MATH 6670 will be an introduction to schemes in algebraic geometry. Originating in the early 1960s, schemes are an elegant generalization of the notion of an algebraic variety, interpolating the algebraic, geometric, topological, and arithmetic aspects of the subject. While we cannot hope for an exhaustive treatment of the theory of schemes in one semester, the goal of the course is to motivate and illustrate some of the basic definitions and concepts of the theory of schemes, with an emphasis on examples and classical constructions.

Textbooks

While we will not follow a specific textbook, good accessible references for the theory of schemes are:

• Algebraic Geometry by Hartshorne
• The Geometry of Schemes by Eisenbud and Harris
• Algebraic Geometry and Arithmetic Curves by Liu
• Foundations of Algebraic Geometry by Vakil

All of them have their strengths and weaknesses and different styles of presentation. For example, Hartshorne is the standard reference, but is a bit terse, especially in comparison to Vakil. Liu is somewhere between the two, but does not assume from the beginning that we are working over an algebraically closed field like Hartshorne does. In the end, it depends on your interests and preferences.

Content-wise, we will be essentially covering the material in Hartshorne, Chapter II, with lots of supplements.

For the working algebraic geometer, building familiarity with the ultimate reference—the volumes of Éléments de Géométrie Algébrique ("EGA")—is useful, as you can find all the results we see in the course (and much more) proven there in great generality.

Announcements

• All homework assignments should be submitted through Gradescope. The entry code for the class is "ZARISKI."
• Here is the questionnaire from the second class. In case you missed the class, please fill out and return it to me as soon as you can.
• The first class takes place on Tuesday, January 22. Due to the snowstorm, it will just be an introduction and motivation to the course and scheme theory.

Homeworks

• Homework 1, due Thursday, January 31.
• Homework 2, due Thursday, February 7.
• Homework 3, due Thursday, February 14.
• Homework 4, due Thursday, February 21.
• Homework 5, due Thursday, February 28.
• Homework 6, due Thursday, March 7.
• Homework 7, due Thursday, March 21.
• Homework 8, due Thursday, March 28.
• Homework 9, due Thursday, April 11.
• Homework 10, due Thursday, April 25.
• Homework 11, due Tuesday, May 7.

Final Assignment

All due Friday, May 17.

• Comprehensive
• Revision
• Mock Homework/Exam

Topics covered

• Week 1 (01/24, 01/26): Why schemes? From varieties to schemes. Presheaves and sheaves. (Eisenbud–Harris I.1, Hartshorne II.1).
• Week 2 (01/29, 01/31): Sheafification, stalks, translating algebraic results into sheaf theory. (Eisenbud–Harris I.1, Hartshorne II.1).
• Week 3 (02/05, 02/07): Functoriality of sheaves, morphisms of ringed spaces, locally ringed spaces, affine schemes, modules and the tilde construction. (Eisenbud–Harris I.1-2, Hartshorne II.2).
• Week 4 (02/12, 02/14): Anti-equivalence between the category of affine schemes and rings, immersions, closed subschemes (Eisenbud–Harris I.2, Hartshorne II.2).
• Week 5 (02/19, 02/21): Constructing general schemes, morphisms, fiber products, relative schemes (Eisenbud–Harris I.2-3, Hartshorne II.2-3).
• Week 6 (02/28): Introduction to valuative criteria, families (Hartshorne II.4, Eisenbud–Harris II.3)
• Week 7 (03/05, 03/07): An introduction to algebraic groups and their representations.
• Week 8 (03/12, 03/14): Separatedness, introduction to quasicoherent sheaves (Hartshorne II.3.4-5, Eisenbud–Harris III.1).
• Week 9 (03/19, 03/21): Local properties of rings, local properties of schemes, local properties of ring homomorphisms and morphisms (Hartshorne II.3-5).
• Week 10 (03/26, 03/28): Projective schemes (Hartshorne II.2, II.5, Eisenbud–Harris III).
• Week 11 (04/09, 04/11): Some quotients, invariants, and a first look at geometric invariant theory.
• Week 12 (04/16, 04/18): Closed subschemes of projective space, Chow's lemma (Eisenbud–Harris III.2, Hartshorne II.4-5).
• Week 13 (04/23, 04/25): Algebraic curves via scheme theory, degree theory for regular 1-dimensional schemes, divisors (Hartshorne II.5-6).
• Week 14 (04/30, 05/02): More on divisors, invertible sheaves, line bundles (Hartshorne II.5-6).
• Week 15 (05/07): Moduli problems and moduli spaces, and where you can go beyond this course.

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