Algebraic Geometry II (Spring 2014)

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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). One of our good intentions is to return toward the end of the course to algebro-geometric questions with a classical flavor.
I have chosen for this course a somewhat experimental set-up. My intention is to use for a subset of the material  (mostly the part involving cohomology of sheaves and homological algebra) 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 of course be given careful instructions about what to cover and they will usually take up where the last of pair of students left of (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 姜清元.

Prerequisites
Familiarity with basic algebraic geometry over an algebraically closed field such as AG I.

Literature
See the webpage of  AG I. We shall further develop the notes of that course but these will not cover everything that we discuss. A separate appendix collects some of the material not particular to Algebraic Geometry such as category theory  and sheaf cohomology. As all of you should know, it always pays to make notes by yourself (on paper is best in general).  And for a completer picture it is indispensable to consult books. (Go to this mathoverflow question if you want to learn which text is favored by other (former) students of algebraic geometry.)

Place and time
Mondays 15:20-16:55 and Thursdays 9:50-11:25 in the Teaching Building 4, room 4301. The first session is Monday Febr 24. Note that there will be no classes on April 7 (Qing Ming Jie holiday) and May 1 (Labour day); the class of June 2 (Dragon Boat festival) will be moved to June 3.

Schedule of student presentations (date, section, speakers)
Mar 20: Derived functors, Sun Ao/Chen Zhangchi, Exercises
Mar 27:Cohomology of Sheaves,  Zheng Zhiwei/Qiu Congling, Exercises
April 3: Cohomology of Noetherian Affine Schemes, Zhang Mingyi/Yuan Beihui, Exercises
April 10:Cech Cohomology, Fu Yulong/Wang Zhiyuan, Exercises
Apr 17-24: Cohomology of Projective Spaces, Xie Ying/Su Xiaoyu,
Exercises
May 08: Ext groups and sheaves, Mi Fangzhou/Cao Zhu, Exercises: 
Hartshorne Ch. II 5.1-a,b,c
May 15-22: Serre duality, Chen Jiaming/Chen Lingchuan, Exercises  (with an erratum)
May 29-June 5: Higher direct images of sheaves, Zheng Zhiwei/Qiu Congling, Exercises
June 12: Flat morphisms, Sun Ao/Chen Zhangchi; Exercises: Hartshorne Ch. II 3.10-a, 3.20-a,d,e,f. 

Teaching assistant
姜清元

Grading
There will be homework exercises, to which essentially the same rule applies as to Algebraic Geometry I: the grades will be ‘sufficient’, ‘in between’ or ‘insufficient’ and must for that purpose be handed in within a week after the mentioned date (by mail or hard copy) to my teaching assistant. 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
To be updated weekly, the starred exercises will be graded and count towards your final  grade

Feb 24: Appendix: finished Example 1.7; Exerc. 80*.
Feb 27: Appendix: representative functors, Yoneda lemma, adjoint functors.
Mar 3:  Appendix: sums and products,  Ch. 3 until Section 5; Exerc. 82* (this exerc. was edited on 3/17).
Mar 6: Ch. 3  until Prop. 5.6; Exerc. 83*.
Mar 10: Ch.3 finished the proof of Lemma 6.7.
Mar 13: Ch. 3 until the valuative criterion of separatedness Thm. 6.17; Exerc. 84*.
Mar 17: Ch. 3 up to Prop. 6.22.
Mar 24: Finished Section 6.
Mar 31: Quasi-coherent and coherent sheaves.
April 15: Locally free  and invertible sheaves.
April 22: Picard group, Cartier divisors.
April 29: Weil divisors.
May 5: Degree of a divisor on a curve, Chow's lemma.
May 12: Serre twisting sheaves. Finished Corollary 10.6.
May 19: Until Definition 10.14.
June 3: Blowing up an ideal.
June 9: Projective birational morphisms.