# Algebraic Geometry II, Summer semester 2019

**Lecturer:**Georg Oberdieck

**Assistant:**Emma Brakkee

**Tutors:**Thorsten Beckmann, Matthew Dawes

Note 1: The room of Thorsten Beckmann's tutorial, Tuesdays 10-12, has been changed from 0.008 to N0.003.

Note 2: The second midterm exam has been cancelled. Instead, we will have a review problem sheet that counts for 100 points.

### Classes

Monday 16:00 c.t. - 18:00, Kleiner Hörsaal (Wegelerstrasse 10)

Thursday 16:00 c.t. - 18:00, Kleiner Hörsaal

First day of class: **April 1**.

### Office hours

Fridays 11-12 am in 1.028.

### Content

The course introduces the modern language of algebraic geometry: sheaves, schemes, cohomology.

### Pre-requisites

Commutative Algebra, Algebraic Geometry I (see syllabus of the first semester).

### Syllabus

https://www.math.uni-bonn.de/~georgo/syllabus2.pdf

### References

- R. Hartshorne, Algebraic Geometry.
- R. Vakil, Foundations of algebraic geometry. Online lectures.
- Qing Liu, Algebraic Geometry and Arithmetic Curves.
- U. Goertz, T. Wedhorn: Algebraic Geometry I.

### Tutorials

Day | Time | Room | Tutor |
---|---|---|---|

Tuesday | 10:00 c.t. - 12:00 | N0.003 | Thorsten Beckmann |

Tuesday | 12:00 c.t. - 14:00 | 0.003 | Matthew Dawes |

Wednesday | 14:00 c.t. - 16:00 | 1.008 | Matthew Dawes |

### Problem sets & midterm exams

There will be one problem set every week.
Solutions to the exercises are to be handed in every Monday before the lecture.
The solutions are submitted individually, group submissions are not allowed.
The problem sets are 50 points each.

There will be two midterm exams that replace the exercise sheets the given week.
They are 45 minutes in class exams, 100 points each.

One has to have half of the total numbers of points to be admitted to the final exam.

### Final exam

The exam is a 2 hour written exam. The dates are as follows.

**Final exam 1:**Wednesday, July 24, 9:00 - 11:00, Großer Hörsaal.**Final exam 2:**Monday, September 23, 9:00 - 11:00, Kleiner Hörsaal.

### List of problem sets and midterm exams

Problem set 1, due April 8.
*Correction:* in the second line of Problem 3, the third term in the short exact sequence should be I/F, not F/I.

Solutions for problem set 1.
*Correction:* new (correct) proof of Problem 1(a).

Problem set 2, due April 15.
*Correction:* in Problem 4(f), the right hand side of the isomorphism should also be cohomology with support in Z.

Solutions for problem set 2.

Problem set 3, due April 29.
*Corrections:* in Problem 3(b), one has to assume that X is noetherian, otherwise the statement is false.
See also
this MathOverflow post.
In Problem 4(c), You may use that H^1(X,O_X*) can be computed using Čech
cohomology.
In the hint for Problem 5, the tensor product of the
complexes is *homotopy equivalent* to the Čech complex for the tensor product.

Solutions for problem set 3.

Practice problems for Midterm 1.

Solutions for practise sheet 1.

Problem set 4, due May 13.
*Correction:* Added some hints for the second part of Problem 4.

Solutions for problem set 4;
for Problem 4 (now complete):
solution for Problem 4.

Problem set 5, due May 20.

Solutions for problem set 5.
*Correction:* corrected the solution for Problem 5.

Problem set 6, due May 27.
*Correction:* In Problem 5, the curve should be proper. In Problem 4, the pullback of L_2 is L_1.

Solutions for problem set 6.

Problem set 7, due June 3.
*Correction:* We changed Problem 4 (the old problem will be on a later problem set).

Solutions for problem set 7.

Review sheet, due June 17.

Solutions for review sheet.
*Correction:* Some corrections and more details in Problems 2, 3, 4c, 7 and 8.

Problem set 8, due June 24.

Solutions for problem set 8.

Problem set 9, due July 1.

Solutions for problem set 9.

Problem set 10, due July 8.
*Correction:* In Problem 2(c), calculate the cohomology of the *structure sheaves* of the fibres.
In Problem 1(c), the equation of the curve should be y^2=x^3+x^2.

Solutions for problem set 10.
*Correction:* Corrected Problem 2.