MATH-535 / 5 credits

Teacher: Schlegel Mejia Sebastian

Language: English

## Summary

This course is aimed to give students an introduction to the theory of algebraic curves, with an emphasis on the interplay between the arithmetic and the geometry of global fields. One of the principle goals will be to understand the geometric formulation of global class field theory.

## Content

• Algebraic curves, line bundles
• Riemann-Roch and Serre duality for curves
• Picard variety of curves
• Adelic language of global fields and geometric class field theory

## Keywords

Algebraic geometry, algebraic curves, global fields, class field theory

## Required courses

• Linear algebra
• Group Theory
• Rings and Modules
• Modern Algebraic geometry

## Recommended courses

• Algebraic topology
• Differential geometry
• Complex Analysis

## Learning Outcomes

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

• Analyze basic problems in algebraic geometry of curves and solve them.
• Use the statements of basic theorems like Riemann-Roch, Serre duality, etc, and understand their proofs
• Reason intuitively about the geometry and topology of curves over the complex and finite fields.

## Teaching methods

2h lectures+2h exercise sessions weekly.

Oral Exam

## Supervision

 Office hours Yes Assistants Yes Forum No

## Bibliography

• Hartshorne, Algebraic Geometry
• R. Miranda, Algebraic Curves and Riemann Surfaces
• J. P. Serre, Algebraic Groups and Class Fields

## In the programs

• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Algebraic geometry III - selected topics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Algebraic geometry III - selected topics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional
• Semester: Spring
• Exam form: Oral (summer session)
• Subject examined: Algebraic geometry III - selected topics
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Type: optional

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