International school on Algebraic Geometry and Algebraic Groups

1. Automorphism groups of projective varieties
Michel Brion

2. Reductive groups
Ngô Bảo Châu

3. Singularities in algebraic geometry
Jungkai Chen

4. Commutative Algebra (CA)
Tran Nam Trung

5. Affine varieties and projective varieties (AV)
 Doan Trung Cuong (tutor Do Van Kien)

6. Algebraic schemes (AS)
Nguyen Dang Hop 

7. Cohomology of schemes (CS)
Nguyen Tat Thang (tutor Pham Thanh Tam)

8. Algebraic groups acting on varieties and their applications (AG)
Joao Pedro dos Santos (tutor Phung Ho Hai) 

 Schedule

 Content

1.Automorphism groups of projective varieties
Michel Brion

Abstract: The talk will survey results, examples and open questions on the automorphism group of a projective variety X. In particular, we will sketch a proof of the fact that Aut(X) has the structure of a "locally algebraic group" or "algebraic group with possibly infinitely many components"). In particular, the set of its connected components is a discrete group, whose structure will be discussed in the talk.

2. Reductive groups
Ngô Bảo Châu

3. Singularities in algebraic geometry
Jungkai Chen

Abstract. I will start from the definition of singularities and then work on various examples in dimension 1, 2 and 3. Especially, I will work on surface A-D-E singularities and quotient singularities. If time permits, I will introduce birational geometry, in which singular varieties appear naturally. 

4. Commutative Algebra (CA)
Tran Nam Trung

1. Commutative Rings:

- Local rings and graduated rings

- Localization & sets of associated prime ideals

- Noetherian and Artinian rings

- Hilbert's basis theorem 

2. Integral dependence:

- Cayley-Hamilton theorem and Nakayama lemma

- Integrally closed domain and Noether's Normalization Theorem

- Hilbert's Nullstellensatz  

3. Dimension of rings 

- Parameter system and dimension of the rings

- Hilbert's polynomial

- Krull's main ideal theorem

4. Regular rings

- Regular sequences and Koszul complex

- Regular rings

- Free resolutions and Hilbert's syzygy theorem

5. Affine varieties and projective varieties (AV)
Doan Trung Cuong 

Content: Relation between a system of polynomials and the set of solutions is basic for algebraic geometry. In this introductory course to affine varieties and projective varieties, I will explain in detail the bridge between algebra and geometry and other relations. The content includes 

- varieties, defining ideal, coordinate ring;

- regular function/map, rational function/map;

- dimension, tangent space;

- degree, genus, etc

Prerequisites: Noether ring, polynomial ring, general topology.

Literature:

- K. Hulek, Elementary algebraic geometry.

- R. Hartshorne, Algebraic geometry (1st chapter).

6. Algebraic schemes (AS)

Nguyen Dang Hop 

1. Sheaves and locally ringed spaces

2. Schemes, affine and projective schemes

3. Proper morphisms and completeness of the projective spaces

4. Divisors and the divisor class groups

Prerequisites: Abstract algebra (groups, rings, modules, fields), Linear algebra.

Literature:

Atiyah-MacDonald, An Introduction to Commutative Algebra.

Robin Hartshorne, Algebraic Geometry

7. Cohomology of schemes (CS)
Nguyen Tat Thang (tutor Pham Thanh Tam)

1) Cohomology of sheaves

- Derived functors

- Cohomology of sheaves

- A vanishing Theorem of Grothendieck

2) Cohomology of affine schemes

- Sheaves of modules, (quasi)-coherent sheaves

- Cohomology of affine schemes

3) Cech cohomology

- Cech cohomology associate to a covering

- Relation of Cech cohomology and sheaf cohomology

4) Serre duality Theorem 

- Sheaves of differentials

- Ext groups and sheaves

- Duality Theorem.

8. Algebraic groups acting on varieties and their applications (AG)
Joao Pedro dos Santos (tutor Phung Ho Hai) 

1. Functors and Yoneda's Lemma. Representability and co-representability. Examples of moduli problems.

2. Group schemes and representations (comodules etc). Examples. Equivariant immersions of affine schemes.

3. Cartier's theorem. Lie algebras.

4. Affine quotients, Quotients by finite group schemes. Linear reductivity and Hilbert's theorem.

5. Linearized line bundles, stability and semi-stability and projective quotients. The null-cone. 

6. Tentative: Moduli space of vector bundles on a curve.

Prerequisites: Completions of rings. Dimension theory for varieties.

Dimension of fibres. (Exercise 3.22 of Ch II of Hartshorne, for example.)

Proj of a graded algebra. The notion of invertible sheaves; pull-back and push-forward of coherent modules.