# Syllabus Math 512

## 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.