A full year of Algebra (the needed Columbia courses are Math
W4041 and 4042) and a semester of Complex Variables (Math
V3007).
It is helpful, but not required or essential, to have some
understanding of the elementary topology of finite
dimensional real vector spaces, as covered in many courses
at Columbia: for example,
Topology W4043, Introduction to Modern Analysis I
W4061 or Analysis and Optimization V2500.
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In this course we study algebraic curves over the complex
numbers from three points of view. The primary
point of view is algebro-geometric, but we will also look at
them through the lens of complex analysis (where they are
known as Riemann surfaces), and finally through the lens of
topology. This
is done very nicely in the book we will be using.
We will first study curves in affine space and then
projective spaces.
This will give us the opportunity of understanding
the advantages of working in projective space, which is
compact. Then
we define our important invariant of the curve embedded in
projective space, the degree. We also
discuss the notion of singular point, and define invariants
associated to singular points
Then we will prove our first big theorem, Bézout's Theorem,
concerning the number of intersections of two curves in P2
(projective space of dimension 2). Then we will
study in detail curves of degree 2 and 3.
Then we will switch to a topological point of view: we study
branched covers of one curve (often projective space of
dimension 1) by another curve. This leads us
to our second big theorem, the degree-genus formula. The genus is
the most important invariant of an algebraic curve. We will also
see that there is a field extension associated to any cover
of curves, which we can study using field theory from
algebra.
Then we switch to a complex analytic point of view, and
study compact Riemann surfaces. This leads us
to the notion of differentials on an algebraic curve and the
notion of line bundle (or linear system). Using these
concepts, we prove the major theorem of the course: the
Riemann-Roch theorem.
Up to this point we will have mainly studied curves in P2. We will
conclude by studying the resolution of plane curve
singularities and embeddings of curves in higher dimensional
projective space.
If time permits we will study the topology of a plane
curve in the neighborhood of a singular point.
Required Text
Complex Algebraic Curves, by Frances Kirwan.
Springer. It will be available at Book Culture bookstore on
112th Street between Broadway and Amsterdam.
This site refers to it simply as
"Kirwan". Reading and homework will be assigned
from it, so you need access to a copy. Copies are
on reserve in the Mathematics Library.
Other recommended books
You can find on the web a short set of lectures notes by
Nigel Hitchin called Algebraic Curves, which "trace a path
through material covered in more detail in Kirwin". You may find
this helpful.
Other useful books, all on reserve in the Mathematics
Library are Robert Walker's Algebraic Curves, William
Fulton's Algebraic Curves, and Rick Miranda's Algebraic
Curves and Riemann Surfaces. Walker's book
is roughly at the same level as Kirwan, but completely
algebraic. Fulton's
book is more advanced, and also completely algebraic. Miranda's
book is much more comprehensive than Kirwan, so that in
first seven chapters there is a lot of interesting material
additional to what we will be covering.
It will also be useful for you to have access to the texts
you used in Algebra and Complex Variables.
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