Math 4300 (Algebraic Curves) Fall 2003 (3 hours)

Oasis title: INTRO TO ALG CURVES.
Prerequisite: MATH 4000/6000 or permission of department.
Class Meetings: TTh 12:30-1:45, Boyd 222.
Instructor: Dr. Bill Rulla, Boyd 523A, 542-2575, rulla@math.uga.edu
Office Hours: 11-12:30 TTh and 3:30-5 W. Also you almost certainly will be able to catch me after class TTh.

Catalogue description: Polynomials and resultants, projective spaces. The focus is on plane algebraic curves: intersection, Bezout's theorem, linear systems, rational curves, singularities, blowing up.

My description: We encountered algebraic curves long before college; for example we are all familiar with the graphs (or ``zero loci'') of y=x2 , x2+y2=1 etc. in R2. These ``plane'' curves generate scads of interesting questions, and greatly influence other areas of math. For instance: Fermat's Last Theorem can be phrased in terms of the existence of points (p,q) on the curves given by xn+yn=1 such that p,q are rational and pq is nonzero. What do these curves look like? Does their geometry reveal anything enlightening?

We will see that more can be said about the geometry of curves in (e.g.) R2 if we regard their equations as taking and giving values in C; in this case their zero loci now live in C2= R4. These ``curves'' are now two dimensional, i.e. are surfaces. What do they look like? Can we get a good picture of them, without being particularly capable of visualizing R4?

Towards this end, we can apply projective geometry, topology, complex analysis, and even field theory. One of the attractions of the subject is that it ties together interesting concepts from these and other areas of undergraduate math (and as far beyond as you want to go...). For instance: Non-trivial maps between these surfaces correspond to finite extensions of their ``function fields,'' so Galois theory can be brought to bear.

For another instance: it is somewhat unsatisfying that certain functions, e.g. log z, sqrt z, etc. can never be both holomorphic and defined on all of C. To resolve this, Riemann constructed surfaces which are ``large'' enough to parametrize the multiple values taken on by these functions (e.g. the surface for sqrt z has points enough to accomodate the two square roots of each nonzero z). These ``Riemann surfaces'' have interesting and non-trivial topology, and furthermore locally look like C (so that we can generalize complex analysis to them). Loads of ongoing research, including topics such as moduli spaces and string theory, bear on them. Amazing fact: there is a natural correspondence between these surfaces and the plane curves we introduced above, which makes much of the study assailable from the (sometimes easier) points of view of algebra and projective geometry.

This is the Riemann surface of the square root function. Click on it to see how it works!

This course will be a gentle introduction to (algebraic) curves in a plane (e.g. R2, C2, CP2), which itself will be an introduction to the subject of algebraic geometry. We will emphasize concrete examples, ground-up reasoning, and intuition.

Topical outline:

  • Algebraic curves: Zero sets of polynomials in two variables defined over Q, R, C, .... Generalization to curves in CP2. Connections between containment and divisibility of defining polynomials.

  • Parametrizability: When can a plane curve be parametrized by rational functions? (the ``Fermat curves'' cannot, for n > 2).

  • Singularities: When is a curve in CP2 a manifold? If it's not a manifold, what kind of singularities can it have? How do people work with singular curves?

  • Bezout's Theorem: We will define a local intersection number and use it (among other things) to express constraints on singularities and intersections of curves.

  • Riemann Surfaces: To an arbitrary (reduced) plane curve one can associate a unique Riemann Surface. In fact, the association is functorial, and in a somewhat surprising way.

  • Elliptic Curves: These are curves of genus one (with a distinguished point). They admit a group structure, i.e. are themselves groups. To make coding theory even more indecipherable, elliptic curves (over fields of finite characteristic) are increasingly being used in place of more mundane groups.

  • Degree-genus formula: What can be said about the genus of the Riemann surface associated to a plane curve of degree d?

  • Riemann-Hurwitz formula: We'll analyze branched covers of Riemann surfaces (a branched cover is almost a ``covering map,'' if you've encountered those before). We can conclude interesting relations between Riemann surfaces by studying the topology of branched covers.

  • Quadratic transforms, resolution of singularities: Can we turn a singular plane curve (over an arbitrary field) into a smooth one? Can we at least improve ``nasty'' singularities?

  • Differentials on Riemann surfaces: Can we differentiate functions defined on a Riemann surface? What good would such derivatives (or other differentials) be?

  • Consequences of Riemann-Roch: How bad can the field of meromorphic functions be on a smooth plane curve? It must be larger and horribly more disgusting than the field of restrictions of rational functions from CP2....(?) We'll also use R.R. to demonstrate associativity of the group law on a plane cubic.

  • Other things! Lots of options here, left up to public opinion.

    Text: Plane Algebraic Curves, by Gerd Fischer; American Mathematical Society (2001). We will cover Chapters 1-3 and parts of 4, 5, and 9 in detail, but will supplement these chapters with ideas, arguments, and examples from (some of) the references listed below.

    Grading: There will be occasional homework assignments to turn in, and a midterm and final.

    References:

    R.~Churchill, and J.~Brown.
    Complex Variables and Applications (4th Ed.).
    McGraw-Hill, Inc. (1984).

    C.H.~Clemens.
    A Scrapbook of Complex Curve Theory (2nd Ed.).
    AMS, Graduate Studies in Mathematics, Vol. 55 (2003).

    D.~Cox, J.~Little, and D.~O'Shea.
    Ideals, Varieties, and Algorithms.
    2nd edition, Springer-Verlag, New York (1996).

    G.~Fischer.
    Plane Algebraic Curves.
    AMS, Student Mathematical Library Vol. 15 (2001).

    O.~Forster.
    Lectures on Riemann Surfaces.
    Springer-Verlag, New York (1981).

    F.~Kirwan.
    Complex Algebraic Curves.
    London Mathematical Society, Student Texts 23
    (1995).

    R.~Miranda.
    Algebraic Curves and Riemann Surfaces.
    AMS, Graduate Studies in Mathematics, Vol. 5 (1995).

    T.~Shifrin.
    Abstract Algebra: A Geometric Approach.
    Prentice-Hall (1996).

    R.~Smith.
    Lecture notes from his plane curves class, 1991.