Homework and Syllabus
Algebraic Geometry
MATH 6170 Spring 2021
| Date | Topics |
Reading |
Homework |
| Friday January 15 | Introduction to algebraic geometry What is algebraic geometry, and what will we cover in this class. |
||
| Monday January 18 |
MLK
DAY |
NO CLASS |
NO CLASS |
| Wednesday January 20 |
Varieties Affine varieties |
Hartshorne I.1 |
|
| Friday January 22 |
Varieties
continued Affine varieties continued |
HW 1 Hartshorne Exercises I.1 |
|
| Monday
January 25 |
Varieties continued Projective varieties |
Hartshorne I.2 |
|
| Wednesday
January 27 |
Varieties continued Projective varieties continued |
||
| Friday
January 29 |
Varieties continued Morphisms |
Hartshorne I.3 |
HW 2 Hartshorne Exercises I.2 |
| Monday February 1 |
Varieties
continued Morphisms continued |
||
| Wednesday February 3 |
Varieties
continued Rational maps |
Hartshorne I.4 |
|
| Friday February 5 |
Varieties
continued Rational maps continued |
HW 3 Hartshorne Exercises I.3 |
|
| Monday February 8 |
Varieties
continued Rational maps continued |
||
| Wednesday February 10 |
Varieties
continued Nonsingular varieties |
Hartshorne I.5 | |
| Friday February 12 |
Varieties
continued Nonsingular varieties continued |
HW 4 Hartshorne Excercises I.4 |
|
| Monday February 15 |
Varieties
continued Nonsingular varieties continued |
||
| Wednesday February 17 |
WELLNESS
DAY |
NO CLASS |
NO CLASS |
| Friday February 19 |
Varieties
continued Intersections in projective space |
Hartshorne I.7 |
HW 5 Hartshorne Excercises I.5 |
| Monday
February 22 |
Varieties continued Intersections in projective space continued |
||
| Wednesday
February 24 |
Varieties continued Intersections in projective space continued |
||
| Friday
February 26 |
Varieties continued Intersections in projective space continued |
HW 6 Hartshorne Excercises I.7 |
|
| Monday March 1 |
Varieties
continued What is algebraic geometry, again? |
Hartshorne I.8 | |
| Wednesday March 3 |
Review
exercises Quadric surfaces 1: birational but not isomorphic to projective space, and projection from a point |
||
| Friday March 5 |
Review
exercises continued Quadric surfaces 2: The Segre Embedding, and the connection to representation theory (Part 1) |
HW 7 Review and Revise Exercises |
|
| Monday
March 8 |
Review exercises
continued Quadric surfaces 3: First introduction to cohomology |
||
| Wednesday
March 10 |
Review exercises
continued Quadric surfaces 4: Connection to representation theory (Part 2) |
||
| Friday March 12 | Review
exercises continued Quadric surfaces 5: Resolving the birational map to projective space, and the blow-up of a point as an introduction to line bundles |
HW 8 Review and Revise Exercises |
|
| Monday March 15 | Review
exercises continued The blow-up of a point as an introduction to line bundles continued |
||
| Wednesday March 17 |
Review
exercises continued Quadric surfaces 7: Singular quadric surfaces, the quadric cone, resolving singularities with blow-ups |
||
| Friday March 19 |
Review
exercises continued Resolving locally planar singularities: blowing-up curves in the plane to resolve singularities |
HW 9 Review and Revise Exercises |
|
| Monday March 22 | SPRING PAUSE Review exercises continued Affine and quasi-affine varieties |
SPRING PAUSE |
SPRING PAUSE |
| Wednesday March 24 | SPRING PAUSE Review exercises continued Affine and quasi-affine varieties continued, and introduction to the Cremona transformation |
SPRING PAUSE |
SPRING PAUSE |
| Friday March 26 | SPRING PAUSE Review exercises continued The Cremona transformation continued |
SPRING PAUSE |
SPRING PAUSE |
| Monday
March 29 |
Vector bundles Introduction and definition, line bundles |
Huybrechts 2.2 | |
| Wednesday
March 31 |
Vector bundles
continued Line bundles and rational maps to projective space |
Huybrechts 2.3 | |
| Friday
April 2 |
Vector bundles
continued Divisors of sections of line bundles, and rational equivalence |
HW 10 Review and Revise Exercises |
|
| Monday April 5 |
Vector
bundles continued Cartier divisors and rational equivalence |
||
| Wednesday April 7 |
Vector
bundles continued Line bundles, divisors, and Cartier divisors |
||
| Friday April 9 | Vector
bundles continued Introduction to tangent bundles |
HW 11 Huybrechts Exercises 2.2.2-9 For 2.2.6-7 assume that X is a variety such that for every affine open subset, the ring of regular functions is a UFD. In all the problems, replace the condition of compactness with projectivity. |
|
| Monday
April 12 |
Vector bundles
continued Tangent bundles constructed geometrically |
||
| Wednesday
April 14 |
Vector bundles
continued Canonical bundles defined, and adjunction |
||
| Friday
April 16 |
Vector bundles
continued Canonical bundles and Kodaira dimension |
HW 12 Huybrechts Exercises 2.3.2, 2.3.3, 2.3.6, 2.3.7 |
|
| Monday April 19 |
Cohomology Introduction to cohomology, cohomology of vector bundles, Serre duality, and Riemann--Roch for curves |
Huybrechts Appendix B Hartshorne III.1-2 |
|
| Wednesday April 21 |
Cohomology
continued Applications to curves, and cohomology of line bundles on projective space. |
Hartshorne III.5 and IV.1 |
|
| Friday April 23 |
Cohomology
continued Vanishing theorems, characteristic classes, and Riemann--Roch. |
Hartshorne III.7,
A.3-4 Huybrechts 5.1-2 |
HW 13 Huybrechts Exercises 2.4.1, 2.4.2, 2.4.4, 2.4.6 |
| Monday
April 27 |
Review |
||
| Wednesday
April 29 |
Review |
||
| Friday April 30 | READING DAY |
NO CLASS |
NO CLASS |
| Sunday
May 2 |
FINAL
EXAM 1:30 - 4:00 PM |
FINAL EXAM |
FINAL EXAM |
I strongly encourage
everyone to use LaTeX for typing homework. If you have
a mac,
one possible easy way to get started is with texshop.
If you are using linux,
there are a number of other possible ways to go, using
emacs, ghostview, etc. If you are using windows,
you're on your own, but I'm sure there's something online.
Here is a sample homework file to use: (the .tex
file, the .bib
file, and the .pdf
file). This site can help you find LaTeX symbols
by drawing: http://detexify.kirelabs.org/classify.html.
You may also want to try https://cocalc.com
(formerly https://cloud.sagemath.com/)
or https://www.overleaf.com
(formerly https://www.sharelatex.com/)
for a cloud version.