Algebraic geometry III - selected topics

MATH-535 / 5 crédits

Enseignant: Rychlewicz Kamil Piotr

Langue: Anglais

Summary

This course is an introduction to intersection theory on algebraic varieties. An important aim of the course is to develop geometric intuition while using the language of schemes developed in the basic algebraic geometry course, thus building a solid foundation for further study.

Content

  • Recap: Divisors, sheaf cohomology and morphisms to Grassmannians, canonical bundles
  • Picard group of a variety. Jacobian variety of a curve
  • Riemann-“Roch theorem and Serre duality for curves
  • Intersection theory on smooth surfaces, numerical equivalence
  • Chow groups and Chow ring
  • Chern classes and Segre classes
  • Chow rings of Grassmannians
  • Bezout theorem
  • Introduction to classical Schubert calculus
  • Hirzebruch-Riemann-“Roch theorem and implications

Potential additional topics, depending on time constraints and™ preferences of the participants:

  • Intersection theory and torus actions: toric varieties and Bial‚ynicki-Birula decomposition
  • Intersection theory on singular varieties, characteristic classes
  • Cycle map and comparison between Chow rings and singular cohomology

Keywords

Algebraic geometry, Chow rings, characteristic classes, cohomology

Learning Prerequisites

Required courses

 

Or equivalent courses on linear algebra, groups, rings, modules and schemes.

 

Recommended courses

 

Learning Outcomes

By the end of the course, the student must be able to:

  • Analyze basic problems in intersection theory and solve them.
  • Reason intuitively about the geometry and topology of algebraic varieties over the complex and finite fields.
  • Use the statements of basic theorems like Riemann-Roch, Serre duality, etc, and understand their proofs
  • Prove basic results concerning geometry of smooth algebraic varieties and their subvarieties
  • Formalize heuristic arguments concerning intersections of subvarieties, including multiplicities
  • Compute Chow rings and numerical invariants of algebraic varieties
  • Apply the methods of intersection theory to recover classical results of Schubert calculus

Teaching methods

2h lectures+2h exercise sessions weekly.

Assessment methods

Oral Exam

Supervision

Office hours Yes
Assistants Yes
Forum No

Resources

Bibliography

We will follow mainly

 

  • William Fulton, Intersection Theory

 

The following also treats the topic and might be useful:

 

  • David Eisenbud and Joe Harris, 3264 and All That: A Second Course in Algebraic Geometry

 

The following books provide the general algebro-geometric background:

 

  • Robin Hartshorne, Algebraic Geometry
  • Qing Liu, Algebraic Geometry and Arithmetic Curves
  • Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry
  • Görtz-Wedhorn, Algebraic Geometry I & II

 

Ressources en bibliothèque

Moodle Link

Dans les plans d'études

  • Semestre: Printemps
  • Forme de l'examen: Oral (session d'été)
  • Matière examinée: Algebraic geometry III - selected topics
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Oral (session d'été)
  • Matière examinée: Algebraic geometry III - selected topics
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel
  • Semestre: Printemps
  • Forme de l'examen: Oral (session d'été)
  • Matière examinée: Algebraic geometry III - selected topics
  • Cours: 2 Heure(s) hebdo x 14 semaines
  • Exercices: 2 Heure(s) hebdo x 14 semaines
  • Type: optionnel

Semaine de référence

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