**Description**:

In this school, we aim at introducing beginning graduate students to algebraic geometry through specific and central themes of the theory. Instead of developing abstract material from the ground, we shall concentrate on obtaining a general picture of the following themes:

- Algebraic curves,
- Jacobian varieties,
- Abelian varieties,
- Algebraic groups.

By relying often on complex manifolds and leaving generality to the side, this school will equip the student with the basic reflexes and mental pictures necessary for understanding some of the most beautiful theories joining algebra and geometry.

More specifically, algebraic curves form an especially rich and accessible class of algebraic varieties. Their theory has its own specific features, and involves Jacobian varieties in an essential way. In turn, Jacobian varieties are classical and important examples of abelian varieties. These feature prominently in algebraic geometry, for their own sake and via geometric constructions such as those of the Picard and Albanese varieties. Algebraic groups provide a natural setting to the theory of Jacobian varieties and their generalizations, for example to singular curves; they also have great interest on their own.

Schedule: October 31-November 11, 2022

We shall have 4 courses, each having 10 hours of lectures + 5 hours tutorial.**Lecturers and tutors**

- Introduction to algebraic curves: Dao Van Thinh
- Jacobian of Curves: Ngo Dac Tuan (tutor Vo Quoc Bao)
- Abelian varieties: Joao Pedro dos Santos (tutor Phung Ho Hai)
- Algebraic groups: Michel Brion (tutor Dao Phuong Bac)

**Participants**: The school aims at postgraduate students (masters and PhDs) and senior undergraduate students from all over Vietnam as well as the neighboring countries.

**Mode of the school is hybrid**: online and offline.

**Financial support**: Depending on the budgets, we will offer financial support covering full or partial expenses for the students to participate in the school, including: travel expenses, lodging and meals. No participation fees are required.

To apply for the support please fill out the form and provide your (temporary) academic records and a letter of recommendation.

**Detailed program**:**Course 1: Introduction to algebraic curves**

Đào Văn Thịnh;

- Divisors, differentials, statement of the Riemann-Roch theorem and some applications.
- Normalisation and correspondence function fields smooth non-singular curves.
- Cohomology.

**References:**

Griffiths book on Algebraic curves**Course 2: Jacobian of Curves**

Ngô Đắc Tuấn;

In this course we develop the algebraic theory of Jacobians for smooth projective curves. We present the construction, some basic properties and if time permits, we study the case of hyperelliptic curves.

- Divisors, lines bundles on curves.
- The Hilbert schemes of points. Symmetric powers.
- The Picard functor and the construction of Jacobians.
- The theta divisor and self-duality of Jacobians.
- Some examples.

**References**

- R. Hartshorne. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
- J. S. Milne. Abelian varieties. In Arithmetic geometry (Storrs, Conn., 1984), pages 103–150. Springer, New York, 1986.

Course 3: Abelian varieties

Joao Pedro dos Santos

- Rigidity lemma and applications: every morphism of abelian varieties is the composition of a group homomorphism and a translation, abelian varieties are commutative.
- Rational maps to abelian varieties,
- Theorem of the cube and applications, symmetric line bundles; projectivity of abelian varieties; structure of n-torsion subgroups. Isogenies.
- Quotients by finite groups: existence and basic properties of the quotient of a quasi-projective variety by a finite group.

* References*:

- Mumford, Abelian Varieties, Chapter II (+ Chapter III for the theorem of the cube)
- Milne, Abelian Varieties. https://www.jmilne.org/math/CourseNotes/AV.pdf
- G. van der Geer and B. Moonen, Abelian varieties, online notes.

**Course 4: Algebraic groups**

Michel Brion

- Definitions and basic examples. Group actions, orbits, closed subgroups generated by images of
- morphisms.
- Structure of connected algebraic groups of dimension one via algebraic curves.
- Statement of Chevalley’s structure theorem: every connected algebraic group is an extension of
- an abelian variety by a connected linear group.
- Semi-abelian varieties and algebraically trivial line bundles on abelian varieties.
- The Albanese morphism to an abelian or semi-abelian variety.
- The Picard variety. Jacobians of singular curves.
- Duality between the Albanese and Picard varieties.

* References*:

- A. Borel. Linear Algebraic Groups. Graduate Text in Mathematics 126, Springer-Verlag, New York, 1991.
- J. S. Milne. Algebraic Groups. Cambridge Studies in Advanced Mathematics 170, Cambridge University Press, 2017.
- J-P. Serre. Morphismes universels et variété d’Albanese. Séminaire Claude Chevalley, Tome 4 (1958-1959), Exposé no. 10, 22 p.
- J-P. Serre. Algebraic groups and class fields. Graduate Texts in Mathematics 117. Springer- Verlag, New York, 1988.