# Course Identification

## Lecturers and Teaching Assistants

## Course Schedule and Location

## Field of Study, Course Type and Credit Points

## Comments

## Prerequisites

A semester course in algebraic geometry: algebraic varieties, morphisms, commutative rings and modules over them, dimension, smoothness.

## Restrictions

## Language of Instruction

## Attendance and participation

## Grade Type

## Grade Breakdown (in %)

## Evaluation Type

**No final exam or assignment**

## Scheduled date 1

## Estimated Weekly Independent Workload (in hours)

## Syllabus

[1] Algebraic curves and their non-singular models

[2] Riemann-Roch theorem - elementary approach

[3] Sheaves, quasi-coherent sheaves, Serre's theorem, coherent sheaves, Nakayama's lemma

[4] Cohomologies

[5] Higher cohomological operations with sheaves. Base change

[6] Divisors, invertible sheaves, Picard group

[7] Riemann-Roch theorem and applications.

## Learning Outcomes

Upon successful completion of this course students should be able to:

[1] Describe the basic notions of commutative algebra and algebraic geometry.

[2] Translate problems from algebra to geometry and vice versa.

[3] Use powerful algebraic techniques in geometric problems.

[4] Solve abstract algebraic problems by using acquired geometric intuition.

[5] Access modern literature in the broad fields of algebra and geometry.

## Reading List

[1] Atiyah-Macdonalds "Introduction to commutative algebra"

[2] Eisenbud "Commutative Algebra With a View Toward Algebraic Geometry"

[3] Kempf "Algebraic varieties"

[4] A course by A. Gathmann

http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf