Sheaves, Cohomology and Vector Bundles
- The course will be held in English.
- The lecture takes on Monday 10:15 - 11:45 in SR 10, Geb. E2 4,
and on Thursday 12:30 - 14:00 in SR 9, Geb. E 2 4. - There will be exercise session on Thursday 10:15 - 11:45 in SR 10, once every two weeks.
The main objective of the lecture is to learn and be familiar with sheaves, vector bundles, and sheaf cohomology, which appear as key materials in algebraic topology and algebraic geometry. A secondary objective is to learn basic algebraic curve theory and Ulrich sheaves.
Very basic knowledge of commutative algebra and algebraic geometry is recommended.
Tentative topics containing:
presheaf, sheaf, divisor, line bundle, vector bundle, projective morphism, derived functor, sheaf cohomology, Čech cohomology, Serre duality, Riemann-Roch formula, Riemann-Hurwitz formula, Petri theorem, Green's (2g+1+p)-theorem, canonical curve, determinantal representation, Ulrich sheaf
| R. Hartshrone | Algebraic Geometry | Springer | 1977 |
| R. Vakil | Foundations of Algebraic Geometry | math216.wordpress.com | |
| E. Arbarello, M. Cornalba, P. A. Griffiths, J. Harris | Geometry of Algebraic Curves I | Springer | 1985 |
| D. Eisenbud | The Geometry of Syzygies | Springer | 2005 |
| D. Eisenbud, F.-O. Schreyer, J. Weyman | Resultants and Chow forms via exterior syzygies | Article published in: J. Amer. Math. Soc. | 2003 |
| A. Beauville | An introduction to Ulrich bundles | Article published in: Eur. J. Math. | 2018 |
Lecture Notes : Week 0, Week 1, Week 2, Week 3, Week 4, Week 5, Week 6, Week 7, Week 8, Week 9, Week 10, Week 11, Week 12, Week 13, Week 14&15.
There will be a small oral exam. Details will be determined later.
An exercise sheet will be uploaded before one week every exercise class.
Exercise 1 (08. 11. 2018)
Exercise 2 (22. 11. 2018)
Exercise 3 (06. 12. 2018)
Exercise 4 (17. 01. 2019)
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Yeongrak Kim Zi. 429, Geb. E2 4 D-66123 Saarbrücken This email address is being protected from spambots. You need JavaScript enabled to view it. |