Algebraic curves

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 Riemann-Roch 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.

Pre-requisites 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	Riemann-Roch Theorem
Week 5	Riemann-Roch 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)