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  
 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 7Week 8, Week 9, Week 10, Week 11, Week 12, Week 13Week 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)


Yeongrak Kim
Zi. 429, Geb. E2 4
D-66123 Saarbrücken
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© AG Schreyer