OVERVIEW

Language: English

AIMS AND CONTENT

LEARNING OUTCOMES

The course objective is to present an elemental introduction to the concepts and methods of Modern Algebraic Geometry.

AIMS AND LEARNING OUTCOMES

The aim of the teaching is to provide the basic knowledge of modern algebraic geometry. Affine sets and their relation to commutative algebra will be seen. After that, the theory of sheaves will be introduced. Finally, the modern theory of algebraic varieties will be seen. At the end of the teaching the student will be able to describe the basic properties of an algebraic variety such as whether it is complete, separate, singular or not. He/she will be familiar with the theory of sheaves. The student will also be able to algebraise geometric problems to solve them rigorously.

PREREQUISITES

The teaching is a natural continuation of the teaching of IGS.  It is advisable to have taken all of the Algebra courses in the Bachelor's degree.

TEACHING METHODS

Teaching style: In presence

SYLLABUS/CONTENT

1. Affine algebraic sets, definitions, examples properties.
2. Noether's Normalization Lemma, Hilbert's Nullstellensatz and the structure theorem for affine sets will be proved.
3. Introduction to sheaves theory with emphasis on sheaves in algebraic geometry.
4. Definition of algebraic varieties with notable examples: projective spaces, product varieties, quotient varieties.  
5. Properties of algebraic varieties separability and completeness, regularity.
6. Proof of Chow's Lemma. 
7. Relationship between algebraic vector fibers and locally free and invertible bundles.

RECOMMENDED READING/BIBLIOGRAPHY

1. George R. Kempf: Algebraic Varieties , Cambridge University Press, 1993.

2. D. Mumford: The Red Book of Varieties and Schemes , Springer, 1999.

3. J. Dieudonne': Cours de geometrie algebrique vol 1 et 2 , Presses Universitaires de France , 1974.

4. J. le Potier: Geometrie Algebrique , DEA de Mathematiques de l' Universite 2001-2002

5. M. Reid: Undergraduate Commutative Algebra , London Math. Soc. Student Texts 29, 1995.

6. I.R. Shafarevich: Basic Algebraic Geometry I, (Second Edition), Springer Verlag, 1994.

7. L. Badescu, E. Carletti, G. Monti Bragadin: Lezioni di Geometria Analitica , Universita` di Genova, 2004 (www.dima.unige.it/~badescu).

8. Ellingsrud, Ottem: Algebraic Geometry 1, PDF online (2021). 

9  Qing Liu: Algebraic Geometry and Arithmetic Curves, Oxford Graduate Texts in Mathematics, 2006.

TEACHERS AND EXAM BOARD

Exam Board

MATTEO PENEGINI (President)

ELEONORA ANNA ROMANO

LESSONS

LESSONS START

The class will start according to the academic calendar.

EXAMS

EXAM DESCRIPTION

Oral Eaxam. 

Students with a certified DSA, disability or other special educational needs are advised to contact the lecturer at the beginning of the course in order to agree on teaching and examination methods that, while respecting the teaching objectives, take into account individual learning methods and provide suitable compensatory tools.

ASSESSMENT METHODS

Oral Examination. During the oral exam, the student should be able to: prove all the theorems presented in class, report correctly all definitions, and solve some simple algebraic geometry exercises such as determining whether a variety is separate, complete or regular.

Exam schedule