About the format of the course
As is clear from its name, this is a continuation of Algebraic Geometry I, taught last Fall. For a general description of the field, see the webpage of that course. The main focus will be now on schemes and their cohomology. For this we shall need some basic category theory and homological algebra and so, at the risk of boring some of you, we are going to spend some time familiarizing ourselves with this stuff. (On the other hand, we hope and expect that you will find this to be useful also outside Algebraic Geometry). Toward the end of the course we intend to return to algebro-geometric questions with a classical flavor.
For a subset of the material (mostly the part involving cohomology of sheaves and homological algebra) we shall use a hybrid format of course and seminar. To be specific: at some point I will teach only one day of the week about schemes, the other day of the week being reserved for two students to do a presentation. The students will be given careful instructions about what to cover and they will usually take up where the last of pair of students left off (as if it were a seminar embedded in a course). The task of these students is not to just present some material, but also to design one or two exercises for their fellow students, which they then have to grade themselves (this all being supervised by me and my teaching assistant.
Prerequisites
Familiarity with basic algebraic geometry over an algebraically closed field such as AG I.
Place and time
Mondays 15:20-16:55 and Wednesdays 9:50-11:25 in Teaching Building 6-B, room 305, beginning Febr. 20.
Homework
Each week I or the presenters give homework in the form of some exercises. Only exercises with an asterisk will be graded (with grade ‘sufficient’, ‘in between’ or ‘insufficient’) and must for that purpose be handed in on Monday the following week (to my teaching assistant 王彬 or to the presenters).
Grading
Essentially the same rule applies as to Algebraic Geometry I: the grades will be ‘sufficient’, ‘in between’ or ‘insufficient’. The final score will be based on these grades and on the quality of your presentation. There will be no written exam.
Material covered and exercises (starred exercises will be graded and count towards H)
Feb. 20: Ch. 3 until Example 1.5. Exerc. 81*, 82.
Feb. 22: Ch. 3 until Theorem-def 1.6
Feb. 27: Appendix: categories, presheaves, sheafification, Exerc. 105*.
March 1: Notion of a scheme.
March 6: Proj construction, Exerc. 85, 86.*
March 8: Functor of points, Exerc. 88*.
March 13: Products, base change, extension of scalars.
March 20: Finiteness properties, Hasse-Weil zeta function.
March 27: Integral schemes, immersions, separation property, valuation rings.
April 1 (instead of April 2, Tomb Sweeping day): Valuative criterion for being separated.
April 9: Properness and the valuative criterion for it.
April 16: Direct and preimage image of a sheaf, adjointness property.
April 24: Lecture by Professor Jean-Pierre Serre (3rd. and final of a series)
April 30 (instead of May 1, Labourer’s day): Sheaf pull-back for modules over ringed spaces, quasi-coherent sheaves.
May 8: Coherent sheaves, scheme theoretic image, Chow’s lemma.
May 15: Picard group, Cartier divisor, Cartier class group
May 22: Weil divisor, Weil class group. Examples: cuspidal and nodal curves.
May 29: no class (Dragon boat festival)
June 5: Serre duality for nodal curves, Grothendieck-Riemann-Roch
Schedule of student presentations (date, section, speakers)
Most sections refer to Chapter 3 of Hartshorne’s book.
March 15: Derived functors, Peigen Li, Yunpeng Zi. Find the exercise here.
March 22-29: Cohomology of Sheaves, Chen Bingyi, Zhenping Gui.
April 5-12: Cohomology of Noetherian Affine Schemes, Jiang Yikai, Chen Rui.
April 19: Cech Cohomology, Zeng Keyou, Qi Renrui.
April 26-May 3: Cohomology of Projective Spaces, Piye Yang, Zhong Yiming.
May 10: Ext groups and sheaves, Wu Zhixiang, Zhang Zhiyu.
May 17-24: Serre duality, Xu Kai, Zao Ruishen.
May 31: The Tate residue, Zao Ruishen.
Teaching assistant
王彬
Literature
I prepared a set of notes which accompanies this course and that is available here. The part relating to this course begins with Chapter 3. It is possible that during the course I make some revisions, so do not print more of it than you need. Below I listed some literature that you might find useful to consult; for other opinions, take a look at this mathoverflow question.
David Eisenbud and Joe Harris: The Geometry of Schemes, GTM 197, Springer. A good introduction to schemes and related notions.
Robin Hartshorne: Algebraic Geometry, Springer Verlag GTM 52, Springer. Still the most widely used introduction to modern algebraic geometry.
Liu Qing: Algebraic Geometry and Arithmetic curves, Oxford Science Publications. The original motivation of the author was to give an exposition of arithmetic surfaces. But the first half of the book is an excellent introduction to schemes and the second half well illustrates the power of the scheme approach.
David Mumford: The Red Book of Varieties and Schemes, Lecture Notes in Mathematics 1358, Springer. This is in fact two rather separate books which have been reprinted in a single volume. Relevant for this course is the part this (nowadays, yellow) lecture note it is named after, which was essentially the first book on schemes meant for students. It is still a very good introduction, written in the author's characteristic style: informality paired with precision.
David Eisenbud: Commutative Algebra with a view toward Algebraic Geometry, GTM 150. A substantial text of about 780 pages. The topic of the subtitle here enters mostly through local properties or via affine varieties. The book has detailed proofs, often accompanied by enlightening discussions. It shows that there is little difference between Commutative Algebra and Local Algebraic Geometry.
Ulrich Görtz, Torsten Wedhorn: Algebraic geometry I, Schemes with examples and exercises, Adv. Lectures in Mathematics. Vieweg + Teubner.
Fu Lei: Algebraic Geometry, a concise introduction (of about 260 p.) to the theory of schemes based on a course taught at the Morningside Center. It is joint publication of Springer and Tsinghua UP and that is reflected by its price here on campus: for 39 元 it is a steal.
And for the brave:
Alexandre Grothendieck-Jean Dieudonné: Éléments de Géométrie Algébrique. Publications Mathématiques de l'IHES. This is the fundamental source. Only 4 chapters of the planned 13 have appeared, but they already comprise about 1500 pages. Go for it if you want rigor and generality (it has been translated into Chinese!).
Johan de Jong et alii: The stacks project. This Wiki based enterprise is becoming the natural successor of EGA as the standard opus of reference for algebraic geometry. It is certainly as rigorous and general and goes well beyond the notion of a scheme. Many of the chapters (98 as of Feb. 2017) rest on only few of the preceding ones, so that often you can just start reading a chapter once you have already some basic knowledge of the field.
Ravi Vakil: MATH 216: Foundations of Algebraic Geometry. These course notes delve into the subject in a true Grothendieck spirit right from the start, yet do this in a way that makes prerequisites minimal. When you have finished working through the 700+ page manuscript you have also learned a lot about category theory and homological algebra. It is on Vakil's website available as a wordpress blog, which means that it cannot be accessed this side of the wall. I therefore put a pdf copy here.
Algebraic Geometry during the Spring of 2017
If you want to learn more about this beautiful subject, then here is a nonexhaustive list of activities:
◦ Informal student seminar on (Mixed) Hodge Theory, run on Wednesday evening 19:00-21:00 in the Jin Chun Yuan West
Building. If you are interested, get in touch with 王彬. Go to its webpage for the program.
◦ Seminar Algebraic Geometry. The present time slot is Thursdays, 14:30-15:30. In charge are Dr Liang Dun and
Dr 许权. The intended audience is faculty or advanced graduate students. Contact one of them if you want to receive
the announcements. Go to its webpage for the program.