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 serve as
preparation for a course in scheme theory (which may be covered in Math
203bc). Math 203bc will be taught by Professor Mark Gross in the Winter
and Spring quarters.
We will study affine and projective algebraic varieties,
and their properties. Among others, we will consider concepts such as
smoothness, singularities, dimension, intersection multiplicities. I hope
to illustrate the general theory with many examples. The goal is to cover
roughly the first chapter (+epsilon) of Hartshorne's book.
Instructor: Dragos Oprea, doprea "at" math.youknowwhere.edu,
AP&M 6101.
Lectures: MWF, 10am10:50am, AP&M 7421.
Office hours:
 W 34, Th 12, AP&M 6101.
I am
available for
questions after
lecture or by appointment. Also, feel free to drop in if you see me in my
office.
Textbook: Robin Hartshorne  Algebraic
geometry.
Prerequisites:
 Some knowledge of modern algebra at the
level
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.
Grading:  There will be no exams for this class. The grade
will be
based entirely on homeworks. The problem sets will be posted online.
Important dates:
 Drop deadline: October 10.
 Withdrawal deadline: October 24.
 Veterans Day: November 11.
 Thanksgiving break: November 2728.
 Last day of classes: December 5.
Tentative syllabus: PDF
Lecture Summaries
 Lecture 1:
Introduction to algebraic geometry. Affine space. Affine algebraic sets.
 Lecture 2: More on affine algebraic sets. Zariski
topology. Noetherian rings and Hilbert Basis Theorem.

Lecture 3: Ideals of affine algebraic sets. There is a 11
correspondence between affine algebraic sets and radical ideals. The weak
and strong Hilbert Nullstellensatz. I showed that the two versions are
equivalent.
 Lecture 4: Resultants. The resultant is in
the ideal spanned by the two polynomials. I showed that the resultant of
two polynomials with a common factor equals 0. I showed how to compute the
resultant in terms of the roots.
 Lecture 5: The proof of
the weak Nullstellensatz. Irreducible topological spaces.

Lecture 6: Prime and maximal ideals. Irreducible algebraic sets
correspond to prime ideals. Irreducible algebraic sets in A^2. Any
Noetherian topological space is union of irreducible components. I showed
an example of an algebraic set which decomposes into union of two
irreducible components.
 Lecture 7: Definition of
dimension and examples e.g. dimension of A^2. Coordinate rings. Morphisms
between affine algebraic sets. Morphisms induce homomorphism between
coordinate rings and conversely.
 Lecture 8: Rational
functions on affine sets. Regular points and domain of rational functions.
Regular functions on quasiaffine sets. Presheaves and sheaves.

Lecture 9: Rational maps. Dominant rational maps. Dominant
rational maps induce homomorphisms between function fields. Morphisms
between quasiaffine sets. Birational maps. Rational varieties. Examples:
conics in A^2 are rational, discussion for cubics in A^2.

Lecture 10: Ringed spaces. Morphisms of ringed spaces. I showed that
this recovers the original definitions of morphisms between affine and
quasiaffine sets. Affine varieties. Prevarieties. Gluing of prevarieties.
The affine line with the double origin.
 Lecture 11:
Varieties. Projective spaces. Projective algebraic sets. Zariski topology.
The projective nullstellensatz. Rational functions on algebraic sets.
Rational maps. Morphisms.
 Lecture 12: Examples of
projective algebraic sets: lines, conics, elliptic curves. Showed that
there is only one conic up to change of coordinates. Discussed how the two
conics in A^2 "close up" to only one conic in P^2. Projective varieties
are prevarieties. Regular functions on P^1 are constant.

Lecture 13: Morphisms of projective algebraic sets are mophisms of
ringed spaces. Gluing of morphisms over affine sets. Polynomial maps and
morphisms to P^m. Example: conics in P^2 are isomorphic to P^1, but their
homogeneous coordinate rings are not isomorphic to those of P^1.
 Lecture 14: More examples of projective varities: rational
normal curves. Veronese embeddings. Complement of hypersurfaces in P^n are
affine. Segre embeddings. Products of projective spaces is a projective
algebraic set. The double ruling of the quadric in P^3.

Lecture 15: More examples of projective varities: the Segre varieties.
Subvarieties of P^n times P^m are given by bihomogeneous polynomials.
Irreducible projective algebraic sets are varieties. The main theorem of
projective varieties.
 Lecture 16: Complete varieties.
Regular functions on complete varieties are constant. More examples of
projective varieties: grassmannians of lines in P^n are cut out by
quadrics. There are 2 lines intersecting 4 general lines in P^3.
 Lecture 17: Dimension theory. I spoke about the projection
from a point to a hyperplane. I compared dimensions via surjective
quasifinite morphisms of projective varieties.
 Lecture
18: Dimension of P^n is n. Intersecting with a projective hypersurface
cuts dimension down by 1. Dimensions of affine varieties.

Lecture 19: Dimension of any variety equals the dimension of any open
subset. Dimension of fibers of morphisms. Dimension of intersections of
affine varieties.
 Lecture 20: Tangent spaces of affine
varieties. Tangent cones. Tangent spaces are intrinsic and can be computed
as the dual of m/m^2. Smoothness.
 Lecture 21: Jacobi
criterion. Examples: elliptic curve, Fermat hypersurface, the twisted
cubic curve. Plane curve singularities and their tangent cones. Ordinary
plane singularities. Resolving plane curve singularities. Blowups started:
the case of A^2.
 Lecture 22: Examples of blowups: nodes
are resolved by one blowup, tacnodes are resolved by two blowups. Points
on the exceptional line correspond to tangent directions, curves with
different tangent directions are separated by blowups. Blowing up general
ideals.
 Lecture 23: The 27 lines on the cubic surface.
 Lecture 24: The 27 lines on the cubic surface
(continued). The cubic surface is rational. The cubic surface is
isomorphic to the blowup of P^2 at 6 points.
 Lecture 25:
Bezout's theorem and intersection multiplicities in P^2. I worked out an
example of two curve intersecting in P^2. Lines and smooth cubics intersect in 3
points. Inflection points. Every cubic has 9 inflection points. Every
smooth nonsingular cubic is isomorphic to E_{lambda}. The jinvariant
of an elliptic curve.
 Lecture 25: I computed
intersection multiplicities of tangent lines with curves (at least 2, at
least 3 at an inflection point). Singular points contribute to
intersection multiplicities at least 2. Any irreducible curve has at most
(d^23d)/2 +1 singular points.
 Lecture 26: Addition on cubic curves and examples. Proof of
associativity. It is enough to prove that if two cubics intersect in 9
points, any cubic passing through 8 of them will pass through the 9th.
 Lecture 27: Divisors on plane curves. Divisors of rational
functions. Degree of divisors of rational functions is 0. Picard group.
Picard group of P^1. Picard group of a smooth cubic is isomorphic to the
cubic endowed with the addition law we defined last time.
 Lecture 28: Graded rings. Hilbert functions. For a collection
of points, the Hilbert polynomial "counts" the multiplicities. In higher
dimensions, the Hilbert polynomials encodes the dimension, degree and
arithmetic genus of the projective variety. Examples.
 Lecture 29: I computed the arithmetic genus of a curve of
degree $d$ from the Hilbert polynomial. Bezout's theorem. Where to go
from here.
Notes: There may be typos in the files below, let me know if you
spot any serious ones.  Complete notes from a course taught in Spring 2008,
courtesy of David Philipson PDF.
 Notes on resultants PDF.
 Notes on irreducibility and dimension PDF.
Additional resources:
Homework 1 due Monday Oct 6 PDF.
Homework 2 due Monday Oct 13 PDF.
Homework 3 due Monday Oct 20 PDF.
Homework 4 due Monday Oct 27 PDF.
Homework 5 due Monday Nov 10 PDF.
Homework 6 due Monday Nov 17 PDF.
Homework 7 due Monday Nov 24 PDF.
Homework 8 due Friday Dec 5 PDF.