Math 203A - Algebraic Geometry
Welcome to Math 203a!
This course provides an introduction to algebraic geometry. Algebraic
geometry is a central subject in modern mathematics, and an
active area of research. It has connections with
number theory, differential geometry, symplectic geometry, mathematical
physics, string theory, representation theory, combinatorics and others.
Math 203 is a two quarter sequence. Math 203a (Fall Quarter) will be
cover affine and projective varieties (roughtly the first 2/3 of the
quarter) and basics of scheme theory (the last 1/3 of the quarter).
The Winter Quarter will cover more scheme theory, sheaf cohomology,
Riemann-Roch, Serre duality.
The course description can be found here .
Instructor: Dragos Oprea, doprea "at" math.you-know-where.edu,
Lectures: MW, 2pm-3:20pm, AP&M 7-421 (will most likely
- Tuesday 2-4 in AP&M 6-101.
Textbook: There is no required textbook. I will follow
Andreas Gathamnn's notes available online
lecture or by appointment. Also, feel free to drop in if you see me in my
Other useful texts are
- Igor Shafarevich, Basic Algebraic Geometry I, Varieties in
- Joe Harris, Algebraic Geometry: a first course.
- David Mumford, Algebraic Geometry I, Complex Projective
More advanced but useful references are:
Notes: There may be typos in the files below, let me know if you
spot any serious ones.
- Robin Hartshorne, Algebraic
- David Mumford, The red book of varieties and schemes.
- Complete notes from a course taught in Spring 2008,
courtesy of David Philipson PDF.
- Notes on resultants PDF.
- Notes on irreducibility and dimension PDF.
- Some knowledge of modern algebra at the
of Math 200 is required. However, I will not assume background in
commutative algebra. Familiarity with complex analysis, basic point set
topology, differentiable manifolds is helpful, but not required. Since it
is hard to determine the precise background needed for this course, I will
be happy to discuss prerequisites on an individual basis. If you are
unsure, please don't hesitate to contact me.
- There will be no exams for this class. The grade
based entirely on homeworks. The problem sets will be posted online.
- Drop deadline: October 9.
- Withdrawal deadline: October 23.
- Veterans Day: November 11.
- Thanksgiving break: November 26-27.
- Last day of classes: December 4.
- Lecture 1: Affine
algebraic sets. Noetherian rings. Zariski topology.
- Lecture 2: Ideals of affine algebraic sets. Hilbert's
Nullstellensatz. Weak form implies strong NS. Started the proof of the
weak NS. Resultants.
- Lecture 3: Finished the proof of the weak Nullstellensatz.
- Lecture 4: Noetherian topological spaces. Irreducible
components. Dimension. Regular functions on affine varieties.
- Lecture 5: Regular functions on affine open sets. Presheaves and sheaves. Ringed spaces. Morphisms of ringed
spaces. Morphisms between affine varieties correspond to morphisms between
coordinate rings. Isomorphisms and examples.
- Lecture 6: Abstract affine varieties. Basic open sets are
affine. Prevarieties. Gluing. The affine line with double origin and the
projective line. Varieties. Projective algebraic sets.
- Lecture 7: Projective Nullstellensatz. Cones. Projective
algebraic sets and cones. Projective algebraic sets are prevarieties.
Polynomial maps and morphisms.
- Lecture 8: Conics in P^2 are isomorphic to P^1 but the
homogeneous coordinate rings are not. Examples of morphisms: Veronese embedding, Segre
morphism. Showed that P^n x P^m is projective. Showed that P^n is a
- Lecture 9: Main theorem of projective geometry. Complete
varieties. Complete varieties have no non-constant regular functions.
Complements of hypersurfaces in P^n are affine. A hypersurface and
projective variety in P^n always intersect.
- Lecture 10: Dimension theory. Projection from a point.
Dimension of P^n.
- Lecture 11: Dimension theory: intersection with hypersurfaces
decreases the dimension by 1. Dimension can be computed on dense open
sets. Dimension of affine varieties.
- Lecture 12: Theorem on the the dimension of fibers. Tangent
spaces and smoothness.
- Lecture 13: Jacobi criterion. Examples of smooth varieties.
Plane curve singularities. Tangent cone. Blowups can be used to resolve
- Lecture 14: Blowups (continued). Birational maps. Rational
varieties. Examples. Elliptic curves are not rational.
- Lecture 15: The 27 lines on the cubic surface.
- Lecture 16: Affine schemes. The spectrum of a ring: the set,
the Zariski topology, basic open sets. Quasicompactness. Examples.
- Lecture 17: Rings of fractions. Sheaves over the basis of a
topology. The structure sheaf on the spectrum of a ring. Stalks of the structure sheaf.
- Lecture 18: Scheme theoretic intersections. Projective
schemes. Locally ringed spaces. General schemes and morphisms.
- Lecture 19: Hilbert function. Examples. Hilbert polynomials.
Hilbert polynomials of hypersurfaces in P^n. Degree-genus formula.
- Lecture 20: Bezout's theorem. Examples.
- Lecture 21: Applications of Bezout: Pascal's theorem, number
of singularities of irreducible curves. Preview of Math 203b.