**Ravi Vakil (vakil@math.stanford.edu)**

**Office: 383-Q**

**spring CA: Donghai Pan pandh@stanford.edu
**

**(winter CA: Arnav Tripathy tripathy@stanford.edu, fall CA: Yang Zhou yangzhou@math.stanford.edu)**

**Notes/text: click here**

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 (winter quarter):** Most Mondays, Wednesdays, and Fridays 9:00-10:20 in 381-U (see below for more). **Warning: ***this isn’t precisely the time listed in axess!*

**Office hours:** Right after class (in general). 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.

**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. - 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 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*. - 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:**

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.**

*Problem set 1 (due Friday October 2):* Do 10 of the following problems from the September 18, 2015 version of the notes: 1.2.C (only if you have seen this material enough to want to get happy with natural transformations of functors), 1.3.B, 1.3.D, 1.3.G, 1.3.H, 1.3.K, 1.3.L, 1.3.N, 1.3.R, 1.3.S, 1.3.U, 1.3.W (mandatory if you haven’t seen this before, as I forgot to mention monomorphisms in class), 1.3.X, 1.3.Y, 1.3.Z (worth two; only for those up for it), 1.4.C, 1.4.E.

*Problem set 2 (due Friday October 9): *Do 10 of the following problems from the September 18, 2015 version of the notes: 1.5.C, 1.5.D, 1.5.E, 1.6.A, 1.6.B, 1.6.C, 1.6.D, 1.6.F, 1.6.H, 1.6.I, 1.6.K, 1.6.L. Read and understand the six-line proof of Proposition 1.6.13. 2.1.A, 2.2.I, 2.2.J, 2.3.B, 2.3.C, 2.3.D, 2.3.H, 2.3.I, 2.4.A, 2.4.B, 2.4.C, 2.4.D, 2.4.E, 2.4.K, 2.4.L, 2.4.M, 2.4.N, 2.4.O, 2.4.P, 2.5.A, 2.5.B, 2.5.D, 2.5.F, 2.5.G.

*Problem set 3 (due Friday October 16):* Do 10 of the following problems from the September 18, 2015 version of the notes. Try to do problems on all topics. 2.6.B (important), 2.6.C, 2.6.G (currently unmotivated and will remain so for a while, so do it only if you particularly feel like it), 2.7.B, 2.7.C, 2.7.D, 3.1.A (only for those with experience with manifolds), 3.2.A (important examples), 3.2.C, 3.2.E, 3.2.G, 3.2.H, 3.2.J (mandatory if you haven’t seen it before), 3.2.K (mandatory if you haven’t seen it before), 3.2.L, 3.2.M (important, even if not that hard), 3.2.N, 3.2.P, 3.2.R (if not seen before), 3.2.S (if not seen before), 3.2.T.

*Problem set 4 (due Friday October 23):* Do 10 of the following problems from the September 18, 2015 version of the notes. Try to do problems on all topics. 3.4.F, 3.4.F, 3.4.H, 3.4.K, 3.5.B, 3.5.C, 3.5.E (mandatory), 3.6.A, 3.6.B, 3.6.E, 3.6.F, 3.6.G(a), 3.6.H, 3.6.J, 3.6.K, 3.6.M, 3.6.O, 3.6.P, 3.6.Q (mandatory if you haven’t seen Noetherian rings before), 3.6.S, 3.7.A, 3.7.B, 3.7.D, 3.7.E (mandatory), 3.7.F, 4.1.A (mandatory), 4.3.A (mandatory), 4.3.B, 4.3.F, 4.3.G.

*Problem set 5 (due Friday October 30):* Do 10 of the following problems from the September 18, 2015 version of the notes. Try to do problems on all topics. 4.4.D, 4.4.E, 4.4.F, 4.5.C, 4.5.D, 4.5.E, 4.5.F, 4.5.H, 4.5.I, 4.5.J, 4.5.K, 4.5.L, 4.5.M, 4.5.N, 4.5.O, 4.5.P, 4.5.Q, 5.1.A, 5.1.B, 5.1.C, 5.1.D, 5.1.E, 5.1.F, 5.1.G, 5.1.H, 5.1.I, 5.2.A, 5.2.B, 5.2.G, 5.2.I, 5.3.A, 5.3.B, 5.3.C, 5.3.D, 5.3.E, 5.3.F, 5.3.G, 5.3.J.

*Problem set 6 (due Friday November 6 (not 7, as originally stated)): * Do 10 of the following problems from the September 18, 2015 version of the notes. Try to do problems on all topics. 5.4.C (mandatory), 5.4.E, 5.4.F, 5.4.H (do this if you at all can), 5.4.I (you may assume 5.4.H), 5.4.J, 5.4.K, 5.4.L, 5.4.M, 5.4.N, 5.5.C, 5.5.D 5.5.E, 5.5.G, 5.5.H, 5.5.I, 5.5.J (mandatory), 5.5.K, 5.5.L, 5.5.M, 5.5.N (but you can prove this in a different way if you want — just alert Yang that you are doing so), 5.5.O, 5.5.P (mandatory), 5.5.R, 6.2.A, 6.2.D (mandatory), 6.2.E (answering Joj’s question), 6.3.B, 6.3.C, 6.3.E, 6.3.F (mandatory), 6.3.G, 6.3.H, 6.3.J, 6.3.M.

*Problem set 7 (due Friday November 13):* Do 10 of the following problems from the September 18, 2015 version of the notes. Try to do problems on all topics. 6.4.A (mandatory), 6.4.B, 6.4.D, 6.4.E, 6.4.F, 6.4.G, 6.5.A, 6.5.C, 6.5.E, 6.5.F, 6.5.G, 6.5.H, 6.5.I, 6.5.J, 6.5.L, 7.1.A, 7.1.B, 7.2.A, 7.2.C, 7.2.D, 7.2.F, 7.2.H (mandatory), 7.3.B, 7.3.C (mandatory), 7.3.G, 7.3.H (mandatory), 7.3.I, 7.3.K (mandatory).

*Problem set 8 (due Friday November 20):* Do 10 of the following problems from the September 18, 2015 version of the notes. Try to do problems on all topics. 7.3.O, 7.3.P, 7.P.Q, 7.3.R, 7.4.A, 7.4.C, 7.4.D, 7.4.F, 7.4.G, 7.4.H, 7.4.I, 7.4.J, 7.4.K, 7.4.L, 7.4.M, 7.4.O, 7.4.P, 7.4.Q (do if possible), 8.1.A, 8.1.B, 8.1.C, 8.1.D, 8.1.E8.1.G, 8.1.H (do if possible), 8.1.I (mandatory), 8.1.M, 8.2.C, 8.2.D, 8.3.E, 8.2.F, 8.2.G, 8.2.I, 8.2.K, 8.2.L, 8.2.M (worth two), 8.2.P.

*Problem set 9 (due Wednesday December 2): * Do 10 of the following problems from the September 18, 2015 version of the notes. Try to do problems on all topics. 8.3.A, 8.3.C, 8.3.E, 8.3.F, 8.3.G, 8.4.B, 8.4.E, 8.4.G (mandatory), 9.1.D-9.1I (only if you want to learn about representable functors), 9.2.A, 9.2.B, 9.2.C, 9.2.D, 92.E, 9.2.F, 9.2.G, 9.2.I, 9.2.K, 9.3.A, 9.3.B (mandatory), 9.3.C, 9.3.D, 9.3.E.

**Winter problem set instructions: Problem sets will be due on Fridays at 3 pm, either (in pdf form) by email to Arnav Tripathy (tripathy@stanford.edu), or (in hard copy) in Arnav’s mailbox, on the first floor of the department.**

*Problem set 10 (due Friday January 22): * Do 6 of the following problems from the December 29, 2015 version of the notes. Try to do problems on all topics. Here are some problems, as of Monday January 11. 10.2.D, 10.2.G, 11.1.B (mandatory), 11.1.C, 11.1.D, 11.1.E, 11.1.G, 11.2.E, 11.2.G, 11.2.I, 11.2.J (worth three), 11.3.C(a)(b), 11.3.C(c)(d), 11.3.D, 11.3.E, 11.3.F, 11.3.I (mandatory), 11.4.C, 11.4.D, 12.1.A, 12.1.B (mandatory), 12.1.C, 12.1.D, 12.1.E, 12.1.F, 12.1.G, 12.1.H, 12.1.I, 12.1.J,

*Problem set 11 (due Friday January 29):* Do 10 of the following problems from the December 29, 2015 version of the notes. Try to do problems on all topics. 12.2.D, 12.2.E, 12.2.I, 12.2.L (do if you can!), 12.2.N, 12.2.O (these last two for those with particular interest in non-algebraically closed fields), 12.3.B, 12.3.C, 12.3.F, 12.3.H, 12.3.I, 12.3.K, 12.3.L, 12.3.M, 12.3.N, 12.6.D, 12.6.E. You can also do problems from the previous set that you have not had a chance to do.

*Problem set 12 (due Friday Feburary 5):* Do 10 of the following prolems from the December 29, 2015 version of the notes. Try to do problems on all topics. 13.1.A, 13.1.B, 13.1.C, 13.1.D, 13.1.E, 13.1.F, 13.1.H, 13.1.K, 13.1.L, 13.1.M, 13.2.A, 13.3.A, 13.3.E (mandatory), 13.3.F, 13.3.G, 13.5.A, 13.5.B, 13.5.F, 13.5.G, 13.7.A, 13.7.B, 13.7.C, 13.7.D, 13.7.E (mandatory), 13.7.F (mandatory), 13.7.G, 13.7.I. If you are particularly interested, feel free to do the problems from 13.8.

*Problem set 13 (due Friday February 12): * Do 10 of the following problems from the December 29, 2015 version of the notes. Try to do problems on all topics. 14.1.C, 14.2.C, 14.2.E (mandatory), 14.2.H, 14.2.I, 14.2.J, 14.2.K, 14.2.N, 14.2.O, 14.2.P, 14.2.Q, 14.2.S, 14.2.T, 15.1.A, 15.1.B, 15.1.C 15.1.D, 15.1.E, 15.2.B, 15.3.B, 15.3.C, 15.3.F (mandatory), 15.4.C, 16.2.A, 16.3.A (mandatory), 16.3.C (mandatory), 16.3.D, 16.3.E (each three parts counts as one problem), 16.3.G (better hint: just use adjointness, and prove this for O-modules), 16.3.H.

*Problem set 14 (due Friday February 19):* Do 10 of the following problems from the December 29, 2015 version of the notes. Try to do problems on all topics. 16.4.C, 16.4.D, 16.4.F, 16.4.G, 16.4.H (encouraged, but not mandatory) 16.4.I (I like this!), 16.4.J, 16.4.O (if you want to read the section on a construction of a proper nonprojective scheme), 16.5.B, 16.6.A (mandatory), 16.6.D, 16.6.G (mandatory), 16.6.H, 17.1.A, 17.1.C, 17.1.D, 17.1.E, 17.1.F, 17.1.G, 17.1.H, 17.2.A, 17.2.B, 17.2.C (mandatory), 17.2.D, 17.2.E, 17.2.F, 17.2.G, 17.2.H, 17.2.I.

*Problem set 15 (due Friday February 26):* Do 10 of the following problems from the most recent version of the notes. Try to do problems on all topics. 17.2.D, 17.2.F, 17.2.G, 17.2.H, 17.2.I, 17.3.A, 17.3.B (mandatory), 17.3.C, 17.3.D, 17.4.A, 17.4.C, 17.4.D (if possible), 17.4.E, 18.1.A, 18.1.B, 18.1.D. One last problem: When you solve 17.3.C (finite morphisms are projective), you will show that finite morphisms are projective by using a specific “Proj”. Show that O(1) for this Proj is actually O.

*Problem set 16 (due Friday March 11).* Do 10 of the following problems from the most recent version of the notes. Try to do problems on all topics. 18.2.A, 18.2.C, 18.2.D, 18.2.F, 18.2.H, 18.2.I, 18.3.A, 18.4.A, 18.4.B (mandatory, unless you already understand this well), 18.4.C (mandatory), 18.4.E, 18.4.F, 18.4.G (for those with complex-analytic background), 18.4.I, 18.4.J, 18.4.K, 18.4.L, 18.4.M, 18.4.M, 18.5.B, 18.5.C, 18.6.B, 18.6.C, 18.6.F (mandatory), 18.6.J, 18.6.K, 18.6.L, 18.6.M, 18.6.M, 18.6.P, 18.6.Q, 18.6.R, 18.6.S, 18.6.T, 18.6.U, 18.8.B, 18.8.E, 18.8.F, 19.1.B, 19.1.C, 19.2.B, 19.4.A, 19.4.D.

**Spring problem set instructions: Problem sets will be due on Fridays at 3 pm, either (in pdf form) by email to Donghai (pandh@math.stanford.edu), or (in hard copy) to me in class.
**

*Problem set 17 (due Friday April 8).* Do 10 of the following problems from the most recent version of the notes. Try to do problems on all topics, but don’t do problems you have seen before. 19.5.A, 19.6.B, 19.7.A, 19.7.D, 19.8.A, 19.8.D, 19.8.E, 19.9.B, 19.9.C, 19.9.D, 19.9.E, 19.10.B, 19.10.E, 19.10.F, 19.10.G, 23.1.A, 23.1.B, 23.1.C, 23.1.D, 23.1.E, 23.2.A, 23.2.B, 23.2.D, 23.2.E, 23.2.F, 23.3.B, 23.3.C, 23.3.D, 23.3.E, 23.3.F, 23.3.G (worth two if you then read and understand the proof of the Grothendieck spectral sequence in 23.3.8), 23.4.B, 23.4.D, 23.4.E, 23.4.F, 23.4.G.

*Problem set 18 (due Friday April 15):* Do 10 of the following problems from the most recent version of the notes. Try to do problems on all topics, but don’t do problems you have seen before. 23.5.C, 23.5.D, 21.2.C, 21.2.D, 21.2.E, 21.2.F, 21.2.H, 21.2.J, 21.2.K (recommended), 21.2.L (recommended), 21.2.M (recommended), 21.2.N, 21.2.O, 21.2.P, 21.2.Q, 21.2.R, 21.2.S, 21.2.T, 21.2.U, 21.3.A, 21.3.D, 21.3.F.

*Problem set 19 (due Friday May 6):* Do 10 of the following problems from the most recent version of the notes. Try to do problems on all topics, but don’t do problems you have seen before. 21.3.G, 21.4.A (mandatory), 21.4.B, 21.4.C (but don’t use Serre duality!), 21.4.D (but ignore the hint for (c) if it isn’t helpful), 214.E, 21.4.G, 21.5.A (mandatory), 21.5.B, 21.5.C, 21.5.D, 21.5.E, 21.5.F, 21.5.G, 21.5.H, 21.5.I, 21.5.K, 21.5.N, 21.5.O, 21.6.B, 21.6.D, 21.6.F, 21.6.G, 21.6.I, 21.7.B, 21.7.C (the field here is assumed to be algebraically closed), 21.7.D, 21.7.I, 21.7.K, 21.7.L, 21.7.M, 24.2.J, 24.2.K, 24.2.M, 24.2.N, 24.3.A, 24.3.B, 24.3.F, 24.3.G, 24.4.B, 24.4.C, 24.4.D, 24.4.H, 24.4.I, 24.4.J, 24.5.A, 24.5.B, 24.5.C, 24.5.D, 24.5.E, 24.5.G, 24.5.I (mandatory), 24.6.B (if you want practice using the Artin-Rees Lemma), 24.6.C (to see if you understand the hardest proof in this topic, that of Theorem 24.6.2).

*Problem set 20 (due Friday May 20):* Do 10 of the following problems from the most recent version of the notes. Try to do problems on all topics, but don’t do problems you have seen before. 25.2.A, 25.2.C, 25.2.D, 25.2.E, 25.2.F, 25.2.H, 25.3.A, 25.3.C, 26.1.A, 26.1.E, 26.2.A, 26.2.B, 26.2.C, 26.2.D, 26.2.E, 26.2.F, 26.2.G, 26.3.B, 26.3.C, 26.3.D, 26.3.E.

**Fall quarter (216A)**

*Monday, September 21: 1.1-1.2 (in the notes). * Welcome! About the class. About algebraic geometry. Some abstract nonsense.

*Wednesday, September 23 and Friday, September 25: 1.3-1.4.* Universal properties, (co)limits.

*Monday, September 28: 1.5-1.6.* Adjoints, abelian categories.

*Wednesday, September 30: 2.1-2.3.3. * Presheaves and sheaves and their morphisms; stalks and germs; examples of sheaves.

*Friday, October 2: 2.3.4-2.5.* Presheaves and sheaves of abelian groups both form abelian categories, but for different reasons. Properties determined by germs/stalks.

*Monday, October 5: 2.6-3.1.* Many perspectives on the inverse image sheaf. Sheaves on a base. Examples of “geometric spaces” as “sets, with topology, and with a sheaf of rings”.

*Wednesday, October 7: 3.2.* Spec A as a set, and the ring of functions on it. Eight examples to tell us how to visualize this information. Affine n-space. “Generic points”.

*Friday, October 9: 3.2-3.3.* Hilbert’s Nullstellensatz. Picturing quotients and localizations. Maps of rings give maps of Spec’s. Nilpotents are the reason functions are not determined by their values at points.

*Monday, October 12: 3.4-3.6.4.* The Zariski topology on Spec A, and properties. Vanishing set, distinguished open sets, radical of ideals, connectedness, irreducibility.

*Wednesday, October 14: * *rest of Chapter 3.* Quasicompactness, closed points, specialization, generization, generic points, irreducible and connected components, Noetherian rings and topological spaces, the function “I(.)” which wants to be inverse to “V(.)”.

*Friday, October 16: 4.1-4.3. * The structure sheaf of an affine scheme. Definition of schemes. Locally ringed spaces.

*Monday, October 19: 4.4.* Many examples of schemes.

*Wednesday, October 21: 4.5. * The Proj construction of projective schemes (over a field, or more generally, a ring).

*Friday, October 23: 4.6-5.3. * Topological properties of schemes, including the new property of quasiseparatedness. The affine communication lemma. Noetherian schemes, and schemes finite type over a field.

*Monday, October 26: 5.4-5.5.* Normality, factoriality, and an introduction to associated points.

*Wednesday, October 28: 5.5 (conclusion).* Associated points and primes.

*Friday, October 30: 6.1-6.3.* Morphisms of schemes.

*Monday, November 2: 6.4-6.5.5. *Maps of graded rings yield maps of projective schemes. Rational maps.

*Wednesday, November 4: 6.5.6-7.1. * Fun with rational maps of varieties. “Good” classes of morphisms. (To do for next day: read 7.2.)

*Friday, November 6: 7.2-7.3.11.* Nakayama’s Lemma and related tricks. Quasicompact morphisms, quasiseparated morphisms, affine morphisms, finite morphisms, integral morphisms.

*Monday, November 9: 7.3.12-7.4.5. *(Locally) finite type morphisms. Chevalley’s Theorem.

*Wednesday, November 11: rest of 7.4. *The Fundamental Theorem of Elimination Theory. Definition of closed embeddings.

*Friday, November 13: 8.1-8.2. * (Locally) closed embeddings/subschemes, and examples. Closed subschemes of projective space.

*Monday, November 16: 8.3. *The scheme-theoretic image, and applications.

*Wednesday, November 18: 8.4.* Effective Cartier divisors, and regular sequences, and regular embeddings.

*Friday, November 20: 9.1-9.3.5.* Fibered products: existence and calculation.

*Monday, November 30: 9.3.6-9.6.* Generalities and genericities. “Reasonable” properties are preserved by base change. Geometric fibers. Products of Proj’s are Proj’s (the Segre embedding).

*Wednesday, December 2: 9.7-10.1. *Normalization. Separatedness.

*Friday, December 4: 10.1-10.3.7.* Quasiseparatedness, and more fun with diagonal morphisms. The locus where two morphisms agree, and the “Reduced-to-Separated” theorem. Properness.

**Winter quarter (216B)**

*Monday, January 4: 11.1.* Sketch of the quarter. Quick review of diagonal morphisms and proper morphisms. Dimension/codimension and initial properties.

*Wednesday, January 6: 11.2-11.3.9. *Dimension in geometric situations: Noether normalization; dimension = transcendence degree; codimension is difference of dimensions. Krull’s Principal Ideal Theorem.

*Friday, January 8: 11.3-12.1. * Krull’s Height Theorem, Algebraic Hartogs’s Lemma. Behavior of dimensions of fibers of morphisms of varieties. The Zariski cotangent space.

*Monday, January 11: 12.1-12.2. *Regularity, and smoothness over a field.

*Wednesday, January 13. *Back to dimension theory: a clever lemma, and proof of Krull’s Height Theorem. Krull’s theorem for the dimension of Noetherian local rings. Completion of proof of upper semicontinuity of fiber dimension. (Some of this isn’t yet in the notes.)

*Friday, January 15: 12.2, 12.3, 12.6. *By popular demand: Grothendieck topologies and faithfully flat descent. Smooth and etale morphisms, and the smooth and flat topologies.

*Wednesday, January 20: 12.6, 12.5, 12.8. *More on smooth and etale morphisms. Discrete valuation rings. Fancy facts worth seeing about regular local rings.

*Friday, January 22: 12.5, 13.1. *Geometry of Noetherian normal schemes. Vector bundles.* * (Problem set 10 due.)

*Monday, January 25: 13.2-13.4. *Quasicoherent sheaves. Using the “distiguished affine base”. Quasicoherent sheaves form an abelian category.

*Wednesday, January 27: 13.5-13.7. *Module-like constructions. Finite type, finitely presented, and coherent sheaves, and their properties.

*Friday, January 29: 14. *Line bundles and Weil divisors. (Problem set 11 due.)

*Monday, February 1: 15.1-16.1. *Quasicoherent sheaves, and projective A-schemes. Pushforwards of quasicoherent sheaves.

*Wednesday, February 3: 16.2-16.4.1. *Pullbacks of quasicoherent sheaves (and more generally, O-modules). Line bundles and maps to projective space.

*Friday, February 5: 16.4-16.6.3. *Why the theorem on line bundles and maps to projective space is so useful. The Curve-to-Projective Extension Theorem. Very ample and ample line bundles.* *(Problem set 12 due.)

*Monday, February 8: 16.6-17.2. *End of proof of equivalence of various definitions of ampleness. Relative Spec and Proj.

*Friday, February 12.* No class. (Problem set 13 due.)

*Wednesday, February 17: 17.3-17.4.3. *More on relative Proj. Projective morphisms. First applications to curves.

*Friday, February 19: 17.4.4-18.1.8.* Degree of a projective morphism from a curve to a regular curve. Desired properties of cohomology (of a quasicoherent sheaf on a quasicompact separated scheme).* *(Problem set 14 due.)

*Monday, February 22: 18.1-18.2.* Defining Cech cohomology, and verifying the desired properties.

*Monday, February 29: 18.3-18.5. *Cohomology of O(m) on projective space. Riemann-Roch, genus, degrees of coherent sheaves. Serre duality.

*Wednesday, March 2: 18.6, 18.8. *Hilbert polynomials, and related classical notions. Higher pushforwards and properties.

*Friday, March 4: 19.1-19.4. *Criterion for a projective morphism to be a closed embedding. Genus 0 curves. Introduction to hyperelliptic curves.

*Monday, March 7: 19.4-19.8. *Hyperelliptic curves. Curves of genus 2, 3, 4, 5.

*Wednesday, March 9: 19.9. *Fun with elliptic curves.

*Friday, March 11: 19.10. *Elliptic curves are group varieties.

**Spring quarter (216C)**

*Monday, March 28: 23.1-23.4.* Tor and Ext. Derived functors, and spectral sequences.

*Wednesday, March 30: 23.5, 21.1. * Derived functor cohomology and Cech cohomology agree. Initial motivation for differentials.

*Friday, April 1: 21.1-21.2.18.* The module of Kahler differentials: construction and universal property. The relative cotangent and conormal sequences.

*Monday, April 4: 21.2.20-21.3.6.* Global definition of differentials. Pullbacks of differentials. Redefinition of smooth varieties.

*Monday, April 11: 21.3.7-21.4. *Many examples.

*Wednesday, April 13: 21.5-21.6. *Understanding smooth varieties through their tangent bundles (e.g. Fano, Calabi-Yau, general type, etc.). Unramified morphisms.

*Friday, April 15: 21.7. *The Riemann-Hurwitz formula.

(No class Monday April 18, Wednesday April 20, Friday April 22.)

*Monday, April 25: 24.1-24.4.4.* Flatness in more (simple) detail. Tor, and ideal-theoretic criteria for flatness.

*Wednesday, April 27: 24.4.5-24.5.8. *Flat=free=projective for finitely presented modules over local rings (and many consequences). Topological consequences of flatness.

*Friday, April 29: 24.5.9-24.7. *Flatness implies constant Euler characteristic. Local criteria for flatness.

(No class Monday May 2, Wednesday May 4, Friday May 6.)

*Monday, May 9: 25. *Smooth and etale morphisms, and flatness.

*Wednesday, May 11 and Friday, May 13: 26. *Depth and Cohen-Macaulayness.

*Friday, May 13.*

*Monday, May 16.*

*Wednesday, May 18.*

*Friday, May 20.*

*Monday, May 23.*

*Wednesday, May 25.*

*Friday, May 27.*

(No class Monday, May 30.)

*Wednesday, June 1.*

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