# MA4A5 Algebraic Geometry

**Lecturer: **Christian BĂ¶hning

**Term(s):** Term 1

**Status for Mathematics students:** List C

**Commitment:** 30 lectures plus assignments

**Assessment:** Assignments (30%), 3 hour written exam (70%).

**Prerequisites:**

A background in algebra (especially MA249 Algebra II) is essential. The module develops more specialised material in commutative algebra and in geometry from first principles, but MA3G6 Commutative Algebra will be useful. More than technical prerequisites, the main requirement is the sophistication to work simultaneously with ideas from several areas of mathematics, and to think algebraically and geometrically. Some familiarity with projective geometry (e.g. from MA243 Geometry) is helpful, though not essential.

**Leads To:
**A first module in algebraic geometry is a basic requirement for study in geometry, number theory or many branches of algebra or mathematical physics at the MSc or PhD level. Many MA469 projects are on offer involving ideas from algebraic geometry.

**Content**:

Algebraic geometry studies solution sets of polynomial equations by geometric methods. This type of equations is ubiquitous in mathematics and much more versatile and flexible than one might as first expect (for example, every compact smooth manifold is diffeomorphic to the zero set of a certain number of real polynomials in R^N). On the other hand, polynomials show remarkable rigidity properties in other situations and can be defined over any ring, and this leads to important arithmetic ramifications of algebraic geometry.

Methodically, two contrasting cross-fertilizing aspects have pervaded the subject: one providing formidable abstract machinery and striving for maximum generality, the other experimental and computational, focusing on illuminating examples and forming the concrete geometric backbone of the first aspect, often uncovering fascinating phenomena overlooked from the bird's eye view of the abstract approach.

In the lectures, we will introduce the category of (quasi-projective) varieties, morphisms and rational maps between them, and then proceed to a study of some of the most basic geometric attributes of varieties: dimension, tangent spaces, regular and singular points, degree. Moreover, we will present many concrete examples, e.g., rational normal curves, Grassmannians, flag and Schubert varieties, surfaces in projective three-space and their lines, Veronese and Segre varieties etc.

**
Books**:

- Atiyah M.& Macdonald I. G., Introduction to commutative algebra, Addison-Wesley, Reading MA (1969)
- Harris, J., Algebraic Geometry, A First Course, Graduate Texts in Mathematics 133, Springer-Verlag (1992)
- Mumford, D., Algebraic Geometry I: Complex Projective Varieties, Classics in Mathematics, reprint of the 1st ed. (1976); Springer-Verlag (1995)
- Reid, M., Undergraduate Algebraic Geometry, London Math. Soc. Student Texts 12, Cambridge University Press (2010)
- Shafarevich, I.R., Basic Algebraic Geometry 1, second edition, Springer-Verlag (1994)
- Zariski, O. & Samuel, P., Commutative algebra, Vol. II, Van Nos- trand, New York (1960)