Lecture: Algebraic Geometry II (Algebraische Geometrie II)
Lecturer: Bernd Siebert
Exercise Class Tutor: Carsten Liese

Course Description:

This class provides a systematic introduction into the modern language of algebraic geometry based on Grothendieck's notion of schemes. Schemes are a generalization of algebraic varieties defined over an algebraically closed field to situations locally modelled on polynomial equations with coefficients in an arbitrary commutative ring. The theory of schemes thus provides a framework also for arithmetic geometry and for dealing with families of algebraic varieties.

The following topics will be discussed. Basic concepts of scheme theory, including cohomological methods: General theory of sheaves, affine schemes, projective schemes, types of morphisms, coherent sheaves, divisors, differentials, cohomology of sheaves, Serre duality, higher direct images, flat and smooth morphisms, formal schemes, base change.

We mostly follow Chapter 2 and 3 in Hartshorne's "Algebraic geometry", including many of the exercises.

Lectures and exercise classes are to be conducted in English, as it is stated on STiNE.

Literature:
 R. Hartshorne: Algebraic Geometry, Springer-Verlag M. F. Atiyah, I. G. MacDonald: Introduction to Commutative Algebra, Westview Press Ravi Vakil: The Rising Sea: Foundations of Algebraic Geometry
Meeting Time and Venue:
 Lecture: Tuesday 14:15 – 15:45, Geom H5 and Thursday 10:15 – 11:45, H6 Exercise class: Thursday 12.15 - 13.45, Geom 142
Examination and Grading: In the form of a closing oral examination, individual arrangement at the end of semester with lecturer.
Problems:
 22. Okt. 2015 : II.1.3, II.1.4, II.1.5, II.1.6, II.1.19 29. Okt. 2015 : II.1.20, II.1.22, II.2.1, II.2.10 5. Nov. 2015: II.2.8, II.2.15, II.2.18, II.2.19 12. Nov. 2015: II.3.1, II.3.2, II.3.3, II.3.4 (you may want to use ex II.2.17) 19. Nov. 2015: II.3.6, II.3.10, II.3.11 26. Nov. 2015: Toric varieties I 3. Dez. 2015: Toric varieties II 10. Dez. 2015: II.4.7, II.4.10, II.5.2, II.5.3 17. Dez. 2015: II.5.4, II.5.10, II.5.13 7. Jan. 2016: II.5.9, II.5.11, II.5.18 14. Jan 2016: II.6.2, II.6.6 (Read the discussion on divisors on curves; pp. 136-140) 21. Jan 2016: II.7.1, II.7.2, II.7.5, II.7.8 28. Jan 2016: II.7.7, II.7.11a, II.7.11b, II.7.12