The course intends to give a first introduction to the basic notions and techniques of algebraic geometry.
We start by introducing the basic objects of algebraic geometry, namely algebraic varieties (affine, projective, general) and the morphisms between them. We then treat basic notions such as dimension, tangent space, differential 1-forms, smoothness. Towards the end we will focus on curves and surfaces, the relation between complex curves and Riemann surfaces, the theorem of Riemann-Roch for curves, and intersection theory on surfaces including a proof of Bezout's theorem for the projective plane.
The prerequisites for this course are the standard undergraduate algebra courses on groups, rings and fields (see for example the course notes Algebra 1 and 2 on http://websites.math.leidenuniv.nl/algebra/ or chapter I-IV of S. Lang's 'Algebra', Springer GTM 211)
In particular, we assume that the student is familiar with the following notions: group, (commutative) ring, field, homomorphism, ideal, quotient ring, polynomial ring, domain, fraction field, principal ideal domain, unique factorization domain, topological space, continuous map, connected, Hausdorff, compact.
We will use the language of categories when useful, and with the necessary explanations. This material is reviewed during the Mastermath "Intensive course on Categories and Modules" that we strongly recommend taking in the first week of the Fall semester. The course will further illustrate the use of the language of categories.
We will use some results from Commutative Algebra without giving proofs. It is recommended to follow the Commutative Algebra course that is offered on the same day at the same location, but this is not necessary.
Rules about Homework / Exam
There will be two take-home assignments during the semester, and a final written exam. For the final grade, the take-home assignments count for 40%, and the written exam for 60%.