Instructor: Emily Cliff.
Lectures: TR 9:30--10:50am, 441 Altgeld Hall.
Office hours: M 10--10:50am and W 11--11:50am, 165 Altgeld Hall (or by appointment).
This course is about sheaf cohomology, following Chapter III of Hartshorne's Algebraic Geometry. We will also cover some topics from the end of Chapter II, namely blow-ups and differentials.
The class time will be divided between lectures and problem-solving. You are expected to bring a copy of Hartshorne to each class, to use in solving problems. (You can download a copy from the library.)
Your grade will be based upon your attendance and participation, and also perhaps some problem sheets. (A policy on problem sheets will be posted soon, based on your responses to the survey.)
From March 12 onwards, we are back in Altgeld Hall. Class will be in 441 Altgeld Hall, and office hours will be in my office.
On Tuesday 6 March (and likely Thursday 8 March), class will meet in Mumford Hall 316S again. Office hours will be relocated to some other location, to be announced by email. Please feel free to email me if you have questions, and we can arrange to meet.
On Tuesday 27 February, class will meet in Mumford Hall room 316S due to the GEO strike. The time is 9:30 to 11 as usual. Same deal for Thursday.
Please fill out the class survey. This is mandatory for anyone attending the class, whether you are registered or not. To receive a link to the survey, send me an email.
in reverse chronological order
| Date | Material covered; exercises assigned |
|---|---|
Tuesday, 1 May. |
Homework 4 due. Hand in solutions to these questions. Towards the classification of ruled surfaces. Read Chapter V.2. Look at these exercises. |
| Thursday, 26 April. |
Ruled surfaces. Read Chapter V.2. Look at these exercises. |
| Tuesday, 24 April. |
Applications of Riemann--Roch; introduction to ruled surfaces. Read Chapter V.1, V.2. Look at these exercises. |
| Thursday, 19 April. |
Riemann--Roch for surfaces. Read Chapter V.1. Look at these exercises. |
| Tuesday, 17 April. |
The canonical embedding; the intersection pairing on surfaces. Read Chapter IV.5; Chapter V.1. Look at these exercises. |
| Thursday, 12 April. |
Every curve can be embedded in P3; every curve is birational to a nodal curve in P2. Read Chapter IV. 3. Look at these exercises. |
| Tuesday, 10 April. |
Homework 3 due. Hand in solutions to these questions. Review of linear systems and maps into projective space. Ample, very ample, and base-point free linear systems (on curves). Review Chapter II.7; read Chapter IV.3. Look at these exercises. |
| Thursday, 5 April. |
Hurwitz's theorem; Frobenius morphism; purely inseparable morphisms of curves. Read Chapter IV.2. Look at these exercises. |
| Tuesday, 3 April. |
The Riemann--Roch Theorem; ramification index for finite morphisms of curves. Read Chapter IV.1. Look at these exercises. |
| Thursday, 29 March. |
Review on abstract curves, divisors on curves, linear systems. Look at these exercises. |
| Tuesday, 27 March. |
The theorem on formal functions; the semi-continuity theorem. Read Chapter III.11, 12. Look at these exercises |
| Thursday, 15 March. |
Flat familes; smooth morphisms. Read Chapter III.9, 10. |
| Tuesday, 13 March. |
Higher direct image sheaves; flat modules and flat morphisms. Read Chapter III.8, 9. Look at these exercises. |
| Thursday, 8 March. |
Identifying dualizing sheaves; higher direct image sheaves. Read Chapter III.7, 8. Look at these exercises. |
| Tuesday, 6 March. |
Homework 2 due. Hand in solutions to these questions. Serre duality for projective schemes. Read Chapter III.7. Look at these exercises. |
| Thursday, 1 March. |
Existence of dualizing sheaves for projective schemes. Read Chapter III.7. There are no extra exercises today; keep working on the ones from last class. |
| Tuesday, 27 February. |
Cohomology and Serre duality of projective space. Read Chapter III.5. Look at these exercises. |
| Thursday, 22 February. |
Cohomology of projective space. Read Chapter III.5. Look at these exercises. |
| Tuesday, 20 February. |
More practice with Ext groups/sheaves and Cech cohomology. Look at these exercises. |
| Thursday, 15 February. |
Ext groups and schemes. Read Chapter III.6. Look at these exercises. |
| Tuesday, 13 February. |
Cech cohomology. Read Chapter III.4. Look at these exercises. |
| Thursday, 8 February. |
Cohomology and affine noetherian schemes. Read Chapter III.3. Look at these exercises. |
| Tuesday, 6 February. |
Homework 1 due. Hand in solutions to these questions. First results on sheaf cohomology: Grothendieck vanishing theorem. Read Chapter III.2. Look at these exercises. |
| Thursday, 1 February. |
Introduction to sheaf cohomology! Read Chapter III.1 and III.2. Look at these exercises. |
| Tuesday, 30 January. |
No class today. |
| Thursday, 25 January. |
Bertini's theorem. Tangent sheaves, canonical sheaves, projective genus. (Still Chapter II.8.) Look at these exercises. |
| Tuesday, 23 January. |
Differentials. Read Chapter II.8. Look at these exercises. |
| Thursday, 18 January. |
Properties of blow-ups. (Still Chapter II.7.) Look at these exercises. |
| Tuesday, 16 January. |
Introduction to the course. Proj, P(E), and blowing up. Read Chapter II.7, and complete the class survey. Look at these exercises. |