Wed. Nov. 14): fibered products of schemes: fibered products exist; computing them in practice; pulling back families and fibers of morphisms; properties preserved by base change; products of projective schemes (the Segre embedding); introduction to separated morphisms. Small update Dec. 11.
- Here is a picture of a quadric with its two rulings of lines (found on Rick Litherland's homepage). Here is another picture, showing 3 lines in the same ruling (part of a larger story that Frank Sottile tells on his webpage). Here is a string model at the University of Arizona.
- Problem set 8 (due Fri. Nov. 30). Updated Nov. 17 (one problem removed).
Class 17 (Mon. Nov. 26): quasiseparatedness, separatedness, rational maps, dominant and birational, examples of rational maps. (Some of this was done in class 18.)
Class 18 (Wed. Nov. 28): proper morphisms.
- Problem set 9.
Class 19 (Mon. Dec. 4): dimension and codimension, integral extensions and the Going-up theorem, Nakayama's lemma.
Class 20 (Wed. Dec. 6): dimension and transcendence degree; images of morphisms and Chevalley's theorem; fun in codimension 1 (Krull, Algebraic Hartogs', ...). Summary of quarter (not in notes).
- Problem set 10.

Winter quarter quadric surface

Class 21 (Fri. Jan. 11): nonsingularity ("smoothness" of Noetherian schemes); the Zariski tangent space; the local dimension is at most the dimension of the tangent space.
Class 22 (Mon. Jan. 14): discrete valuation rings: dimension 1 Noetherian regular local rings.
Class 23 (Wed. Jan. 16): valuative criteria for separatedness and properness.
Class 24 (Fri. Jan. 18): vector bundles and locally free sheaves; useful constructions for locally free sheaves; the distinguished affine base (toward quasicoherent sheaves).
No class Mon. Jan. 21 (Martin Luther King Day)
Class 25 (Wed. Jan. 23): quasicoherent sheaves (two or three definitions); quasicoherent sheaves form an abelian category; module-like constructions for quasicoherent sheaves.
Class 26 (Fri. Jan. 25): more module-like constructions; finiteness conditions on quasicoherent sheaves (finite type and coherent sheaves).
Class 27 (Mon. Jan. 28): quasicoherent sheaves of ideals and closed subschemes. Line bundles: some line bundles on projective space; line bundles from effective Cartier divisors. Corrected Feb. 19 (thanks Nathan!).
Classes 28 and 29 (Wed. Jan. 30 and Fri. Feb. 1): invertible sheaves and Weil divisors.
Class 30 (Mon. Feb. 4): the quasicoherent sheaf corresponding to a graded module; invertible sheaves (line bundles) on projective A-schemes; generated by global sections, and Serre's theorem; every quasicoherent sheaf on a projective A-scheme arises from a garded module.
- Problem set 11 (due Fri. Feb. 15).
Class 31 (Wed. Feb. 6): Pullbacks and pushforwards of quasicoherent sheaves.
Class 32 (Fri. Feb. 8): invertible sheaves and maps to projective schemes.
- Problem set 12 (due Wed. Feb. 20).
Classes 33 and 34 (Mon. Feb. 18 and Wed. Feb. 20): relative Spec and Proj, and projective morphisms.
Classes 35 and 36 (Fri. Feb. 22 and Mon. Feb. 25): Cech cohomology of quasicoherent sheaves and its important properties; cohomology of line bundles on projective space.
Class 37 (Wed. Feb. 27): applications of cohomology: Hilbert polynomials, Hilbert functions, degree, arithmetic genus.
Class 38 (Fri. Feb. 29): higher direct image sheaves, properties and fun applications.
no class Mar. 3-7
Class 39 (Mon. Mar. 10): from Tor to derived functor cohomology in general.
FARS seminar ("class 39.5"): Spectral sequences: friend or foe? (Steven Sam at Berkeley warns: "I'm pretty sure the im should be changed to coker in (4) on page 6 (and other parts of that page)." Thanks, I'll fix it when I get a chance!)
Class 40 (Wed. Mar. 12): derived functor cohomology, and the Leray spectral sequence. Don't worry, notes are still coming!
- Problem set 13 (due Sun. Mar. 23).

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.

Possible future topics (in some possible order):

differentials

base-point-free, ample, very ample line bundles

flatness

smooth, etale, unramified

blowing up

Serre duality

cohomology and base change

formal functions

Zariski's main theorem

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