Math 216: Foundations of algebraic geometry

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 an approximation. (See here for an earlier version.)

This course is for mathematicians intending to get near the boundary of current research, in algebraic geometry or a related part of mathematics. It is not intended for undergraduates or people in other fields; for that, people should take Maryam Mirzakhani's class, or else wait for the next incarnation of Math 216 (which will vary in style over the years).

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.

Office hours: Because of the nature of this class, I'd like to be as open as possible about office hours, and not have them restricted to a few hours per week. So if you would like to chat, please let me know, and I'll be most likely happy to meet on a couple of days' notice. For a canonical time: I'm happy to meet immediately after class, but this won't be great for people going to Brian Conrad's course. If people aren't talking to me much, I'll institute more official office hours.

Course assistant: Jack Hall (jhall@math, office 380-U). He will grade your problem sets, and will also have office hours Monday and Friday 11 am - noon.

References: I have some hope of periodically releasing chapters of notes, perhaps roughly once per week or a little less often. (The most recent version should be outside my office door. I will also give each new version out in class as they become available.) You should take notes yourself, and not count on these. The notes from the class two years ago are available here. It may be useful having Hartshorne's Algebraic Geometry, and possibly Mumford's Red Book of Varieties and Schemes (the original edition is better, as Springer introduced errors into the second edition by retyping it). Mumford (2nd ed) may be availble online (with a Stanford account) from Springer. 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. 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. The grader is Jack Hall. There will be a problem set most weeks. The grade will depend only on the problem sets.

• Mon. Sept. 21: introduction. Why you shouldn't take this class. What I'm trying to do in this class. What algebraic geometry is about. That's too hard, so at least what this course is about: why many notions (geometric, arithmetic, algebraic, complex-analytic, ...) can be understood in terms of "geometric spaces", and constructions related to them. Example: Mordell's Conjecture (Faltings' Theorem). A little bit of category theroy: objects, morphisms, source, target, identity, isomorphism, automorphism, examples (sets, vector spaces, A-modules, abelian groups, rings, topological spaces, partially order sets, the open sets of a topological space, subcategory.
• Wed. Sept. 23: covariant functor, forgetfunctor, contravariant functor, opposite category, full and faithful functors. Universal properties: tensor product and its properties, initial object, final object, zero object, fibered product, monomorphism, epimorphism, coproduct.
  • Problem set 1 due Friday Oct. 2.
• Fri. Sept. 25: small category, diagram indexed by an index category, limit (inverse limit, projective limit), colimit (direct limit, injective limit),adjoint functors, tensor product is adjoint to Hom for A-modules.
• Mon. Sept. 28: groupification and other adjoint constructions "projecting categories into smaller ones", additive category, homomorphism, additive functor, kernel, cokernel, monic, subobject, epi, quotient object, abelian category, image, complex, exact, (co)homology, long exact sequence, left-exact and right-exact (additive) functor, exact (additive) functor, fun facts about interactions of (co)limits, adjoints, and left/right-exactness.
• Wed. Sept. 30 ((pre)sheaves): motivation (differentiable or continuous or arbitrary functions on a real manifold), germ, stalk, presheaf, sections of a (pre)sheaf over an open set, sheaf, identity axiom, gluability axiom, restriction of a sheaf, skyscraper sheaf, pushforward (direct image) sheaf.
• Thurs. Oct. 1: Drop by my office 3:20-4:10 if you'd like to discuss complexes, homology, or other abelian category related things.
  • Problem set 2 due Friday Oct. 9.
• Fri. Oct. 2: constant presheaves, locally constant sheaves, sheaf of sections of a continuous map, direct image / pushforward sheaf, ringed spaces and O_X-modules, morphisms of (pre)sheaves, presheaves of abelian groups (etc.) form an abelian category, kernels of sheaves of abelian groups (and cokernel worries), sections are determined by germs, compatible germs, morphisms are determined by stalks.
• Mon. Oct. 5: sheafification (universal property; construction by compatible germs), subsheaves, quotient sheaves; sheaves of abelian groups and O_X-modules form abelian categories; left exactness of global section functor and pushforward; tensor products of O_X-modules; the inverse image sheaf (adjoint description and colimit construction), examples.
• Wed. Oct. 7: base of topology, sheaf on a base, sheaves on a base are same as sheaves (including morphisms). Toward schemes: set, topology, sheaf of functions, and more motivation from manifolds. Affine schemes: Spec A as a set, and a preliminary example. Functions, and values at a point.
  • Problem set 3 due Friday Oct. 16.
• Fri. Oct. 9: the affine line over various fields, Spec Z, affine n-space, the primes of C[x,y,z], Nullstellensatz (first version), maps of rings induce maps of spectra (as sets), examples (quotient rings, maps of affine complex algebraic varieties, localizations), nilpotents, functions aren't determined by their values at points, nilradical.
• Mon. Oct. 12: Zariski topology, vanishing set, radical, radical ideal, maps of rings induce continuous maps of spectra, distinguished open sets, preliminary Noetherian discusion.
  • Problem set 4 due Friday Oct. 23.
• No class Oct. 14 or 16. Instead, read to the end of the chapter 3 notes given out on Mon. Oct. 12 (and outside my office door). Words you should be comfortable with (in this setting): irreducible, closed point, quasicompact, specialization, generization, generic point, Noetherian topological space, Hilbert basis theorem, A[[x]], irreducible component, Noetherian induction, connected, conected component, the function I from subsets of Spec A to ideals of A. Jack is happy to meet on Thursday (as well as his usual office hours); e-mail him if you'd like to set up a time.
• Mon. Oct. 19: the structure sheaf of Spec A (by defining the sections over D(f) as the localization of A at those functions nonvanishing in D(f), and showing this is a sheaf on the base); the O_{Spec A}-module M-tilde, isomorphism of ringed spaces, affine scheme, scheme, isomorphism of schemes, open subscheme, affine open subset/subscheme.
• Wed. Oct. 21: stalks of the structure sheaf; schemes are local ringed spaces; examples (plane minus origin, gluing schemes, affine line with doubled origin, the projective line, projective space).
  • Problem set 5 due Friday Oct. 30.
• Fri. Oct. 23: topological properties of schemes: irreducible, irreducible component, closed point, specialization, gener(al)ization, generic point, connected, connected component, quasicompact as before; quasiseparated. Reducedness and integrality. The affine communication lemma.
• Mon. Oct. 26: (locally) Noetherian scheme, reducedness is affine-local, A-schemes (schemes over a ring A), (locally of) finite type A-schemes, normal, factorial.
• Wed. Oct. 28: associated points of schemes, and fuzzy mathematics; embedded point, rational function, domain of definition, regular at a point, , total fraction ring. Generic points are associated points; if X is reduced, then it has no embedded points; functions are determined by their germs at associated points; zero divisors are precisely those functions vanishing at an associated point of X Primary ideals, (minimal) primary decmposition (and its "uniqueness"), associated primes (of an ideal, of a ring).
  • Problem set 6 due Friday November 6.
• Fri. Oct. 30: morphisms of schemes as maps of local-ringed spaces. Complex schemes, or more generally k-schemes (where k is a field), or more generaly A-schemes (A is a ring), or more generally S-schemes (S is a scheme).
• Mon. Nov. 2: picturing schemes (generic points, nonreduced behavior). Mining an example: maps to A^1_Z correspond to global functions. Generalization 1: representable (contravariant) functors, uniqueness of the representing object up to unique isomorphism, functor of points, A-valued points of a scheme. Generalization 2: group schemes (and more generally group objects in a category). Examples: A^n, G_m, mu_p.
• Wed. Nov. 4: group schemes via functors of points. Useful types of morphisms: quasicompact, quasiseparated, open immersion, affine.
  • Problem set 7 due Friday November 13. (You can pick up the notes outside my office door after class.)
• Fri. Nov. 6: finite morphisms, closed immersions and closed subschemes, criterion for a sheaf of ideals to come from a closed subscheme.
• Mon. Nov. 9: locally principal closed subschemes (and effective Cartier divisors), closed immersions in projective space (hypersurface, degree, hyperplane, quadric, cubic, quartic, quintic, ..., line, conic curve), locally closed immersions/subschemes, scheme-theoretic image (determined affine-locally for affine morphisms or reduced source).
• Wed. Nov. 11: scheme-theoretic closure of a locally closed subscheme, induced (closed sub)scheme structure on a closd subset, reduced version (reduction) of a scheme, morphisms (locally) of finite type/presentation, quasifinite.
  • Problem set 8 due Friday November 20. (You can pick up the notes outside my office door after class.)
• Fri. Nov. 13: projective schemes, projective coordinates, homogeneous ideal, finitely generated graded ring (idiosyncratic definition), irrelevant ideal, Proj construction, projective distinguished open set, (quasi-)projective A-scheme, (quasi-)projective k-variety.
• Mon. Nov. 16: affine and projective cone of a projective A-scheme; maps of graded rings and maps of projective schemes; linear space, line, plane, n-plane, hyperplane; analogues of results for affine schemes; the nth Veronese subalgebra of a graded ring, and the Veronese embedding; classifying plane conics; rational normal curves.
• Wed. Nov. 18: examples: Veronese embedding, rulings on the quadric surface, weighted projective space. Fibered products: the building blocks (affines; open immersions); fibered products exist. In notes: Reinterpretation of the existence argument in terms of representable functors.
• Fri. Nov. 20: computing fibered products in practice; pulling back families; fibers of morphisms; every reasonable property is preserved by base change; how to fix properties not preserved by base change (geometric points, geometric fibers, geometrically connected, geometrically irreducible, geometrically reduced); products of projective A-schemes are projective A-schemes (the Segre embedding). (Expected: Fibered products continued.)
• Mon. Nov. 30 and Wed. Dec. 2. (Expected: Separated morphisms.)
• Fri. Dec. 4. (Expected: Proper morphisms.)