Winter 2009

**Synopsis:** Algebraic geometry combines methods of abstract algebra (especially commutative algebra and the theory of polynomial rings) and geometry. The central idea is to study geometric objects which can be described as the zero sets of polynomials (for instance, the unit circle is the zero set of the polynomial *x*^{2}+*y*^{2}-1) by studying the polynomials themselves. It turns out that there exist deep connections between the properties of polynomials and polynomial rings and the geometry of the zero sets. For example, the circle has only one ''piece'' because the polynomial *x*^{2}+*y*^{2}-1 is *irreducible* whereas the zero set of the polynomial (*x*-1)(*y*-2) consists of two ''pieces'' (the lines *x=1* and *y=2*) because this polynomial factors. Algebraic geometry has connections with many fields of mathematics including mathematical physics, complex analysis, topology and number theory.

**Brief Overview:** Below is a list of topics that may be covered in the course. The first three will be covered in detail. The remaining will be treated as time permits.

- Commutative algebra: During the course, we will review some results from commutative algebra needed for our study of algebraic geometry. These include the topics of rings and ideals, Noetherian rings, Spec of a ring, Hilbert's Nullstellensatz, rings of fractions and localization. Some results will be stated without proof.
- Affine algebraic sets/varieties: Zariski topology, affine algebras, examples.
- Projective algebraic sets/varieties: Graded rings and ideals, projective space, examples.
- Sheaves: Ringed spaces, structure sheaves, sheaves of modules.
- Dimension: Topological and Krull dimesion.
- Tangent spaces and singular points
- Bézout's Theorem: Intersection points of plane curves.

**Prerequisites:** Students should be comfortable with the notions of groups, rings (especially polynomials rings) and fields (e.g. MAT 5141 or an equivalent course). MAT 5327 (Topics in Algebra - Commutative Algebra) will be offered in Fall 2008 and would be excellent preparation for MAT 5150. However, MAT 5327 is not a prerequisite for MAT 5150 and all necessary results from commutative algebra will be reviewed (without proof).

**Course Text:** Daniel Perrin, *Algebraic Geometry: An Introduction*, Springer, 2007. From on-campus computers, an electronic copy of this book can be downloaded free of charge from www.springerlink.com (search for the author and then click on the "books" link on the right).

**Course Outline:** A more detailed syllabus can be found
here. Note that we will not always cover all of the material of a
given section as it is presented in the book. You are only
responsible for the material covered in class. Thus, it is
important that you take good class notes.

**Course Webpage:** There are two main online locations for
information regarding the course:

- https://alistairsavage.ca/mat5150. This is the course home page. It will be updated regularly and contains important material for the course. Announcements will also be posted here.
- Virtual Campus. Students can access their grades via Virtual Campus.