Math 512. Modern Algebraic Geometry Instructor Syllabus
Varieties
- 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)
Sheaves
- Presheaves and sheaves (3 hours, Section II.1)
- Direct and inverse images (1 hour, Section II.1)
Schemes
- 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.