The goal of the course is to introduce concepts and techniques of algebraic geometry, up to a level where students can begin to study current research literature.
This is a second course in Algebraic Geometry, at the level of second year MSc or beginning PhD, for students who have decided to continue studying in this area. The subject matter is roughly chapters 2 and 3 of Hartshorne’s book: locally ringed spaces, schemes, commutative algebra, coherent sheaves, cohomology. However, all this will be treated with more usage of categorical language, and with some more advanced concepts and tools. For example, Grothendieck topologies are necessary for rigid analytic geometry and for etale cohomology, and are useful anyway as a unifying concept. Derived categories are nowadays a standard tool in homological algebra and therefore in geometry. If all goes well we will be able to illustrate the use of derived categories by giving a very nice proof of Serre duality for smooth projective morphisms. Apart from Hartshorne’s book we will use Johan de Jong’s ``stacks project’‘.
2 hours lecture and 1 hour problem session per week
probably a take home assignment plus oral exam
Recommended: R. Hartshorne. Algebraic Geometry. Springer GTM 52
Mastermath Algebraic Geometry, or equivalent