General information
Course type  AMUPIE 
Module title  Algebraic curves 
Language  Enghlish 
Module lecturer  prof. dr hab. Wojciech Gajda 
Lecturer's email  gajda@amu.edu.pl 
Lecturer position  professor 
Faculty  Faculty of Mathematics and Computer Science 
Semester  2024/2025 (summer) 
Duration  60 
ECTS  6 
USOS code  06S2MA04F16224 
Timetable
Module aim (aims)
This is an introductory course in algebraic geometry aimed at students of mathematics of all specializations. We will start with setting up firm foundations and discuss in detail theory of curves over complexes and then more generally over any algebraically closed field. Next we turn to affine and projective algebraic varieties and explain morphisms, local rings, function fields, dimension and nonsingular varieties. This will allow us to prove RiemannRoch over an algebraically closed field. The main aim of the course is to introduce students to the basics of algebraic geometry, explore a large variety of different examples, and prove some important theorems of the trade.
Prerequisites in terms of knowledge, skills and social competences (where relevant)
Knowledge of abstract algebra at the level of an undergraduate level: Groups, Rings an Fields will be very helpful.
Syllabus
Syllabus: 

Week 1 
Conics and Cubics 
Week 2 
Higher degree curves 
Week 3 
Bezout Theorem 
Week 4 
RiemannRoch Theorem 
Week 5 
RiemannRoch II 
Week 6 
Affine varieties I 
Week 7 
Affine varieties II 
Week 8 
Affine varieties III 
Week 9 
Smoothness 
Week 10 
Dimension 
Week 11 
Projective varieties I 
Week 12 
Projective varieties II 
Reading list
 William Fulton, Algebraic Curves, An Introduction to Algebraic Geometry, Benjamin (1969)
 Philip Griffiths, Introduction to Algebraic Curves, AMS (1989)
 Thomas Garrity et. al., Algebraic Geometry, A problem Solving Approach, AMS (2010)
 Klaus Hulek, Elementary Algebraic Geometry, AMS (2003)
 I.R.Shafarevich, Basic Algebraic Geometry I, II, Springer (1994)
 Robin Hartshorne, Algebraic Geometry, Springer (1977)
 Ulrich Goertz, Torsten Wedhorn, Algebraic Geometry I, Vieweg (2010)