Syllabus Math 512


Math 512. Modern Algebraic Geometry Instructor Syllabus


  • Affine and projective varieties (3 hours, Sections I.1-I.2) 
  • Morphisms (2 hours, Section I.3)
  • Rational maps (2 hours, Section I.4)
  • Nonsingular varieties (1 hour, Section I.5)


  • Presheaves and sheaves (3 hours, Section II.1)
  • Direct and inverse images (1 hour, Section II.1)


  • Spec of a ring and the Zariski topology (2 hours, Section II.2)
  • Reduced and nonreduced schemes, irreducible components, and integral schemes (2 hours, Section II.3)
  • Proj of a graded algebra (2 hours, Section II.2)
  • Separated and proper morphisms (2 hours, Section II.4)

Sheaves of Modules

  • Coherent and quasicoherent sheaves on affine and projective schemes (3 hours, Section II.5)
  • Vector bundles (2 hours, Section II.5)

Divisors and Invertible Sheaves

  • Weil and Cartier divisors (3 hours, Section II.6)
  • The divisor class group and the Picard group (3 hours, Section II.6)
  • Linear systems, ample invertible sheaves, and projective morphisms (3 hours, Section II.7)
  • Blowing up (3 hours, Section II.7)

Kähler differentials

  • Non singularity and Bertini’s theorem (3 hours, Section II.8) 
  • The canonical sheaf (2 hours, Section II.8)

(1 hour leeway) 

Suggested Text:

R. Hartshorne, Algebraic Geometry. Section numbers above are drawn from this text.

Recommended assessment: 

Regular homework (7-10 assignments) and final project (read one or more papers from the literature, write a 5-10 page expository paper on the topic, present it to the class).

Approved by the GAC February 2014.