Positivity in Algebraic Geometry

Advanced course: Summer Semester 22/23.

HU-Berlin for the BMS

The course will be based on Lazarsfeld's book "Positivity in Algebraic Geometry I" and can be viewed as a course on Birational Geometry. Burt Totaro does a pretty good job explaining the importance of this book here.

See the schedule below for what we have covered. Feel free to email me with any questions.

Lectures

Mon 9-11 (3.008)

Tues 15-17 (3.011)

Exercises

Mon 11-13 (3.008)

Exercise Sheet

Schedule

17.04.23. Introduction and course planning

19.04.23. 1.1. A Divisors and Line Bundles, and 1.1.B. Linear Series (pg 1-15)

24.04.23. 1.1.C Intersection numbers and numerical equivalence, 1.1.D Riemann-Roch (pg 15-23)

25.04.23. 1.2A. Cohomological Properties (pg 24-33)

1.05.23. 1.2.B NumericalProperties (pg 33-39)

8.05.23. 1.3 Q-divisors and R-divisors (pg 44-50)

9.05.23. Exercise class

15.05.23. Nef line bundles and divisors: 1.4A Definitions, 1.4B Kleiman's Theorem (pg 50-59)

16.05.23. 1.4C Cones (pg 59-65)

22.05.23. 1.5. Examples (pg 73-77)

23.05.23. 1.5. Examples (pg 77-81)

30.05.23. 1.5E. The Cone Theorem (pg 83-86), Projective bundles, Cones on ruled surfaces (pg. 70-73)

5.06.23. 1.7. Ampleness relative to a mapping (pg. 94-98)

6.06.23. 1.8. Castelnuovo-Mumford Regularity (part I) (pg. 98-102)

12.06.23. Projective bundles, examples, cones on ruled surfaces, nefness relative to a mapping

13.06.23. Exercise class

19.06.23. 1.8 Castelnuovo-Mumford Regularity (part II) and Linear series basic definitions (pg. 121-123) 

26.06.23. Linear series: 2.1.A Basic definitions (pg. 121-128),  2.1.B. Semiampleness (pg. 128-133)

27.06.23. 2.1.B. Semiampleness (ctd), 2.1.C Iitaka fibrations (pg. 133-139)

3.07.23. 2.2.A. Basic properties of big divisors (pg. 139-145)

4.07.23. Exercise class

10.07.23. 2.2.B. Pseudoeffective and big cones (pg. 145-148), 2.2.C. Volume of Big Divisor (pg. 148-157), 2.3.A. Zariski's construction (pg. 158)

11.07.23. 2.3.E. Cones of cycles on surfaces and Zariski decompositions (pg. 167-172)

17.07.23. Chapter 4: Summary of Vanishing Theorems (pg. 239-267)

18.07.23. Exercise class