WHAT: Introduction to Algebraic
geometry through toric geometry.
Algebraic geometry is a very broad
and abstract branch of mathematics, and it is somehow gained a reputation of
being unaccessible. This year we are going to experiment with approaching the
general ideas in modern Algebraic geometry through the gateway of Toric
Geometry. Toric varieties are compactifications of an algebraic torus, and this
additional structure creates a beatiful connection to discrete geometry and
combinatorics which then makes computations available. And hopefully having a
large collection of examples and opportunities to get our hands dirty will help
in internalizing the more general and abstract concepts in Algebraic Geometry.
The book we will follow is one click away:
HOMEWORK (really? Really!) Rules: You are
allowed to work in groups AND to write one solution for each group. Also,
don’t expect me to grade the homework *I may or may not*, but do bring to
my attention problems that you would like feedback on.
1.Due February 9th: Exercises
6.1.6, 6.1.8 (a)(b), 6.1.9.
2.Due March 2nd:Exercises 6.2.9, 6.3.5, 6.4.6.
There are several good
exercises in the book and we will be mostly drawing from that pool!
3.William Fulton. Introduction to Toric Varieties.
4.Joe Harris. Algebraic geometry, a first course.
5.David Eisenbud. Commutative algebra with a view towards algebraic
geometry.
6.Miles Reid. Undergraduate Algebraic Geometry.
7.Igor Shafarevich. Basic Algebraic Geometry.
MY LECTURE PLAN: this is just
my log of the classes, so it is in no way meant to be a seriously written thing.
But I do collect here your questions and notes to myself about how to answer
them. Making it available in case it can be helpful. Here.
FIRST
SEMESTER STUFF (FOR MEMORIES AND REFERENCES)
READINGS
1.Sep 2nd (6am) Section 1.1
2.Sep 3rdSection 1.2
3.Sep 15thSection 1.3
4.Sep 19th Section 2.0
5.Sep 22nd Section 2.1, 2.2
6.Sep 29th Section 2.3, 2.4
7.Oct 6th Sections 3.0, 3.1
8.Oct 13th Sections 3.2, 3.3
9.Oct 20th Sections 3.4, 4.0
10.Oct 27thSection 4.1
11.Nov 3rd Sections 4.2, 4.3
12.Nov 10th Sections 5.0, 5.1, 5.2
13.Nov 17th Sections 5.3, 5.4
POSSIBLE PROJECTS:
1.Work out the example of a
complete but not projective toric variety in Section 4.2 (Tim H.)
2.Exercise 4.2.13 on tropical geometry
3.C^3/Z_3 (Douglas)
4.C^3/(Z_2+Z_2) and 4 crepant
resolutions (Ben S.)
5.Line bundles on P^2 (Nand)
6.Reductive groups (Josh)
7.Toric varieties are
Cohen-Macaulay (Zach)
8.Losev-Manin spaces (Ben C.)
9.Mirror quintic (Example
5.4.10) (Tim M.)
10.Proper toric schemes over a DVR (Joan)
HOMEWORK (really? Really!) Rules: You are
allowed to work in groups AND to write one solution for each group. Also,
don’t expect me to grade the homework *I may or may not*, but do bring to
my attention problems that you would like feedback on.
3.Oct 10th Exercises 2.3.5,
2.3.8 (further think about what is the projective variety they define, and what
are the maps that embed it in some projective space) , 2.4.1, 2.4.6
4.Nov 3rd (A) Work out completely
the blowup of P^3 at a torus invariant line. (B) Prove that the blow-up of P^2 at two torus fixed points is
isomorphic to the blow up of P^1xP^1 at one torus fixed point.
5.Nov 7th Exercises 4.1.2, 4.1.4
There are several good
exercises in the book and we will be mostly drawing from that pool!
CLASS GALLERY:
Characters, one parameter
subgroups, and orbits with and without limits.