Math 672/673: Toric Geometry

 

felipe

 

 

 

WHEN: MWF 12 – 12:50 pm

WHERE: ENG E 205 // in the oval with good weather

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:

ksmithbook

 

Cheatsheet for the First Semester

 

FINAL PROJECT (TeX source)

 

READINGS

1.   Jan 23rd Section 6.0

2.   Jan 26th  Section 6.1

3.   Feb 2nd Sections 6.2, 6.3

4.   Feb 9th Sections 6.4, 7.0

5.   Feb 16th Section 7.1

6.   Feb 23rd Sections 7.2,7.3

7.   April 8th Section 12.1

8.   April 10th Section 12.2

9.   April 13th Section 12.3, 12.4

10.                    April 20th Section 12.5, 13.1

 

 

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!

 

VARIOUS OTHER REFERENCES FOR THIS CLASS:

1.   Irena Swanson, Craig Huneke. Integral Closures of Ideals, Rings and Modules.

2.   John Milne. Notes on Algebraic Geometry.

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 3rd   Section 1.2

3.   Sep 15th Section 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 27th  Section 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.

1.   Sep 15th Exercises 1.1.6, 1.1.9, 1.1.11, 1.1.15.

2.   Sep 26th Exercises 1.2.8, 1.2.10, 1.3.12, 1.3.13

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.

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Quotient space (L) and non-normal variety (R)

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Cartier divisors and Picard Groups:

photo 2.JPG 

Sheaves of sections of a divisor:

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Quotients by tori:

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Total Coordinate Rings and their grading:

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Extended example: Second Hirzebruch Surface.

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