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Advanced Algebraic Geometry*
| Credits | 8 credit points |
| Instructors | Edixhoven, S.J. (Universiteit Leiden) |
| edix@math.leidenuniv.nl | |
| Aim | To learn a variety of concepts and techniques of algebraic geometry, so that state of the art literature in this area becomes accessible. |
| Description | 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''. |
| Organization | Three 45 minute lectures, weekly. |
| Examination | Oral examinations on January 11 and 13, 2012 in room 236, Snellius building in Leiden. |
| Literature | R. Hartshorne. Algebraic Geometry. Springer GTM 52. Johan de Jong's stacks project: http://www.math.columbia.edu/algebraic_geometry/stacks-git/ |
| Prerequisites | A first course in Algebraic Geometry on algebraic varieties over algebraically closed fields is an absolute necessity. The mastermath courses Algebraic Geometry taught in recent years by Looijenga, Taelman and Edixhoven indicate the level and content of such a course. See: http://www.math.uu.nl/people/looijeng/algmtk2010.pdf http://www.math.leidenuniv.nl/~edix/teaching/2010-2011/AG-mastermath/ag.pdf |
| Remarks | *This is a course offered by WONDER. It is an advanced master and beginning graduate student level course. Students cannot apply for travel costs for this course which will be held in Leiden. More information: http://www.math.leidenuniv.nl/~edix/teaching/2011-2012/AAG/ |