As time permits, we will discuss the statement of the Riemann Roch theorem.

]]>- Read up through page 11 completing the section on intersection numbers. The intersection number is the dimension of a vector space. Why is that vector space finite dimensional?
- Review projective plane curves, translating what Milne does back into the language of Proj(S.).
- Learn the statement of Bezout’s theorem.
- Read curves of degree two. Is it really true that C can’t be singular?
- Read up to the end of page 20.

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Read the introduction about some of the uses of elliptic curves (including sphere packings and cryptography!)

pages 5-6 should be review.

- Read the definition of a nonsingular curve.
- Work example 1.5 to get a test for a Weierstrass polynomial to define a nonsingular curve.
- Read Prop 1.8 on intersection numbers.
- Read up through page 11 completing the section on intersection numbers
- Review projective plane curves, translating what Milne does back into the language of Proj(S.).
- Learn the statement of Bezout’s theorem.

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- We will discuss a later result (8.6) from Humphreys: let G be an affine algebraic group over an algebraically closed field k, then G is isomorphic to a closed subgroup of GL(n) (over k) for some n.
- 7.3. Let G be an affine algebraic group. Show that G^0 is a normal subgroup of finite index.
- 7.4. Show that the Zariski closure of a subgroup is a subgroup.
- 7.4.B. Show that the kernel and images are closed subgroups, and that the image of the connected component of the identity is the connected component of the image.
- 8.1. State what an action is in the category of k-varieties.
- Show that GLn acts on P^n, on the flag variety.

We will start on our discussion of elliptic curves as time permits.

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- Read the section 6 on complete varieties (pages 45-46) and prepare to discuss Proposition 6.1.
- Read and prepare the proof that P^n is complete.
- Learn the definition of G_m, G_a, GL(n), SL(n), O(n), SO(n), Sp(2m) as algebraic groups. In each case, describe the morphism from A to A\otimes_k A that is dual to the multiplication mu:GxG->G, where G = Spec(A).
- 7.3. Let G be an affine algebraic group. Show that G^0 is a normal subgroup of finite index.
- 7.4. Show that the Zariski closure of a subgroup is a subgroup.
- 7.4.B. Show that the kernel and images are closed subgroups, and that the image of the connected component of the identity is the connected component of the image.
- 8.1. State what an action is in the category of k-varieties.

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- Read 6.4. (automorphisms of the projective line) and prepare the proof that that the automorphism group is PGL(2,k).
- Read the section 6 on complete varieties (pages 45-46) and prepare to discuss Proposition 6.1.
- Read the proof that P^n is complete.
- Show how the product of two elliptic curves has the form Proj(S_.).
- Learn the definition of algebraic group and abelian variety.

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- Work through the details of the construction of the Grassmannian as a k-scheme.
- Work the the construction of the flag variety
- Read 6.4. (automorphisms of the projective line) and prepare the proof that that the automorphism group is PGL(2,k).
- Read the section 6 on complete varieties (pages 45-46).

For elliptic curves, we will use Milne’s book on elliptic curves.

For the theory of linear algebraic groups, we will use Humphreys book.

We will start with Humphrey’s book, moving quickly through a review of algebraic geometry from the first section of the book. Skim through the first 12 pages, which should all be familiar.

- Let k be a field. Check that in the category of affine k-schemes, the (categorical) product of two affine lines (A^1) over k is the plane A^2.
- Read the construction of page 13 (Prop. 1.7) that identifies the product of two projective spaces with a closed subset of a larger projective space. We can use this to give a provisional definition of the product of projective spaces.
- page 14. Learn the definition of the Grassmannian and flag variety.
- Skim to section 2.5.
- We will discuss the “Hausdorff axiom” in Section 2.5.

Monday will cover the algebraic interlude (Section 7.2: Lying Over and Nakayama).

After that, we will return to 7.1, then continue with 7.3, and so on.

Read the definitions of integral ring homomorphism and integral extension of rings. Read Lemma 7.2.1.

- Exercise 7.2.A.
- Exercise 7.2.B.
- Exercise 7.2.C.
- Exercise 7.2.D.

Understand the statement of the Lying over theorem (7.2.5) and its geometric translation in 7.2.6. The proof is optional.

- Exercise 7.2.F (The going-up theorem is an important exercise).

Read the two versions of Nakayama’s lemma. As time permits, work versions 3 and 4 of Nakayama’s lemma in Exercises 7.2.G and 7.2.H. As time permits, work 7.2.I and 72.J.

Understand Grothendieck’s lesson in 7.0.1 about replacing properties of objects with properties of morphisms. Read 7.1.1. about the three properties to expect of reasonable morphisms. Learn the definition of open embedding and open subscheme. Make sure that you are clear about the distinction between “local on the target”, “local on the source”, and “affine-local on the source” (in Definition 7.1.2).

- Exercise 7.1.A. Open embeddings are satisfy the first two properties of being a reasonable morphism.
- Exercise 7.1.B. Open embeddings satisfy the third property as well. We have not covered fibered products of schemes. For this exercise, you need to construct the fiber product in a special case. I suggest that you spend time working through this fibered product.
- Exercise 7.1.C. Open embeddings and locally Noetherian schemes.

Move forward to Section 7.3 on a gazillion finiteness conditions on morphisms. This is a long section that will spill over into Friday.

Read 7.3.1. What is the definition of a quasicompact morphism? Can you state what a quasicompact scheme (in the old terminology) is in terms of a morphism? With each new property of morphism ask yourself if the definition is (affine) local on the source or target. I suggest skipping the parts about quasiseparated morphisms unless there is time to spare at the end of class.

- Exercise 7.3.A.
- Exercise 7.3.B.a.
- Exercise 7.3.C.a

Read the definition of an affine morphism. I suggest reading over the proofs on pages 209-210 at home, but not to spend time in class with them unless there are specific questions.

Section 7.3 is called a “gazillion finiteness conditions” on morphisms. Give yourself time to sort out all of these different conditions. Read 7.3.8. What is a finite morphism? What is the difference between a finite B-algebra and a finitely generated B-algebra?

- Exercise 7.3.G. Finiteness is affine-local on the target.
- Work through the three examples in class of finite morphisms on page 211, especially example 3. Try to understand the third example both intuitively and scheme-theoretically.
- Exercise 7.3.H. Finite morphisms to a point.
- Exercise 7.3.I. Finite morphisms compose.
- Exercise 7.3.J. Finite morphisms are projective.
- Exercise 7.3.K. Finite morphisms have finite fibers.

Move forward with definitions about and exercises on integral morphisms, morphisms of finite type, and quasi-finite morphisms as time permits.

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