Lectures:
Mon 11:00-13:00 (RUD 25 1.115)
Wed 11:00-13:00 (RUD 25 1.115)
Exercise session (Ignacio Barros):
Mon 13:00-15:00 (RUD 25 1.011) (starting in week two 24/10)
Office hours:
Mon 15:00-16:00 (RUD 25 1.429)
This course will assume a basic familiarity with the language of varieties and projective geometry as introduced in a first course on algebraic geometry. The majority of the course will be on developing the basics in the theory of schemes, sheaves and line bundles on varieties, ampleness and embeddings in projective space, Kaehler differentials, sheaf cohomology and Riemann-Roch. Time permitting we will discuss selected topics from birational geometry.
17 Oct Dies academicus (i.e. no lectures)
Lecture 1 (19 Oct): Recap on varieties. Intro to scheme theory.
Lecture 2 (24 Oct): Affine schemes [G02 5.1].
Lecture 3 (26 Oct): Morphisms [G02 5.2]
Lecture 4 (31 Oct): Schemes vs varieties [G02 5.3]
Lecture 5 (2 Nov): Fibre products [G02 5.4]
Lecture 6 (7 Nov): Projective schemes [G02 5.5]
Lecture 7 (9 Nov): Projective schemes [G02 5.5]
Lecture 8 (14 Nov): Hilbert functions and Bezout [G02 6.1, 6.2]
Lecture 9 (16 Nov): Divisors on curves [G02 6.3]
Lecture 10 (21 Nov): Plane cubics [G02 6.4, 6.5]
Lecture 11 (23 Nov): Sheaves of O_X modules and sheafification [G02 7.1]
Lecture 12 (28 Nov): Quasi-coherent and coherent sheaves [G02 7.2]
Lecture 13 (30 Nov): Quasi-coherent and coherent sheaves [G02 7.2]
Lecture 14 (5 Dec): Locally free sheaves [G02 7.3]
Lecture 15 (7 Dec): Differentials [G02 7.4]
Lecture 16 (12 Dec): Line bundles on curves [G02 7.5]
Lecture 17 (14 Dec): Riemann-Hurwitz [G02 7.6]
Lecture 18 (4 Jan): Riemann-Roch for curves [G02 7.7]
Lecture 19 (9 Jan): Sheaf cohomology [G02 8.1]
Lecture 20 (11 Jan): Sheaf cohomology [G02 8.2]
Lecture 21 (16 Jan): Riemann-Roch [G02 8.3]
Lecture 22 (18 Jan): Cohomology of line bundles on projective space [G02 8.4]
Lecture 23 (23 Jan): Cartier divisors [G02 9, Hartshorne II.6]
Lecture 24 (25 Jan): Weil divisors in general [G02 9, Hartshorne II.6]
Lecture 25 (30 Jan): Ample and very ample divisors [Hartshorne II.5, II.7]
Lecture 26 (1 Feb): Intersection theory on surfaces [Hartshorne V.1] (Daniele was kind enough to type up some notes about divisors and the intersection product.)
Lecture 27 (6 Feb): Riemann-Roch and the Hodge index theorem [Hartshorne V.1]
Lecture 28 (8 Feb): -Cancelled-
Lecture 29 (13 Feb): A view to birational geometry of surfaces and classification [Hartshorne V.3, V.5-V.6]
Problem sheets
Sheet 1 to be discussed on 24/10
Sheet 2 to be discussed on 31/10
Sheet 3 to be discussed on 07/11
Sheet 4 to be discussed on 14/11
Sheet 5 to be discussed on 21/11
Sheet 6 to be discussed on 28/11
Sheet 7 to be discussed on 05/12
Sheet 8 to be discussed on 12/12
Sheet 9 to be discussed on 09/01
Sheet 10 to be discussed on 16/01
Sheet 11 to be discussed on 23/01
Sheet 12 to be discussed on 30/01
Sheet 13 to be discussed on 06/02
Exam
The exam will be on the 15th of February 2017.
The second exam will be on the 5th of April 2017.
References
[G02] Gathmann: We will primarily follow chapter 5 onwards from Gathmann's older notes on varieties and schemes.
[G14] Gathmann: His new notes on varieties will be the main source of material on varieties that I'll assume.
Hartshorne: "Algebraic Geometry"
Eisenbud-Harris: "The geometry of schemes"
Harris: "Algebraic Geometry"
Vakil: notes "Foundations of algebraic geometry"
Mumford: "The red book of varieties and schemes"
Mumford-Oda: "Algebraic Geometry II"
Liu: "Algebraic geometry and arithmetic curves"