## 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.