Introduction to Algebraic Geometry 

General Information

• Instructor: 李时璋 (shizhang at amss dot ac dot cn)
• Time: Tuesday & Thursday 13:30--16:30
• Location: 中关村校区S104


• Main reference: Robin Hartshorne, Algebraic Geometry.


• Ultimate reference: The stacks project.


• Other textbooks: Ulrich Görtz & Torsten Wedhorn, Algebraic Geometry with examples and exercises I: Schemes & II: Cohomology of Schemes; Rick Miranda, Algebraic curves and Riemann surfaces; 刘青, 代数几何和算术曲线; Phillip Griffiths & Joseph Harris, Principles of Algebraic Geometry; Ravi Vakil, The rising sea; Claire Voisin, Hodge Theory and Complex Algebraic Geometry I & II.


• Online resources: Milne, Edixhoven & Holmes & Kret & Taelman, Kerr, Mihalcea, Borisov.


• Prerequisites: It will be helpful to have some basic knowledge on set theory, point set topology, commutative algebra, and category theory.

Lectures

The goal is to introduce the language of schemes & cohomology theory, we shall cover 2nd & 3rd chapters of Hartshorne's book. Time permitting, we shall also talk about curves and Riemann surfaces.

Schedule:

• Week 1, 9/3 & 9/5. Items covered:  Intro to AG, presheaf, sheaf (9/3). Sheafification, affine scheme (9/5).
• Week 2, 9/10 & 9/12. Items covered:  Affine scheme (cont'd), locally ringed spaces, schemes, qcoh sheaves on Spec (9/10). Proj construction (9/12).
• Week 3, 9/19. Items covered:  Properties of schemes, properties of morphisms.
• Week 4, 9/24. Items covered:  Properties of morphisms (cont'd), fiber product.
• Week 5, 10/8 & 10/10. Items covered:  Fiber product (cont'd), (quasi-)separated schemes/morphisms (10/8). Separated and proper schemes/morphisms (10/10).
• Week 6, 10/15 & 10/17. Items covered:  Midterm Exam part 1, proper morphisms (cont'd), quasi-coherent sheaves (10/15). Actual midterm exam part 2 and recap (10/17).
• Week 7, 10/22 & 10/24. Items covered:  Quasi-coherent sheaves (cont'd) (10/22). Functor of points of projective spaces (10/24).
• Week 8, 10/29 & 10/31. Items covered:  Coherent sheaves on projective schemes (10/29). Chow's lemma, proper pushforward of coherent sheaves (10/31).
• Week 9, 11/5 & 11/7. Items covered:  Formal function theorem. Stein factorization. Normal schemes (11/5). Zariski's main theorem (11/7).
• Week 10, 11/12 & 11/14. Items covered:  Characterize closed immersion into projective space, Divisors (Weil and Cartier).
• Week 11, 11/26 & 11/28. Items shall be covered:  Reminder on divisors. Line bundles, maps between projective spaces. Blowups.
• Week 12, 12/3 & 12/5. Items shall be covered:  Unramified maps, etale maps. Differential forms, smoothness.
• Week 13, 12/10 & 12/12. Items shall be covered: 
• Week 14, 12/24 & 12/26. Items shall be covered: 

Exercises

Very often, we shall have a 幸运观众 come to board and solve an exercise from the above.

Exams

Midterm Exam.

Grading

TBD