Prerequisites


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.

Course Objectives


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, B├ęzout'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.



Homework


Homework will be scheduled irregularly, and forms an essential part of the course.  Many of the problems will come from Kirwan.


Method of Evaluation


There will be one in-class midterm and a final.  Grades will be assigned on the basis of class participation, homework, and performance on the midterm and the final.