Schemes

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The topics we will discuss are: Sheaves on topological spaces, ringed spaces, sheaves of modules, injective sheaves, cohomology of sheaves, locally ringed spaces, schemes, morphisms of schemes, properties of morphisms of schemes, closed immersions, open immersions, immersions, separated morphisms, morphisms of finite type, morphisms of finite presentation, finite morphisms, proper morphisms, invertible sheaves, Weil divisors, effective Cartier divisors, Cartier divisors, projective morphisms, projective spaces. Hopefully we will be able to say something about cohomology of projective space, duality, and prove the Riemann-Roch theorem for curves in the ``correct generality''.

I will be using the book by Robin Hartshorne, Algebraic Geometry.

It is strongly encouraged to go to the lectures, which are on Monday and Wednesday 2:40-3:55 in Mathematics 507.

Problem sets will appear here. Please find below the current set.

The TA for the course is Thibaut Pugin. His email address is pugin (you know where). He will grade the homeworks and also be available for questions (within reason). Feel free to arrange with him personally a method of delivering your homework (e.g., via email). Just make sure he gets it by the appropriate deadline.

Grades are deterined by a method known as the italian restaurant method (aka bistromath).

The final will be a written exam, intended mainly to see how much you actually got out of the course.

Here are the weekly problem sets. The overal file containing all the exercises is at exercises.tex, exercises.dvi, and exercises.pdf. You can see the labels listed below in the margins of the text and you should also be able to search the document with your viewer. Please do not refer to the exercises by number since they will probably change over time. Feel free to collaborate but write up your own answers. Please hit the refresh button on your browser to make sure you have to latest list and the latest exercises file.

• First set due Wednesday, February 4 (in class). Of the following list, please do:
  at least 1 of the first 2,
  at least 4 of the middle 9, and
  at least 1 of the last 2.
  • exercise-not-locally-generated-by-sections
  • exercise-quotient-j-shriek-Z
  • exercise-one-point
  • exercise-two-points
  • exercise-three-points
  • exercise-quasi-compact-closed-point
  • exercise-morphism-does-not-extend
  • exercise-closed-subscheme-does-not-extend
  • exercise-morphism-integral-schemes-surjective-stalks-not-closed
  • exercise-fibre-product-affines-not-affine
  • exercise-not-geometrically-integral
  • exercise-quasi-coherent-specialization-points
  • exercise-O-module-specialization-points
• To keep up with the course I suggest you read the following material (or the corresponding material in Hartshorne):
  • In schemes.tex, or schemes.dvi, or schemes.pdf please read the sections up to and including the section on the valuative criterion of universal closedness for quasi-compact morphisms. I will review this criterion on Monday morning, February 9.
  • In morphisms.tex, or morphisms.dvi, or morphisms.pdf please read the section on closed immersions which reviews what I did in the Lecture on Thursday, February 4, and corrects the example I tried to give at the end of that lecture.
• Second set due Wednesday, February 11 (in class). Of the following list, please do:
  • exercise-not-geometricaly-reduced
  • 5 exercises out of the section on morphisms (if you do not see a section entitled ``Morphisms'' in the file, then hit the reload button and re-down-load the file)
• I included the universal closedness and separatedness of the projective line as examples in schemes.tex, or schemes.dvi, or schemes.pdf. Please check it out (search the document for ``projective line'' to find them).
• Third set due Wednesday, February 18.
  • From the section ``Tangent spaces'' do 4 of the exercises (try to be brief, also do not always pick the easiest ones).
  • Do exercise exercise-extend-quasi-coherent
  • Do exercise exercise-simple-examples-invertible
• Fourth set due Wednesday, February 25.
  • From the section ``Proj and projective schemes'' do 5 exercises.
• To keep up with the course, please read Hartshorne, Chapter II, section 5.
• Fifth set due Wednesday, March 4: Please do all the exercises in the section entitled ``Morphisms from surfaces to curves''.
• Sixth set due Wednesday, March 11: If you are behind with exercises please catch up.
• Seventh set due Wednesday, March 25: Do 5 of the exercises from the section on divisors.

Handwritten lecture notes handwritten by Qi You: Algebraic Geometry.