Math 673: Algebraic Geometry






WHEN: MWF 12 – 12:50 pm

WHERE: ENG E 204 // in the oval with good weather…that may take a while.

WHAT: The time has finally come when I have to face my fear of Schemes! Yikes!




Togliatti Surface

(quintic with 31 ordinary double points)



Algebraic geometry in the modern age...when the dictionary between algebra and geometry has been tightened to generalize into an algebraic/categorical framework the intuitive geometric notions that let to the origin of the subject. Pluses, you can prove a lot more stuff, and you can really prove it! Minuses, the subject becomes somewhat know what is the geometric intuition you are trying to generalize. What we will attempt to do in this class is to introduce the theory of Schemes by seeing how it arises as a natural generalization of the material we covered last semester. The book we will mostly follow is a very friendly introduction to the subject. It should be freely available on SpringerLink. If you don’t know how to get it, ask Douglas.




HOMEWORK (really? Really!)

Feb 1st. Exercises I-2, I-3, I-4, I-17.

Feb 8th. Exercises I-21, I-24, I-25, I-28.

Feb 15th. In class homework. Exercises I-30,...,I-35.

Feb 22nd. For each of the following rings R, describe the points of Spec(R), then compute the local dimension at every point, and the Zariski cotangent space. Observe whether the scheme is reduced, irreducible, smooth:

a)   R= C[x,y]/(y-x^2)

b)  R= C[x,y]/(y^2-x^2+x^3)

c)   R= C[x,y]/(xy)

d)  R= C[x,y]/(y^2,x^2)

e)   R= C[x,y]/(x^2)

Mar 1st. Exercises I-42, I-43, I-44.

Mar 8th. Exercises I-45, I-46, I-48, I-49.

Mar 29th. Exercises II-9, II-10, II-11, II-12.

Apr 5th. Exercises II-15, II-23.

Apr 22nd. Exercises III-1, III-2, III-3, III-4, III-5.



1.   Robin Hartshorne. Algebraic geometry made hard.

2.   Joe Harris. Algebraic geometry, a first course.

3.   David Eisenbud. Commutative algebra with a view towards algebraic geometry.

4.   Miles Reid. Undergraduate Algebraic Geometry.

5.   Igor Shafarevich. Basic Algebraic Geometry.