
Course unit
ALGEBRAIC GEOMETRY 1
SC02119737, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/03 
Geometry 
8.0 
Course unit organization
Period 
Second semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
4.0 
32 
68.0 
No turn 
Lecture 
4.0 
32 
68.0 
No turn 
Prerequisites:

Many results are based on results from commutative algebra. Basic knowledge of commutative algebra (corresponding to roughly the first half of the commutative algebra course) is recommended. 
Target skills and knowledge:

Knowledge of the basic concepts, constructions and techniques of algebraic geometry. Competence in relating the different properties of algebraic varieties and the main theoretical results about them. Problem solving skills in algebraic geometry. 
Examination methods:

Written exam, possibly taking homework assignments into account. 
Assessment criteria:

Mastering the key techniques and concepts of algebric geometry.
Competence in applying the theoretical results on algebraic varieties and their properties in specific examples, for instance in the solution of exercises.
Problem solving skills in algebraic geometry. 
Course unit contents:

This course is intended as a foundational course in algebraic geometry, starting from the basics of the subject and progressing to more avanced techniques such as the study of sheaves and schemes.
Contents:
Affine varieties.
The Zariski topology.
The sheaf of regular functions on a variety.
Morphisms of varieties.
Projective varieties.
Dimension of a variety.
Introduction to schemes. 
Planned learning activities and teaching methods:

Lectures. Homework, in the form of weekly exercise sheets. The weekly exercise sheets are discussed during problem sessions. 
Additional notes about suggested reading:

The course is based on Andreas Gathmann's lecture notes at TU Kaiserslautern, available online at
https://www.mathematik.unikl.de/~gathmann/de/alggeom.php
The structure of the course will follow the 2003 version of Gathmann's notes, with some material added or substituted from the version of 2014.
There are weekly exercise sheets available on the Moodle page of the course.
Additional references:
 for the parts about affine varieties and varieties in projective space, a good complementary reference is
I. R. Shafarevich, Basic algebraic geometry. 1. Varieties in projective space. Second edition. Translated from the 1988 Russian edition and with notes by Miles Reid. SpringerVerlag, Berlin, 1994. xx+303 pp. ISBN: 3540548122
 for the part about schemes and sheaves one may refer to
I. R. Shafarevich, Basic algebraic geometry. 2.Schemes and complex manifolds. Second edition. Translated from the 1988 Russian edition and with notes by Miles Reid. SpringerVerlag, Berlin, 1994. xiv+269 pp. ISBN: 3540575545
I. G. Macdonald, Algebraic geometry. Introduction to schemes. W. A. Benjamin, Inc., New YorkAmsterdam 1968 vii+113 pp. 
Textbooks (and optional supplementary readings) 

Innovative teaching methods: Teaching and learning strategies
 Lecturing
 Problem based learning
 Problem solving
Innovative teaching methods: Software or applications used
 Moodle (files, quizzes, workshops, ...)
 Latex
Sustainable Development Goals (SDGs)

