Schemes, Spring 2017
Professor A.J. de Jong,
Department of Mathematics.
This is the webpage of the graduate course
"Spring 2017 Mathematics GR6262 ARITH and ALGEBRAIC GEOMETRY".
Tuesday and Thursday, 11:40 AM - 12:55 AM in Room 407 Math.
Grading will be based on homework and a final exam.
The TA is Remy van Dobben de Bruyn. He will be in the help room ??.
We will use the Stacks project as our main reference, but
of course feel free to read elsewhere. If you see a four character
alphanumeric code, like
then this is a link to a chapter, section, exercise, or a result
in the Stacks project.
Please keep up with the course by studying the following material
as we go through it.
Part I: basics
- Locally ringed spaces
(skip closed immersions)
- Affine schemes
- Quasi-coherent modules on affines
- Fibre products of schemes
- Quasi-compact morphisms
- Separation axioms
- Functoriality for quasi-coherent modules
Part II: cohomology
You can start with reading Hartshorne, Chapter III, Sections 1, 2.
The Stacks project defines the derived category before
discussing higher derived functors. Either first read a bit about
the derived category and then read the links below or try reading
the sections listed below anyway (without reading the definition of the
derived category) and see what parts of them make sense with the
definitions given in the lectures and see if you can prove the statements.
- Injective and projective resolutions
- Derived functors
- Spectral sequences
- exact couples
- differential objects
- filtered differential objects
- filtered complexes
- double complexes
- Cech cohomology
- Cech cohomology on presheaves
- Cech cohomology and cohomology
- Alternating Cech complex
Part III: cohomology of quasi-coherent modules on schemes
- Cohomology of quasi-coherent modules on schemes
- Cohomology of projective space
- Cohomology of coherent sheaves on Proj
- Quasi-coherence of higher direct images
- Euler characteristics
- Hilbert polynomials
- FYI: Numerical intersections
Part IV: coherent duality
You can start with reading Hartshorne, Chapter III, Sections 6, 7.
Another place to read is the chapter "Proof of Serre duality" in
The Stacks project has a discussion on the level of derived categories
following ideas of Neeman and Lipman. This is probably impossible
to grok without a serious effort, so I suggest you attend the lectures
to help limit what you should read.
- Overview of a duality theory
- Dualizing module on proper over "good" Noetherian local ring
Part V: Curves
Please follow along by reading Hartshorne, Chapter IV.
The following is more a list of topics.
- Varieties and rational maps
- Types of varieties
- Degrees on curves
- The chapter on Algebraic Curves
Please do the exercises to keep up with the course:
- Due 1-24 in class:
(be sure to open the
and read the definitions)
- Due 1-31 in class:
029U (not part 2),
02FH (read definition preceding exercise),
02AW (answer as much as you can)
- Due 2-7 in class: Do 3 exercises from
Section 0293, do
069S part 3, and do
- Due 2-14 in class: Do at least 4 exercises from
- Due 2-21 in class: Try at least 4 exercises from
- Due 2-28 in class: Try at least 3 exercises from
- Due 3-7 in class: Try at least 3 exercises from
- Due 3-21 in class: Try at least 3 exercises from
- Due 3-28 in class: Try at least 3 exercises from
- Due 4-4 in class:
- Due 4-11 in class:
- Due 4-18 in class: Give a list of all possible pairs
(degree of L, dimension of H^0(X, L)) where X is a smooth
projective genus 5 curve and L is an invertible sheaf of degree
between 0 and 8 inclusive. For each pair try to give an example
to show that it occurs.
- No further homeworks.
Most of this will be discussed in the lectures:
- Chapter on sheaves
- Chapter on sheaves of modules
- Locally ringed spaces
- Bases and sheaves