WS 2021

Commutative Algebra and Algebraic Geometry

Algebraic geometry is the study of systems of polynomial equations and their geometric counterparts, namely, algebraic varieties. Commutative algebra is the subfield of algebra that deals with commutative rings and their modules; it provides the tools to effectively analyse algebraic varieties.

In this introductory course, we will consider a special family of algebraic varieties, namely, projective ones: they are the zero sets of homogenous polynomials in an extension of the standard affine space, which is called the projective space. The algebraic couterpart to projective varieties are homogeneous ideals and graded modules. We will study the basic properties of these objects and of the maps between them.

The exercise course is taught by Professor Josef Schicho. You can find information about his course at this link.

Language: the course will be taught in English.

Time: Tuesdays, 8:30-10:00 (for precise dates, please check on KUSSS)

Place: room S2 054 in Science Park 2

Content of the course (tentative) :

Exam: the grade will be based on an oral exam (80% of the grade) and on a short essay of 2-3 pages on a topic agreed with the teacher, to be handled at latest 5 days before the oral exam (20% of the grade).

  1. Affine plane and affine curves.
  2. The projective space.
  3. Prime, radical, homogenous ideals.
  4. Projective varieties.
  5. Primary decomposition and irreducible components.
  6. Graded modules and the Hilbert polynomial.
  7. Degree and dimension of varieties.
  8. Examples of projective varieties (Segre, Veronese, Grassmannians).
  9. Morphisms between varieties.