International school on algebraic geometry and algebraic groups

• Michel Brion (University of Grenoble)
• Phùng Hồ Hải (Institute of Mathematics, VAST)
• Joao Pedro dos Santos (University of Montpellier)

• Institute of Mathematics, VAST
• International Centre for Research and Post-graduate Training in Mathematics, IM-VAST
• International Research Laboratory France-Vietnam in Mathematics and its Applications (CNRS-VAST-VIASM)
• Vingroup Innovation Foundation

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.

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

1. R. Hartshorne. Algebraic geometry. Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
2. 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:

1. Mumford, Abelian Varieties, Chapter II (+ Chapter III for the theorem of the cube)
2. Milne, Abelian Varieties. https://www.jmilne.org/math/CourseNotes/AV.pdf
3. 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:

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

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