• Mastermath course ALGEBRAIC GEOMETRY

    Instructors: Carel Faber (Universiteit Utrecht) Gerard van der Geer (Universiteit van Amsterdam)

    E-mail: c.f.faber@uu.nl G.B.M.vanderGeer@uva.nl

    Aim: The course intends to give a first introduction to the basic notions and techniques of algebraic geometry

    Description: This introduction begins with varieties defined over an algebraically closed field. We treat basic notions like dimension, tangent space and smoothness. We continue with divisors, line bundles and differential forms. We shall treat a number of basic results on algebraic curves. We also intend to give a first encounter with schemes.

    Organization: There will be three hours of lectures every week. Exercises will be given so that students can test their progress.

    Examination: There will a written final exam. During the course there will be a few diagnostic tests that do not count for the final grade.

    Literature: We shall use the course notes of Ben Moonen as the basic text. In the last two lectures we also use the addendum by Robin de Jong. Besides that we shall refer the students to Hartshorne's book 'Algebraic Geometry' and the book 'Geometry of Schemes' by Eisenbud and Harris. For those who want to pursue in algebraic geometry it may be recommended to buy these books, esp. the book by Hartshorne

    Treated in week 6: affine space, Zariski topology, Hilbert basis theorem, Hilbert Nullstellensatz, affine variety. Exercises Chapter 1: 1.3, 1.4, 1.6, 1.8, 1.9.

    Treated in week 7: Syllabus 2.1-2.3. Atiyah-Macdonald 2.4-2.6.

    Treated in week 8: Chapter 2 and 3 of the syllabus.

    Treated in week 9: Chapter 4 of the Syllabus. Exercises 4.1-4.4.

    Treated in week 10: Chapter 5 till Section 4. Exercises 5.1-5.5.

    Treated in week 11 and 12: Chapter 6.1 and 6.2; Prop. 5.21. Also: Atiyah-McDonald: summary of Chapter 11. Exercises: 5.2 5.3 5.4 5.5 6.4 6.5 ; 5.7 en 5.8.

    Treated in week 13: Chapter 6: blowing up ; Chapter 7: properness, completeness, till 7.9. Exercises: 6.1-- 6.5. 7.1.

    Here is a diagnostic test.

    Treated in week 14: Chapter 7 (up to 7.16).

    Treated in week 15: Chapter 8 (up to 8.8). Also Commutative Algebra: integral dependence, Going-up Thm, Noether Normalization Thm. Exercises: 7.5, 7.6, 8.1.

    Week 16 is a week off: we shall continue again in week 17.

    Treated in week 17: Chapter 8 (up to including 8.19) and Appendix A4 (up to Dedekind rings). Exercises: 8.1, 8.3.

    Treated in week 18: Chapter 8.

    No lectures on May 6. We resume at May 13. The last lecture is on May 20.

    Treated in the last two lectures divisors on curves (roughly covered by the addendum).

    Here is a second diagnostic test.

    Exam: 10 June, 14:00-17:00, IWO 4.04C (Rood) (Meibergdreef 29, 1105 AZ Amsterdam Zuid-Oost)

    Here you find exam and solutions.