Math 536 (Algebraic Geometry 2), Fall 2024, Lev Borisov

10:20am-11:40am, M Th, Hill Center 423, Busch



Text:  Hartshorne "Algebraic Geometry"
 

Office Hours: I will be in my office 240 Hill Center on Busch Campus at TBA. These times may change, in which case the new times will be announced on my home page. Other good times to talk are right before or right after the class. It is also possible to ask questions by email. Emails will be generally answered within 24 hours, often sooner. The more detailed your email question is, the more detailed the reply message will be. My email is borisov at math dot you know what.
 

Homework: Suggested homework problems are listed in the schedule of lectures below. Homework will not be mandatory but you are encouraged to complete it, since this is the only way to make sure you understand the material. Collaboration on homework is strongly encouraged.
 

Disability Services: phone: (848)445-6800, Lucy Stone Hall, Suite A145, Livingston Campus, 54 Joyce Kilmer Avenue, Piscataway, NJ 08854, https://ods.rutgers.edu/
Rutgers University welcomes students with disabilities into all of the University's educational programs. In order to receive consideration for reasonable accommodations, a student with a disability must contact the appropriate disability services office at the campus where you are officially enrolled, participate in an intake interview, and provide documentation: https://ods.rutgers.edu/students/documentation-guidelines. If the documentation supports your request for reasonable accommodations, your campus disability services office will provide you with a Letter of Accommodations. Please share this letter with your instructors and discuss the accommodations with them as early in your courses as possible. To begin this process, please complete the Registration form on the ODS web site at: https://ods.rutgers.edu/students/registration-form. Students with disabilities requesting accommodations should present a Letter of Accommodations to the instructor as early in the term as possible. See here for details.
 

Student wellness services: Counseling, ADAP & Psychiatric Services (CAPS), phone: (848)932-7884, 17 Senior Street, New Brunswick, NJ 08901. CAPS is a University mental health support service that includes counseling, alcohol and other drug assistance, and psychiatric services staffed by a team of professionals within Rutgers Health services to support students' efforts to succeed at Rutgers University. CAPS offers a variety of services that include: individual therapy, group therapy and workshops, crisis intervention, referral to specialists in the community, and consultation and collaboration with campus partners.
 

Violence Prevention & Victim Assistance (VPVA) phone: (848)932-1181, 3 Bartlett Street, New Brunswick, NJ 08901, www.vpva.rutgers.edu. The Office for Violence Prevention and Victim Assistance provides confidential crisis intervention, counseling, and advocacy for victims of sexual and relationship violence and stalking to students, staff and faculty. To reach staff during office hours when the university is open or to reach an advocate after hours, call (848)932-1181.
 

Schedule of Lectures


Sep. 5. Presheaves and sheaves. Section II.1
Homework: 1. Check the universal property of sheafification.
2. Go through the proof of Proposition 1.1 in Section II.1 of Hartshorne.
3. Prove that for a morphism of sheaves, the presheaf kernel is already a sheaf.
4. Prove that for a morphism of sheaves F->G, the sheafification of the presheaf image is naturally a subsheaf of G.

Sep. 9. Presheaves and sheaves -- continued. Section II.1
Homework: II.1.14, II.1.15, II.1.16.

Sep. 12. Definition of schemes and morphisms of schemes. Section II.2
Homework:
Check that Zariski topology on Spec(A) is indeed a topology.

Sep. 16. Definition of schemes and morphisms of schemes -- continued. Section II.2
Homework: II.2.3, II.2.7, II.2.10

Sep. 19. Basic properties of schemes, open and closed embeddings, fiber products. Section II.3
Homework: II.3.1

Sep. 23. Basic properties of schemes, open and closed embeddings, fiber products -- continued. Section II.3
Homework: II.3.9, II.3.10, II.3.11 (if ambitious)

Sep. 26. Separated and proper morphisms. Section II.4
Homework: II.4.2, II.4.3

Sep. 30. Review.

Oct. 3. Sheaves of modules. Quasi-coherent sheaves. Invertible sheaves. Section II.5

Oct. 7. Sheaves of modules. Quasi-coherent sheaves. Invertible sheaves -- continued. Section II.5
Homework: II.5.2, II.5.7, II.5.10(a,b)

Oct. 10. Sheaves of modules. Quasi-coherent sheaves. Invertible sheaves -- continued further. Section II.5

Oct. 14. Weil Divisors. Section II.6

Oct. 17. Cartier Divisors. Section II.6
Homework: II.6.1

Oct. 21. Divisors on curves. Section II.6

Oct. 24. Projective morphisms. Section II.7

Oct. 28. Projective morphisms -- continued. Section II.7

Oct. 31. Differenials. Section II.8

Nov. 4. Derived Functors. Section III.1

Nov. 7. Cohomology of sheaves. Section III.2

Nov. 11. Cohomology of sheaves -- continued. Section III.2

Nov. 14. Cohomology on a Noetherean affine scheme. Section III.3

Nov. 18. Cech cohomology. Section III.4

Nov. 21. Cech cohomology -- continued. Section III.4

Nov. 25. The cohomology of projective space. Section III.5

Nov. 26. Review.

Dec. 2. Ext groups and sheaves. Section III.6

Dec. 3. Ext groups and sheaves -- continued. Section III.6

Dec. 5. Serre Duality. Section III.7

Dec. 9. Review.