Algebraic Geometry

Time: the second semester

Course Classification: elective course

Credit: 3 hours

Period: 54 hours

Previous courses: Abstract Algebra

Syllabus:

(1) Sheaves: presheaves, the sheaf associated to a presheaf, stalks, injectivity and surjectivity between sheaves, abelian categories of presheaves and sheaves

(2) Schemes: locally ringed spaces, affine schemes and schemes, morphism to affine schemes, projective schemes, rational points on schemes

(3) Properties of schemes: reduced and irreducible, Noetherian scheme, finite type, dimension, fibred product, fibres, pass from finite field to complex field

(4) Separated and proper morphism: separated morphism, valuative criterion, proper morphism, properness of projective morphism

(5) Sheaves of modules: quasi-coherent sheaves and coherent sheaves, the affine and projective, scheme case, global generated property, finiteness of coherent sheaves

(6) Divisors: Weil divisor, divisor on curves, Cartier divisor, invertible sheaves, Chow groups

References:

[1] R. Hartshorne, Algebraic Geometry, Springer-Verlag, GTM 52.

[2] Q. Liu, Algebraic Geometry and Arithmetic Curves, Oxford University Press, 2002.

[3] J. P. Serre, How to use finite fields for problems concerning infinite fields, ArXiv: 0903.0517.