## 2017-18 course

Ravi Vakil (vakil@math.stanford.edu)

Office:  383-Q

General information

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. We do not live in an ideal world.

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 wait for a later 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.

Email list: I have a class email list that I’ll use occasionally, to let you know about things like changed class times and problem set corrections. If you are on the list and want to be off it, or vice versa, please let me know.

Time and place (216B, winter 2018): Mondays, Wednesdays, and Fridays 9-10:20, in Hewlett 103 starting Friday Feb. 9.  (Earlier:  420-245.)

Office hours:  Winter quarter office hours Mondays 12:30-2:30, Wednesday 2:30-3:30.

References:

• The notes based on earlier versions of this class, and on many useful comments from people around the world, are available here. They will be updated throughout the year. I would very much like comments, suggestions, and corrections.
• Encyclopaedic background: Johan de Jong’s stacks project has in my mind become essentially the universal reference for algebraic geometry, and becoming more so with every edit. It is free, comprehensive, well-written, philosophically well thought through, searchable, and (important for a reference) modular (when you look something up, you can read “around it” to understand the proof).
• Other more “text-like” references: It may be useful having Hartshorne’s Algebraic Geometry, and possibly Mumford’s Red Book of Varieties and Schemes (the first edition is better, as Springer introduced errors into the second edition by retyping it). Mumford’s second edition is available online (with a Stanford account) from Springer. Two other excellent sources: Qing Liu’s Algebraic Geometry and Algebraic Curves, and Gortz and Wedhorn’s Algebraic Geometry I.
• 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. But no need to read them.
• 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.

Lecture outlines

Fall quarter (216A)

Monday, September 25: 1.1-1.3 (in the notes). Introduction to the class. Some category theory abstracting what you know well. Categories, functors, universal properties. Posets=partially ordered sets. Localization, tensor products.

Wednesday, September 27: 1.3-1.4. Fibered products, products, coproducts, monomorphisms, Yoneda’s lemma. Limits and colimits of diagrams in a category. All limits in the category of sets exist. Filtered colimits of sets exist. All colimits of A-modules exist.

Friday, September 29: 1.4-1.5. Adjoints. Key examples (1.5.D and 1.5.D). Groupification of abelian semigroups.

Monday, October 2: 2.1-2.2. Presheaves and sheaves. Germs of a sheaf at a point. The stalk of a sheaf at a point.

Wednesday, October 4: 2.2-2.3. Examples of sheaves: restriction of sheaves, (locally) constant sheaves, morphisms to X sections of a map, pushdforward. Morphisms of (pre)sheaves.

Friday, October 6: 1.6, 2.3-2.4, beginning of 1.6.  Properties determined at the level of stalks.  Compatible germs, and sheafification.

Monday, October 9:  rest of 1.6.  Abelian categories.

Wednesday, October 11:  2.5, 2.7, 2.6.  Brief introduction to the inverse image sheaf.

Friday, October 13:  2.6, beginning of 3.1.  The inverse image (different points of view).  Beginning to think about schemes.  The underlying set of an affine scheme.  (Examples next day!)  Problem set 1 due.

Monday, October 16.  3.1-3.2.  Examples of the underlying set of the spectra of various rings.   Statement of the Nullstellensatz (and Zariski’s Lemma).  Quotients and localizations induce subsets of Spec’s.

Wednesday, October 18:  3.2-3.5.  Maps of rings induce maps of Spec’s as sets.  Functions are not determined by their values at points, and the reason is nilpotents.  The Zariski topology (on Spec A), and Vanishing set V(.).  A base for the Zariski topology on Spec A:  the Doesn’t-vanish sets D(f).

Friday, October 20 (taught by Sean Howe):  3.5-3.6.  Problem set 2 due.  Functions on Spec A.  Picturing ${\mathbb{A}}^2_{\mathbb{C}}$:   closed points, generic points, open sets, closed sets, principal open sets.  The topology on Spec A.  Connectedness, irreducibility.

Monday, October 23 (taught by Sean Howe):  3.6-3.7.    Noetherian topological spaces, decomposition into irreducible components, Noetherian induction, Noetherian rings and modules, Quasicompactness, closed points, specialization, generisation, generic point of a closed set.  $I(S)$, the bijection between radical ideals and closed subsets, prime ideals and irreducible closed subsets, minimal prime ideals and irreducible components.

Wednesday, October 25 (taught by Brian Conrad):  4.1-4.2.  The structure sheaf on the distinguished base of Spec A.  The structure sheaf.

Friday, October 27 (taught by Brian Conrad):  4.3.  Isomorphism of ringed spaces, affine scheme, scheme, functions on open subsets of a scheme, Zariski topology on scheme.  Problem set 3 due.

Monday, October 30:  4.3-4.4.  Locally ringed space, residue field at a point $\kappa(p)$, the line with the doubled origin, the projective line.

Wednesday, November 1:  4.4-4.5.  Projective space.  Graded rings, and the Proj construction.  Projective A-schemes.

Friday, November 3:  5.1-5.2.  Topological properties:  connected, connected component, (ir)reducible, irreducible component, quasicompact, generization/specialization, generic point, Noetherian topological space, closed point.  Quasiseparated, reduced, integral scheme; function field.   Introduction to the Affine Communication Lemma.

Monday, November 6:  5.3.   Affine Communication Lemma.  A-schemes, (locally) Noetherian schemes, finite type A-schemes.  Degree of a closed point of a finite type k-scheme.  (Quasi)affine and (quasi)projective varieties.

Wednesday, November 8:  5.4, 6.1, 6.2.  Normality and factoriality.   Various motivations for what morphisms of “geometric spaces” should be.  Morphisms of ringed spaces.   Morphisms of locally ringed spaces, and schemes.

Friday, November 10:  6.3-6.4.     The category of A-schemes.   Many examples of discussing morphisms without excess cutting-into-affines.  Some maps to projective space.  Maps of graded rings, and maps of projective schemes.  Problem set 4 due.

Monday, November 13:  6.5.  Rational maps (from reduced schemes):  rational maps, dominant, birational rational maps.  For irreducible affine varieties, dominant rational maps are “the same as” inclusions of function fields in the opposite direction.

Wednesday, November 15 (taught by Pablo Solis):  7.1-7.2.  Metrics for being “reasonable”: local on target, stable under composition, stable under base change.  E.g. open embeddings.  Affine morphisms, integral homomorphisms, and the adjugate matrix trick.

Friday, November 17 (taught by Pablo Solis):  7.2-7.3.   First glimpses why base change is important.  Integral ring maps, Nakayama.  Finite morphisms.  (Ravi says:  skip the finite presentation discussion, as it is mangled.)  Problem set 5 due.

(Monday, November 20 – Friday, November 24:  no class, Thanksgiving break.)

Monday, November 27:  7.1-7.3 continued.   Including, proof of the QCQS lemma.

Wednesday, November 29 (taught by Pablo Solis) and Friday December 1:  7.4.  Chevalley’s Theorem and elimination of quantifiers.  Problem set 6 due.

Monday, December 4 and Wednesday December 6.  Associated points (or associated primes, or associated irreducible closed subsets), and proof of key facts about them.   Length. (Not the argument currently in section 5.5.)

Friday, December 8:  7.4.  Proof of the Fundamental Theorem of Elimination Theory using linear algebra.  Proof of Chevalley’s Theorem using the Fundamental Theorem of Elimination Theory (not the argument currently in the notes, but may be there before long).  Looking back over what we’ve done in the course.

Winter quarter (216B)

Monday, January 8:  8.1 and 8.2.  Closed embeddings, with many examples, including in projective space.

Friday, January 12: 8.2-8.3. More projective geometry. The Veronese embedding. Rulings on the quadric surface. The (closed sub)scheme-theoretic image (reasonable only when the source is reduced or the morphism is quasicompact — a bit complicated!). The (reduced) subscheme structure on a closed subset.

Wednesday, January 17: 9.1-9.3. Fibered products exist in the category of schemes. Doing business with fibered products in practice. Fibers of morphisms, and more generally, pulling back families.

Monday January 22: 9.4, 9.6, 9.7. Properties preserved by base change. “Fixing” properties not preserved by base change. The Segre embedding. Normalization.

Wednesday January 24: 10.1 Separatedness. Classes of morphisms defined in terms of properties of the diagonal. Quasiseparatedness.

Friday January 26: 10.2-10.3. “The locus where two morphisms from X to Y agree”, the Reduced-to-Separated Theorem, and proper morphisms.

Monday January 29: 11.1-11.3. Dimension, its relation to transcendence degree. Krull’s Principal Ideal Theorem, and initial consequences. (Caution: my presentation of 11.3 differs from the current version of the notes.)

Wednesday January 31: 11.3-11.4. More on Krull’s Theorem. Krull’s Height Theorem. General comments on the behavior of dimensions of fibers of reasonable morphisms.

Friday February 2: 11.4, 12.1. Uppersemicontinuity of fiber dimension. The (Zariski) (co)tangent space.

Monday February 5: 12.1, the regularity part of 12.2 and 12.3, 12.5, 12.8. Regularity and nonsingularity.

Wednesday February 7:  12.2, 12.3, 12.6, 12.7, 13.1.  Smoothness over a field.  Regularity vs. smoothness.  Smooth and etale morphisms.  Valuative criteria for separatedness and properness.   Initial thoughts on vector bundles.

Friday February 9: 13.1-13.5. Vector bundles vs. locally free sheaves. Definition of quasicoherent sheaves. The distinguished affine base, and how to think of them as “a module for every ring”. Quasicoherent sheaves form an abelian category.

Monday February 12: 13.5-13.7. From constructions and facts about modules to constructions and facts about quasicoherent sheaves.

Wednesday February 14: 13.7-14.2.4. Torsion-free and torsion sheaves (especially on regular curves). Line bundles on projective space. From line bundles with rational sections to Weil divisors, and the injectivity of the “div” map for Noetherian normal schemes. The quasicoherent sheaf O(D).

Friday February 16: 14.2-15.1. More on the divisor – line bundle correspondence. Computing Picard groups in actual examples.

Monday February 19:  no class (Presidents’ Day).

Wednesday February 21: 15.2-15.3, 16.4-16.5. The line bundles O(m) on projective schemes. Globally generated quasicoherent sheaves, and base-point-free line bundles. Line bundles and maps to projective space.

Friday February 23: 16.5, 16.1-16.3. The Curve-to-projective extension theorem. Pulling back quasicoherent sheaves more generally. (And pushing forward quasicoherent sheaves by qcqs morphisms.)

Monday February 26: 16.6. (Very) ample line bundles, and characterizations of ampleness.

Wednesday February 28: 17.1-17.3. Relative Spec and Proj, and projective morphisms.

Friday March 2:  no class (I will be at MSRI).

Monday March 5: no class (I’ll be at a natural sciences town hall meeting).

Wednesday March 7.

Friday March 9.

Monday March 12.

Wednesday March 14.

Friday March 16.

Homework

You can wave your hands all you want, but it still won’t make you fly.  — Mark Kisin

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. There will be a problem set most weeks. Your grade will depend on the problem sets.

Collaboration is encouraged, but you should give credit for ideas that are not your own.  (You will not penalized for that.)  Do not do any problems that you already know how to do or that you would find easy.  If you have seen some of the material before, and thus don’t have many problems to choose from, please let me know, and we can work out an alternative arrangement.

216A:

• Problem set 1 (due Friday October 13):  Do 10 of the following problems from the June 4, 2017 version of the notes:
• 1.2.B, 1.3.A, 1.3.D, 1.3.E, 1.3.F, 1.3.I, 1.3.N, 1.3.O, 1.3.P, 1.3.R, 1.3.V, 1.4.B, 1.4.E, 1.4.F, 1.5.E, 1.5.F, 1.5.G, 1.5.H, 2.2.A, 2.2.I, 2.3.A, 2.3.B
• required to be done by the end of problem set 5:  1.3.C, 1.3.G (if you haven’t seen tensor products before), 1.3.H, 1.3.Q, 1.3.S, 1.3.X, 1.3.Y, 1.4.A, 1.4.C, 1.5.C, 2.2.G, 2.2.J, 2.3.C,
• only if you have the background, or want to learn about something:  1.2.D, 1.3.Z.
• Problem set 2 (due Friday October 20):  Do 10 of the following problems from the June 4, 2017 version of the notes:
• 2.3.I, 2.3.J, 2.4.E, 2.4.F, 2.4.I, 2.4.J, 2.4.K, 2.4.L, 2.4.M, 2.4.N, 2.4.P, 2.5.A, 2.5.B, 2.5.C, 2.5.D, 2.5.E, 2.5.F, 2.5.G, 2.5.H, 2.6.C, 2.6.D, 2.6.E, 2.6.F, 2.7.D.
• any of the earlier required problems not yet done.
• required to be done by the end of problem set 5:  2.3.E, 2.4.A, 2.4.C, 2.4.D, 2.6.B, 2.7.A, 2.7.B, 2.7.C.
• only if you have the background, or want to learn about something:  3.1.A, 3.1.B.
• Problem set 3 (due Friday October 30):  Do 10 of the following problems from the June 4, 2017 version of the notes:
• any of the earlier required problems not yet done (note that I’ve changed the instructions for them, so you just need to finish them by the end of problem set 5)
• 3.2.A, 3.2.C, 3.2.D, 3.2.E, 3.2.G, 3.2.H, 3.2.I, 3.2.L, 3.2.M, 3.2.N, 3.2.Q, 3.2.R, 3.2.S, 3.2.T, 3.4.C, 3.4.D, 3.4.E, 3.4.F, 3.4.H, 3.4.K.
• required to be done if you haven’t seen them before:  3.2.J, 3.2.K, 3.2.J, 3.2.K.
• required to be done by the end of problem set 5:  3.2.O or 3.2.P;  3.4.I, 3.4.J.
• Problem set 4 (due Friday November 10):  Read all of the problems, and be familiar with their contents.  Do 10 of the following problems from the June 4, 2017 version of the notes:
• any of the earlier required problems not yet done
• 3.5.B, 3.5.C, 3.5.E, 3.6.B, 3.6.E, 3.6.G, 3.6.I, 3.6.J, 3.6,K, 3.6.O, 3.6.R (or 3.6.U), 3.6.S, 3.6.T, 3.7.D, 3.7.E, 3.7.F, 3.7.G, 4.1.A, 4.1.B, 4.1.D, 4.3.A, 4.3.B, 4.3.F (required), 4.3.G (required), 4.4.A, 4.4.D, 4.4.F, 4.5.A, 4.5.C, 4.5.D, 4.5.E, 4.5.I, 4.5.K
• Problem set 5 (due Friday November 17):  Read all of the problems, and be familiar with their contents.  Do 10 of the following problems from the June 4, 2017 version of the notes:
• any of the earlier required problems not yet done
• 4.5.O, 5.1.B, 5.1.E, 5.1.F, 5.1.I, 5.2.A, 5.2.C, 5.2.E, 5.2.F, 5.2.H, 5.2.I, 5.3.A, 5.3.B, 5.3.C, 5.3.E, 5.4.I, 5.4.J, 6.2.A, 6.2.C, 6.2.D, 6.3.B, 6.3.C, 6.3.E, 6.3.J, 6.3.I, 6.3.M, 6.3.N, 6.4.D, 6.4.E, 6.4.G
• challenge problems:  5.1.H, 5.4.M
• required to be done if you haven’t seen them before:  5.4.A, 5.4.F
• required to be done by the end of Problem set 6:  5.4.H, 6.3.F, 6.4.A
• Problem set 6 (due Friday December 1):  Read all of the problems, and be familiar with their contents.  Do 10 of the following problems from the November 18, 2017 version of the notes (which are probably the same):
• any of the earlier required problems not yet done
• 7.2.A, 7.2.B, 7.2.C, 7.2.I, 7.3.A, 7.3.B, 7.3.C, 7.3.D, 7.3.F, 7.3.G, 7.3.H, 7.3.J, 7.3.K, 7.3.L, 7.3.M.
• required to be done:  7.1.B.
• required to be done if you haven’t seen them before:  7.2.F, 7.2.G, 7.2.H.

216B:

• Problem set 1 (due Friday January 26): Read all of the problems, and be familiar with their contents.  Do 15 of the following problems from the November 18, 2017 version of the notes:
• 8.1.D, 8.1.F,  8.1.J, 8.1.K, 8.1.L, 8.2.A, 8.2.B, 8.2.C, 8.2.D, 8.2.E, 8.2.F, 8.2.G, 8.2.H, 8.2.J, 8.2.N, 8.2.O, 8.2.P, 8.2.Q, 8.3.A, 8.3.C, 8.3.D, 8.3.E, 8.3.G, 8.4.B.
• required to be done:  8.1.G, 8.1.H, 8.1.M.
• 9.1.A, 9.2.A, 9.2.B, 9.2.E, 9.2.G, 9.2.H, 9.2.I, 9.2.J, 9.2.K, 9.3.D or 9.3.E, 9.3.G.
• Problem set 2 (due Friday February 2): Read all of the problems, and be familiar with their contents.  Do 10 of the following problems from the November 18, 2017 version of the notes. Try to do some from each section.
• 9.4.E, 9.5.C, 9.5.D, 9.5.E, 9.6.B, 9.7.D, 9.7.H, 9.7.I, 9.7.J, 9.7.K, 9.7.L, 9.7.P, 10.1.A, 10.1.D, 10.1.E, 10.1.F, 10.1.J, 10.1.L, 10.1.K, 10.1.M, 10.2.A, 10.2.B, 10.2.C, 10.2.D, 10.2.E, 10.2.H, 10.3.B, 11.1.C, 11.1.D, 11.1.E, 11.1.F, 11.1.G, 11.1.H, 11.1.I, 11.1.J, 11.1.K.
• required to be done: 9.2.F, 10.2.G
• Problem set 3 (due Friday February 9): Read all of the problems, and be familiar with their contents.  Do 8 of the following problems from the November 18, 2017 version of the notes. Try to do some from each section.
• (don’t do any you did last week) 11.1.C, 11.1.D, 11.1.E, 11.1.F, 11.1.G, 11.1.H, 11.1.I, 11.1.J, 11.1.K, 11.2.D, 11.2.H, 11.2.I, 11.2.J, 11.3.C, 11.3.E, 11.4.A, 11.4.C, 11.4.F, 11.4.G, 11.4.H.
• strongly recommended: 11.1.B, 11.2.A, 11.3.C, 11.3.F (if you haven’t seen this before).
• Problem set 4 (due Friday February 16):  Read all of the problems, and be familiar with their contents.  Do 10 of the following problems from the November 18, 2017 version of the notes. Try to do some from each section.
• (don’t do any you did last week) 11.4.A, 11.4.C, 11.4.F, 11.4.G, 11.4.H, 12.1.A, 12.1.C, 12.1.D, 12.1.E, 12.1.F, 12.1.H, 12.2.A, 12.2.C (except remove the finite type hypotheses, replacing them with as little as you can manage), 12.2.F, 12.2.G, 12.2.N, 12.2.O, 12.3.B, 12.3.C, 12.3.D, 12.3.E, 12.3.F, 12.3.G, 12.3.H, 12.3.I, 12.3.L, 12.3.M, 12.3.N, 12.5.G, 12.5.I, 12.6.A, 12.6.B, 12.6.C, 12.6.D, 12.7.B, 12.7.C, 13.1.A
• particularly recommended: 12.1.B, 12.1.I, 12.5.H
• if you haven’t seen Discrete Valuation Rings before (or want more practice): 12.5.A, 12.5.B
• if you haven’t seen the Artin-Rees lemma before, then these are highly recommended: 12.9.A, 12.9.B
• Problem set 5 (due Friday February 23):  Read all of the problems, and be familiar with their contents.  Do 10 of the following problems from the November 18, 2017 version of the notes. Try to do some from each section.
• 13.1.B, 13.1.C, 13.1.E, 13.1.F, 13.1.K, 13.2.A, 13.2.B, 13.3.A, 13.3.B,
13.3.C, 13.3.D (even though it was discussed in class), 13.3.G, 13.5.A, 13.5.B, 13.5.C, 13.5.D, either 13.5.E or 13.5.F, 13.5.G, 13.5.H, 13.6.C, 13.6.D,
13.7.A, 13.7.G, 14.1.D, 14.2.C.
• particularly recommended: 13.2.C, 13.3.F, 13.3.H, 13.4.A, 13.6.A, 13.7.E, 13.7.F, 13.7.K, 14.1.C, 14.2.E.
• for those with arithmetic background: 13.1.L, 13.1.M
• Problem set 6 (due Friday March 2, although I won’t pick it up until Monday March 5):  Read all of the problems, and be familiar with their contents.  Do 10 of the following problems from the November 18, 2017 version of the notes. Try to do some from each section.
• 14.2.J, 14.2.R, 14.2.U, 15.1.A, 15.1.C, 16.3.C, 16.3.D. 16.3.H,
16.5.B
• particularly recommended: 14.2.I, 14.2.T, 15.2.B, 16.3.A, 16.3.B, 16.3.E (do three of the ten parts, but they should be interesting ones)
• for those with arithmetic background: 14.2.S
• Problem set 7 (due Friday March 9):  Read all of the problems, and be familiar with their contents.  Do 10 of the following problems from the November 18, 2017 version of the notes. Try to do some from each section.
• 16.6.C, 16.6.D, 16.6.H17.1.A, 17.1.D, 17.1.E, 17.1.F, 17.1.G, 17.2.A, 17.2.B, 17.2.C, 17.2.D, 17.2.E, 17.3.A, 17.3.C
• particularly recommended: 16.6.G, 17.1.H, 17.3.B