MAT 589 Syllabus Introduction to Algebraic Geometry Spring 2020

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 28
  Summary of the classical ideal—variety correspondence.
  
  
• Thursday, January 30
  Summary of commutative ring theory.
  
  
• Tuesday, February 4
  Sheaves. Section II.1
  
  
• Thursday, February 6
  Locally ringed spaces. Section II.2
  Problem Set 1 due in lecture.
  
  
• Tuesday, February 11
  Affine schemes and Proj. Section II.3.
  
  
• Thursday, February 13
  Properties of morphisms. Various sections.
  Problem Set 2 due in lecture.
  
  
• Tuesday, February 18
  Fiber products and separatedness. Section II.4.
  
  
• Thursday, February 20
  Valuative criteria. Section II.4.
  Problem Set 3 due in lecture.
  
  
• Tuesday, February 25
  Proper morphisms and Chow's Lemma. Section II.4.
  
  
• Thursday, February 27
  Quasi-coherent sheaves. Section II.5.
  Problem Set 4 due in lecture.
  
  
• Tuesday, March 3
  More about quasi-coherent sheaves. Section II.5.
  
  
• Thursday, March 5
  Quotients and Proj.
  Problem Set 5 due in lecture.
  
  
• Tuesday, March 10
  Projective morphisms and divisors. Sections II.6 and II.7.
  
  
• Thursday, March 12
  Differentials. Section II.8.
  Problem Set 6 due in lecture.
  
  
• Tuesday, March 24
  Ample and very ample divisors. Section II.7.
  
  
• Thursday, March 26
  Homological algebra. Section III.1.
  Problem Set 7 due in lecture.
  
  
• Tuesday, March 31
  Properties of derived functors. Section III.2.
  
  
• Thursday, April 2
  Cohomology of quasi-coherent sheaves on a Noetherian affine. Section III.3.
  Problem Set 8 due in lecture.
  
  
• Tuesday, April 7
  Čech cohomology. Section III.4.
  
  
• Thursday, April 9
  Čech and sheaf cohomology. Section III.4.
  Problem Set 9 due in lecture.
  
  
• Tuesday, April 14
  Cohomology on projective space. Section III.5.
  
  
• Thursday, April 16
  Duality on projective space. Section III.5.
  Problem Set 10 due in lecture.
  
  
• Tuesday, April 21
  Serre duality I. Sections III.6 and III.7.
  
  
• Thursday, April 23
  Serre duality II. Section III.7.
  Problem Set 11 due in lecture.
  
  
• Tuesday, April 28
  Lefschetz theorems and differentials. Various sections.
  
  
• Thursday, April 30
  Dualizing and canonical sheaves. Section III.7.
  Problem Set 12 due in lecture.
  
  
• Tuesday, May 5
  Zariski's Main Theorem. Section III.11.
  
  
• Thursday, May 7
  Semicontinuity and base change. Section III.12.
  Problem Set 13 due in lecture.
  
  

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