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This is the first semester of a two-semester sequence on Algebraic Geometry.
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English
English [CC]
FREE
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Description
The goal of the course is to introduce the basic notions and techniques of modern algebraic geometry. It covers fundamental notions and results about algebraic varieties over an algebraically closed field; relations between complex algebraic varieties and complex analytic varieties; and examples with emphasis on algebraic curves and surfaces. This course is an introduction to the language of schemes and properties of morphisms
Course content
- Course Introduction, Zariski Topology Unlimited
- Affine Varieties Unlimited
- Projective Varieties, Noether Normalization Unlimited
- Grassmannians, Finite and Affine Morphisms Unlimited
- More on Finite Morphisms and Irreducible Varieties Unlimited
- Function Field, Dominant Maps Unlimited
- Product of Varieties, Separatedness Unlimited
- Product Topology, Complete Varieties Unlimited
- Chow’s Lemma, Blowups Unlimited
- Sheaves, Invertible Sheaves on P{{}} Unlimited
- Sheaf Functors and Quasi-coherent Sheaves Unlimited
- Quasi-coherent and Coherent Sheaves Unlimited
- Invertible Sheaves Unlimited
- (Quasi)coherent Sheaves on Projective Spaces Unlimited
- Divisors and the Picard Group Unlimited
- Bezout’s Theorem Unlimited
- Abel-Jacobi Map, Elliptic Curves Unlimited
- Kähler Differentials Unlimited
- Smoothness, Canonical Bundles, the Adjunction Formula Unlimited
- (Co)tangent Bundles of Grassmannians Unlimited
- Riemann-Hurwitz Formula, Chevalley’s Theorem Unlimited
- Bertini’s Theorem, Coherent Sheves on Curves Unlimited
- Derived Functors, Existence of Sheaf Cohomology Unlimited
- Birkhoff–Grothendieck, Riemann-Roch, Serre Duality Unlimited
- Proof of Serre Duality Unlimited
Instructor
Massachusetts Institute of Technology
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