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18.725

Algebraic Geometry I

3-0-9 units · Graduate

Introduces the basic notions and techniques of modern algebraic geometry. 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. Introduction to the language of schemes and properties of morphisms. Knowledge of elementary algebraic topology, elementary differential geometry recommended, but not required.

Fall 2015 · Algebraic Geometry

Topics

  • Course Introduction, Zariski Topology
  • Affine Varieties
  • Projective Varieties, Noether Normalization
  • Grassmannians, Finite and Affine Morphisms
  • More on Finite Morphisms and Irreducible Varieties
  • Function Field, Dominant Maps
  • Product of Varieties, Separatedness
  • Product Topology, Complete Varieties
  • Chow’s Lemma, Blowups
  • Sheaves, Invertible Sheaves on P 1
  • Sheaf Functors and Quasi-coherent Sheaves
  • Quasi-coherent and Coherent Sheaves
  • Invertible Sheaves
  • (Quasi)coherent Sheaves on Projective Spaces
  • Divisors and the Picard Group
  • Bezout’s Theorem
  • Abel-Jacobi Map, Elliptic Curves
  • Kähler Differentials
  • Smoothness, Canonical Bundles, the Adjunction Formula
  • (Co)tangent Bundles of Grassmannians
  • Riemann-Hurwitz Formula, Chevalley’s Theorem
  • Bertini’s Theorem, Coherent Sheves on Curves
  • Derived Functors, Existence of Sheaf Cohomology
  • Birkhoff–Grothendieck, Riemann-Roch, Serre Duality
  • Proof of Serre Duality

Readings

Fall 2003 · Algebraic Geometry

Topics

  • Introduction
  • Recap of Last Time
  • Projective Space (cont.)
  • Presheaves
  • Review of things not covered enough (Topics: Fibers, Morphisms of Sheaves)
  • Homework Review
  • Projective Varieties
  • A Review on Projective Varieties
  • Applications
  • Recap on the Applications
  • Homework 5 Problem
  • Back to Dimension
  • Fibers of Morphisms
  • Fiber Products
  • Completeness
  • Chow’s Lemma
  • Curves
  • What is Next?

Readings

  • Mumford, David. The Red Book of Varieties and Schemes. Vol. 1358, Lecture Notes in Mathematics. New York: Springer-Verlag. ISBN: 354063293X. Includes the Michigan Lectures (1974) on Curves and Their Jacobians.
    Required
  • Shafarevich, I. R., and M. Reid. Basic Algebraic Geometry I. 2nd revised and expanded ed. Berlin; New York: Springer-Verlag Telos. ISBN: 0387548122.
    Listed reading
  • Hartshorne, Robin. Algebraic Geometry. New York: Springer-Verlag, 1977. ISBN: 0387902449.
    Listed reading
  • Eisenbud, David. Commutative Algebra With a View Toward Algebraic Geometry. Vol. 150, Graduate Texts in Mathematics. New York: Springer-Verlag, 1994. ISBN: 0387942696.
    Listed reading
  • Atiyah, M. F., and I. G. MacDonald. Introduction to Commutative Algebra. Reading, Mass.: Addison Wesley Publishing Company, 1969. ISBN: 0201407515.
    Listed reading
  • Mumford, David. The Red Book of Varieties and Schemes. Vol. 1358, Lecture Notes in Mathematics. New York: Springer-Verlag. ISBN: 354063293X. Includes the Michigan Lectures (1974) on Curves and Their Jacobians.
    Listed reading

Prerequisites & course information

Prereq: None. Coreq: 18.705 G (Fall) 3-0-9 units

Sources