MATH 465/565: Introduction to Algebraic Geometry

Instructor: Evan Bullock
Office: Herman Brown 408
Office Hours: Wednesday 3:30-5:00pm, Thursday 1:30-3:00pm, or by appointment.
Email:
Time: 01:00PM - 01:50PM MWF
Location: HB 453
Textbook:  Shafarevich, Basic Algebraic Geometry I, available from the campus store
TA: Letao Zhang

Handouts

For those not familiar with tensor products, I have written a brief summary of the definition of the tensor product (over a field).  For more information, including the more general definition for modules, see the section from Atiyah & MacDonald or from Dummit & Foote about tensor products.

Here's the section on symmetric and alternating tensors, also from Dummit & Foote.

Algebraic Geometry: A First Course, by Joe Harris, has a very nice treatment of Veronese and Segre varieties and Grassmannians.

I've written some lecture notes on Grassmannians.

I've collected some additional problems on dimension.

Homework


Homework 1, due Friday, January 21
Homework 2, due Friday, January 28
Homework 3, due Monday, February 7
Homework 4, due Friday, February 11
Homework 5, due Friday, February 18
Homework 6, due Friday, March 11
Homework 7, due Wednesday, March 23  (see notes)
Homework 8, due Friday, April 1 (hint added to #7)
Homework 9, due Friday, April 8 (hints added to #1 and #6, hint in #2a replaced with a hint that will be more helpful in part b)
Homework 10, due Friday, April 15 (hint added to #6b)
Homework 11, due Friday, April 22 (extra credit problem added)

Homework will be assigned weekly, due on Fridays unless announced otherwise.   Homework will be very important in this class, and you should you work on every assigned problem: in some cases, results developed in the homeworks may be used in class and likewise some results may be stated without proof in class with the proof left to you on the homework. 

I encourage you to talk to other students about the homework problems, but you must write up your own solutions and they should reflect your own understanding. 

If you'd like to typeset your homeworks on a computer, I would strongly suggest learning to use LaTeχ.  The only difference between 465 and 565 is that students in Math 565 are required to typeset all of their homeworks in LaTeχ.  Please include figures in your homework write-ups whenever you think they might be useful to the grader in understanding your solutions.  Your figures may be hand-drawn even if you are signed up for 565.

Description


The topic of the course is algebraic varieties (common zero sets of polynomial equations) in affine and projective space.

Topics may include: plane algebraic curves, Hilbert's Nullstellensatz, regular and rational maps, products of quasi-projective varieties and the Segre embedding, completeness of projective varieties, dimension theory, degree, Bézout's theorem, tangent space and tangent cone, blow-ups and resolution of singularities, Grassmannians and the Plücker embedding.

Prerequisites


Math 463/563, or equivalent, is a prerequisite for this class: students should be familiar with rings, fields, and Galois theory.

Exams

There will be an open-book take-home midterm and a closed-book take-home final:

Midterm Exam 1, due Friday, February 25 (solutions to selected problems)

Grades

Homework counts as 60% of your grade.  The final exam counts for 30% the midterm counts for 10%. 

Disability Support

If you have a documented disability that will impact your work in this class, please contact me to discuss your needs. Additionally, you will need to register with the Disability Support Services Office in the Allen Center.

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