# Algebraic Curves

### The class meets Tuesdays and Thursdays, 1:30 - 2:45pm at 413, Keller Hall.

Office: 501, Keller Hall (5-th floor)
e-mail: pavel(at)math(dot)hawaii(dot)edu
Usually, I respond to e-mail messages within a day. If I do not, that means that I do not have a reasonable answer, and am still thinking.
Office hours: By appointment, whenever the participants have questions to ask or discoveries to report.

Lecture Notes

I am typing the notes while doing the class. For this reason, the pdf file is always under construction.

Besides the exercises marked so in the notes, a very important exercise (the most important for me) is to read the notes carefully, locate and report to me everything you find questionnable in the text.
These may be inconsistencies, ambigous notations, inaccurate, misleading or plainly wrong statements/sentences, missing justifications, etc.; grammatical errors, and typos are less valuable findings.

Class goals and content

Primary goals of this class are educational. The class is devoted to the Riemann - Roch theorem. This theorem plays a central role not only in algebraic geometry, but, more broadly, in contemporary mathematics. Originally, the theorem has been proved as a theorem about Riemann surfaces, and, from this prospective, belongs to Analysis. Uniformization theorems tell us that every compact Riemann surface is an algebraic curve defined over $\mathbb C$.

The Riemann - Roch theorem has been widely generalized in the context of algebraic geometry. Contemporary ways to understand this theorem in the context of schemes and sheaf cohomology require a lot of prerequisites and will not be considered in this class.

In this class, we discuss Weil's proof of Riemann - Roch theorem for algebraic curves over an arbitrary algebraically closed field.

This is an entirely algebraic proof which does not require much of prerequisites. At the same time, this proof may serve as a gateway to algebraic geometry. As a warning, I am not and algebraic geometer; I simply believe that an educated mathematician must get exposed to central ideas of math.

In a nutshell, the class covers Chapter 1 from S. Lang, Introduction to algebraic and abelian functions. We do some preparations in order to start covering this material, and may throw in some applications after this principal material is covered.

Prerequisites
This is a "Topics in Algebra" class, thus knowledge of graduate abstract algebra is assumed. While the students are not assumed to remember every theorem, their ability to read and understand an arbitrary excerpt from (any) graduate algebra textbook is assumed. At the same time, the amount of algebra needed is limited by just several specific topics. Brave students without very solid command in graduate algebra are welcome. The class may be harder for these students but they will certainly benefit more.

Formally, the knowledge of complex analysis is not required for understanding any single argument in this class. However, the theory under consideration stems from the theory of compact Riemann surfaces, and a background in complex analysis is very useful to catch the intuition behind the sunjects under discussion.
The class will be much more interesting for those students who can and will constantly consult the book by R. Miranda, Algebraic Curves and Riemann Surfaces. I will insistently make parallels trying to underscore the fact that the two treatments actually consider "the same" subject. The leading idea is that, although analytic methods are not available when working over an arbitrary (algebraically closed) field, all theorems remain true as soon as alternative purely algebraic methods of proof are developed. Geometric intuition coming from the classical theory of Riemann surfaces is indispensable in understanding and visualizing the theorems.

Contents
• Some basics definitions from algebraic geometry
Algebraic curves are defined as algebraic varieties of dimension one. We thus have to briefly overview the notion of algebraic sets, varieties, and their dimension.

There is abundant literature where these basic notions are treated, typically, on the very first few pages of a text.
The most standard source is R. Hartshorne, Algebraic geometry; one finds a bit simpler and an easier readable version in J. Silverman, The arithmetic of elliptic curves.

• Discrete valuation rings and their extensions
An algebraic curve (as any curve) may be thought of as the set of its points. In the case of an algebraic curve, these "points" are discrete valuation rings, and we need some facts about ring extensions for such rings to start with.
This part is pure algebra; several theorems from S. Lang, Algebra will be discussed in detail.

• The Riemann - Roch theorem
This is the principal part of the class: all preliminaries are done, and we now closely follow Chapter 1 of S. Lang, Introduction to algebraic and abelian functions.
Although all proofs are self-contained and purely algebraic, the parallelism with the theory of Riemann surfaces is emphasized, and constant references to the book by R. Miranda, Algebraic Curves and Riemann Surfaces, are made for the sake of promotion of geometric intuition.

• Some applications
Some (fairly standard) immediate applications of the Riemann - Roch theorem will be considered. The primary source here is again Chapter 1 of S. Lang, Introduction to algebraic and abelian functions,
though some other sources may be also used.

Textbooks

In the description above, a few books were mentioned, and there may be other texts used occasionally. I have electronic versions of all texts which I am going to use, and may share these with students if there is a need. To make it clear, the primary textbook in this class is
Chapter 1 of S. Lang, Introduction to algebraic and abelian functions,
and I will constantly make parallels with the classical theory of Riemann surfaces as it is exposed in
R. Miranda, Algebraic Curves and Riemann Surfaces
which may be considered as a supplementary textbook.