Introduction to Algebraic Varieties

Systems of polynomial equations in many variables define algebraic varieties, which are the basic objects of study in algebraic geometry. In this course we introduce some fundamental notions of algebraic geometry, such as coordinate rings, local rings, function fields, and affine and projective algebraic varieties. We develop the necessary algebraic tools and we prove the Hilbert’s Nullstellensatz, the core theorem that links the algebraic and the topological sides of algebraic varieties.

The course focuses on the study of one dimensional algebraic varieties, that is, algebraic curves. Plane curves will be discussed in detail as main concrete examples. For them we prove a theorem of Bezout that answers the question: How many points lie in the intersection of two plane curves?
We study morphisms and rational maps between arbitrary curves, and singularities (these occur for example when two branches of the same curve intersect). The goal of the course is to prove the birational classification of algebraic curves via the correspondence between smooth projective curves and function fields of transcendence degree one.

The final grade is calculated according to the results of hand-in exercises (20%) and a final exam (80%).

The prerequisites for the course are: linear algebra, groups, rings and fields. The courses Linear algebra, Groups or Introduction Groups and Rings and the course Rings and Galois are sufficient, although Galois theory is not necessary.

Language of the course: English

Linear algebra (WISB107 en WISB108) and Introduction to groups and rings (WISB124). Recommended: Fields and Galoistheory (WISB224) and Modules and Representations (WISB223). See the courseplanner (cursusplanner.uu.nl) for the contents of those courses: select Faculty of Science and then the programme of the bachelor Mathematics of the most recent year.

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