MAT 589 Syllabus Introduction to Algebraic Geometry Spring 2017

The schedule for topics covered in lecture is as follows. Please do the assigned reading prior to lecture. This will make the lectures more effective for you.

• Tuesday, January 23
  Summary of the classical ideal—variety correspondence.
  
  
• Thursday, January 25
  Summary of commutative ring theory.
  
  
• Tuesday, January 31
  Sheaves. Section II.1
  
  
• Thursday, February 2
  Locally ringed spaces. Section II.2
  Problem Set 1 due in lecture.
  
  
• Tuesday, February 7
  Affine schemes and Proj. Section II.3.
  
  
• Thursday, February 9
  Properties of morphisms. Various sections.
  
  
• Tuesday, February 14
  Fiber products and separatedness. Section II.4.
  Problem Set 2 due in lecture.
  
  
• Thursday, February 16
  Valuative criteria. Section II.4.
  
  
• Tuesday, February 21
  Proper morphisms and Chow's Lemma. Section II.4.
  Problem Set 3 due in lecture.
  
  
• Thursday, February 23
  Quasi-coherent sheaves. Section II.5.
  
  
• Tuesday, February 28
  More about quasi-coherent sheaves. Section II.5.
  Problem Set 4 due in lecture.
  
  
• Thursday, March 2
  Quotients and Proj.
  
  
• Tuesday, March 7
  Projective morphisms and divisors. Sections II.6 and II.7.
  Problem Set 5 due in lecture.
  
  
• Thursday, March 9
  Differentials. Section II.8.
  
  
• Tuesday, March 21
  Ample and very ample divisors. Section II.7.
  Problem Set 6 due in lecture.
  
  
• Thursday, March 23
  Homological algebra. Section III.1.
  
  
• Tuesday, March 28
  Properties of derived functors. Section III.2.
  Problem Set 7 due in lecture.
  
  
• Thursday, March 30
  Cohomology of quasi-coherent sheaves on a Noetherian affine. Section III.3.
  
  
• Tuesday, April 4
  Čech cohomology. Section III.4.
  Problem Set 8 due in lecture.
  
  
• Thursday, April 6
  Čech and sheaf cohomology. Section III.4.
  
  
• Tuesday, April 11
  Cohomology on projective space. Section III.5.
  Problem Set 9 due in lecture.
  
  
• Thursday, April 13
  Duality on projective space. Section III.5.
  
  
• Tuesday, April 18
  Serre duality I. Sections III.6 and III.7.
  Problem Set 10 due in lecture.
  
  
• Thursday, April 20
  Serre duality II. Section III.7.
  
  
• Tuesday, April 25
  Lefschetz theorems and differentials. Various sections.
  Problem Set 11 due in lecture.
  
  
• Thursday, April 27
  Dualizing and canonical sheaves. Section III.7.
  
  
• Tuesday, May 2
  Zariski's Main Theorem. Section III.11.
  Problem Set 12 due in lecture.
  
  
• Thursday, May 4
  Semicontinuity and base change. Section III.12.
  
  

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