What is this course about?

A first characterization of Algebraic Geometry is: the study of the common zero sets of a collection of polynomial equations in a given number of indeterminates. These polynomials were originally assumed to have real coefficients so that the zero set would lie in real n-space. After the advent of complex numbers, it was soon noticed that the theory became much more manageable if these coefficients were taken to be complex numbers (for instance a polynomial in a single real variable of degree d has at most d real solutions, but if we take the coefficients complex, then the number of complex solutions is exactly d, if we are willing to count solutions with multiplicity). It had also been observed that things became even better if we include “solutions at infinity”. This means looking for solutions in a somewhat bigger space then complex n-space, namely complex projective n-space. This was followed by the observation that for much of the theory the only property of the complex numbers that intervened was that they make up an algebraically closed field. We thus arrive at the point of what is called Projective Algebraic Geometry: the study of common zero sets of systems of homogeneous polynomials in n+1 variables x_{0},...,x_{n} with coefficients in an algebraically closed field.

However, if it so happens that the polynomials have their coefficients in a smaller field that is not algebraically closed such as the field of rational numbers, then it makes sense (and there may be good reason) to ask for solutions with coefficients in that field. But this is often a subtle issue which usually involves Galois theory, even when the field is that of the real numbers and this explains why it was not a good idea to start out that way. Things becomes even more complicated if the algebraically closed field is replaced by a ring, for instance the ring of integers. Such questions are by no means uninteresting, as many natural questions in number theory can be stated that way. In the 1950s it was gradually recognized that in order to accommodate this kind of generality, a complete rebuilding of the foundations was called for. This foundational work was carried out during a relatively short period (1958-1970) under the leadership of A. Grothendieck. The tools and language developed by him (with his notion of scheme taking the place of an algebraic variety) and the underlying way of looking at things are now universally accepted as the framework to work in (and to formulate results about) algebraic geometry. At the same time, experience has taught us that the scheme setting is ill-suited for a first acquaintance with algebraic geometry, and this is why most of this course is concerned with Algebraic Geometry over an algebraically closed field.

Prerequisites

Basic commutative algebra concerning rings and modules and a bit of Galois theory. The Tsinghua Honors Algebra course is a perfect preparation.

Literature

This course will be based on my lecture notes:

Eduard Looijenga: *Algebraic varieties*, Surveys of Modern mathematics **15**, Higher Education Press, Beijing (2019). (Current price: 69 元.)

You are encouraged to obtain a copy of this book. I appreciate you reporting any inaccuracies. In fact, any feedback is welcome; for example, if let me know if you feel that at certain places the exposition is unclear. An older set of notes is also available here. Below I list 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 as rigorous and general and goes well beyond the notion of a scheme. Many of the chapters (95 as of Dec. 2015) 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.

Place and time

Tuesdays 13:30-15:05, Fridays 13:30-15:05 in Room 6B310 (Teaching Building 6), beginning Sept. 10.

Final exam arrangement

It has two parts:

Part 1: a grade based on your home work assignments or the grade for an oral exam (if you go for this option, then make an appointment by Dec. 13)

Part 2: an Essay of about 10 pages about a topic closely connected with the course. The proposal for a topic must be made by Sunday, Dec. 8 and it requires my approval (which you get no later than Dec. 10). The Essay (preferably submitted as a file) must be handed in by Jan. 3.

Your final grade will become available the week of Jan 6-Jan 10.

Homework

Each week I 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 by Tuesday the following week to my teaching assistant Wang Bin (王彬). It is possible to get your final grade without submitting any homework, but you have to tell me before Dec. 15, as this may involve an oral exam.

Grading

The homework grades can contribute towards your final grade as follows: the grade (named H) for your homework is determined by taking the average of all the weekly grades except for the two worst ones, using the rule: sufficient=10, in between=7, insufficient=5 and with failure to hand in a starred (*) homework exercise resulting in a zero grade. If you get grade W for thuntilse written exam, then your final grade is computed according to the rule 0.6H+0.4W or W (rounded off appropriately), whichever is highest, provided W is at least 6.

Material covered and exercises (starred exercises will be graded and count towards H)

Here you will find a brief indication what was done when, plus the home work problems for that week. Exercise numbers and section numbers refer to my lecture notes.

Sept. 10: Up to (and including) Prop.1.2.3. Exerc. 1*,2*,4*.

Sept. 17: Up to (and including) Lemma 1.2.14. Exerc. 8, 10*

Sept. 20: Up to Prop. 1.3.5. Exerc.11*. (Exerc. 11b must be modified: make these the ideals of R contained in 𝔭 but

containing the (ideal of) elements of R annihilated be an element in R-𝔭.)

Sept. 24: Up to (an including) Example 1.4.4. Exerc. 15*

Sept. 27: Up to Remark 1.4.11. Exerc. 18*, 19*.

Sept. 29: (by 闫旗军) Finished Subsection 1.5.

Oct. 8: Finished Subsection 1.7. Exerc. 22, 28*, 29.

Oct. 11: Finished Subsection 1.8. Exerc. 31*, 32.

Oct. 15: Almost finished Prop. 1.9.6. Exerc. 34, 35*.

Oct. 18: Finished Subsection 1.10. Exerc. 36, 37*.

Oct. 22: Up to (an including) Definition 1.1.5. Exerc. 39

Oct. 25: Up to (an including) Lemma 1.10.4.

Algebraic Geometry during the Fall of 2019

Here I will list activities in case you want to learn more about Algebraic Geometry.

Graduate seminar *K3 surfaces and related topics*.

Every Tuesday evening 19:00-21:00 (begins Oct. 15), conference room 3, Jinchunyuan West building (2nd floor).

Contact 钟一鸣 for more information.

Beijing Algebraic Geometry Colloquium

Seminar Algebraic Geometry Tsinghua