Math 203A  Algebraic Geometry
Welcome to Math 203a!
Course description:

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 three quarter sequence. Math 203a will
cover affine and projective varieties corresponding roughly to the first
chapter of Hartshorne.
The course description can be found
here.
Instructor: Dragos Oprea, doprea "at" math.youknowwhere.edu,
AP&M 6101.
Lectures: MWF, 10:0010:50, AP&M 7421.
Office hours:
 Wednesday 12:30pm in AP&M 6101. The office hours will be hybrid, the zoom link can be found in Canvas
I am
available for
questions after
lecture or by appointment. Also, feel free to drop in if you see me in my
office.
Textbook: I will roughly follow
Andreas Gathamnn's notes available
online.
I recommend that you also consult Shafarevich's Basic
Algebraic Geometry and Hartshorne's Algebraic
Geometry.
Other useful texts are
 Joe Harris, Algebraic Geometry: a first course.
 David Mumford, Algebraic Geometry I, Complex projective
varieties
 David Mumford, The red book of varieties and schemes
 Ravi Vakil, The Rising Sea
Additional resources:
Prerequisites:
 Some knowledge of modern algebra at the
level of Math 200 is required. I will try to keep the algebraic
prerequisites to a minimum. Familiarity with basic
point set topology, complex analysis and/or differentiable manifolds is
helpful to get some
intuition for the concepts. 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.
Grading:  There will be no exams for this class. The grade
will be
based entirely on homeworks and regular attendance of lectures. The
problem sets are mandatory and are a very important part of the course.
The problem sets are submitted via Gradescope.
Important dates:
 First class: Friday, September 23.
 Veterans Day: Friday, November 11.
 Thanksgiving break: November 2425.
 Last class: Friday, December 2.
Lecture Summaries
 Lecture 1: Introduction. Affine algebraic sets. Zariski
topology. Ideals of affine algebraic sets. Weak and strong Nullstellensatz 
Notes
 Lecture 2: Weak and strong
Nullstellensatz. Correspondence between radical ideals and affine
algebraic sets. Irreducibility 
Notes
 Lecture 3: Noetherian spaces. Irreducible components.
Krull dimension. Coordinate rings. Fraction field  Notes
 Lecture 4: Regular
functions on affine
varieties and their open subsets. Basic open sets and their regular functions. Examples Notes
 Lecture 5: Presheaves and sheaves. Stalks. Ringed spaces.
Examples  Notes
 Lecture 6: Morphisms between affine varieties.
Corespondence with morphisms between coordinate rings.
Isomorphisms. Examples  Notes
 Lecture 7: Rational maps, dominant maps, birational isomrphisms,
correspondence with fraction fields  Notes
 Lecture 8: Abstract affine varieties. Basic open sets are abstract affine. Prevarieties. Gluing. Projective line and line with double origin  Notes
 Lecture 9: Closed immersions. Morphisms of prevarieties. Regular functions on the projective line. Morphisms from prevarieties to affine varieties. Products  Notes
 Lecture 10: Products of affine varieties. Products of prevarieties. Universal properties  Notes
 Lecture 11: Varieties. Projective algebraic sets. Zariski topology
 Notes
 Lecture 12: Plane conics. Projective closure. Cones and projective varieties. Projective Nullstellensatz. Regular functions. Homogeneous coordinate rings
 Notes
 Lecture 13: Projective varieties as ringed spaces. Morphisms. Examples. Rational normal curve. Veronese morphism. The Segre morphism  Notes
 Lecture 14: Segre embedding. Projective varieties are separated. Grassmannians and Plucker embedding  Notes
 Lecture 15: Proper/complete varieties. Projective varieties are proper. Morphisms from complete varieties are closed  Notes
 Lecture 16: Regular functions on complete varieties are constant. Dimension theory for projective varieties. Projection from a point  Notes
 Lecture 17: Comparing the dimension of a variety to that of the projection. Dimension of projective space. Noether normalization  Notes
 Lecture 18: Finite maps and some properties. Geometric form of Noether normalization  Notes
 Lecture 19: Dimension and transcendence degree of fraction field. Dimension of arbitrary varieties. Krull's Hauptidealsatz. Applications
 Notes
 Lecture 20: Krull's Hauptidealsatz and its proof  Notes
 Lecture 21: Theorem of dimension of fibers. Tangent spaces and tangent cones
 Notes
 Lecture 22: Intrinsic nature of the tangent space. Smoothness. Jacobi criterion. The singular locus is open and dense
 Notes
 Lecture 23: More on tangent spaces. Bertini's theorem. Introduction to blowups  Notes
 Lecture 24: Blowups of affine varieties. Exceptional set. Strict transform  Notes
 Lecture 25: More on blowups. Dimension of the exceptional set. Blowups of projective varieties and some examples. Invariants of projective varieties. Some motivating questions
 Notes
 Lecture 26: The Hilbert function and some examples. First properties. The Hilbert polynomial  Notes
 Lecture 27: The Hilbert polynomial. Degree. Arithmetic genus. Examples. The degreegenus formula  Notes
 Lecture 28: Global and local Bezout theorem. Intersection multiplicities  Notes
 Lecture 29: Applications of Bezout's theorem. Divisors on curves. Principal divisors have degree zero. Upper bound on the number of singularities of plane curves. Review and topics for Math 203b  Notes
Homework:
Homework 1 due Friday, Sept 30  PDF
Homework 2 due Friday, October 7  PDF
Homework 3 due Monday, October 17  PDF
Homework 4 due Monday, October 24  PDF
Homework 5 due Monday, October 31  PDF
Homework 6 due Monday, November 7  PDF
Homework 7 due Monday, November 14  PDF
Homework 8 due Monday, November 21  PDF
Homework 9 due Friday, December 2  PDF