Syllabus for 'Advanced Algebraic Topology'



Syllabus for the course 0366503501

"Algebraic Geometry, I"

(2020-21, Fall semester)

Lecturer: Prof. E. Shustin

Introduction to algebraic geometry.

Affine varieties: Algebraic sets and the Zariski topology. Hilbert’s Nullstellensatz. Irreducibility and dimension.

Functions, morphisms and varieties: Functions on affine varieties. Sheaves. Morphisms between affine varieties. Prevarieties. Varieties.

Projective varieties: Projective spaces and projective varieties. Cones and the projective Nullstellensatz. Projective varieties as ringed spaces.

Dimension: The dimension of projective varieties. The dimension of varieties. Blowing up. Smooth varieties. The 27 lines on smooth projective cubic surface.

Schemes: Morphisms and locally ringed spaces. Schemes and prevarieties. Fiber products. Projective schemes.

Applications of scheme theory: Hilbert polynomials. Bezout’s theorem. Divisors on curves. The group structure on a plane cubic curve. Plane cubic curves as complex tori.

Sheaves theory: sheaves and sheafification. Quasi-coherent sheaves. Locally free sheaves. Differentials. Line bundles on curves. The Riemann-Hurwitz formula. The Riemann-Roch theorem.

Prerequisites: Linear Algebra 1,2, Algebra B-1,2.

Bibliography:

M. Atiyah, I. Macdonald, Introduction to commutative algebra, Addison-Wesley (1969).

D. Bump, Algebraic geometry, Wiley (1998)

S. D. Cutkosky. Introduction to Algebraic Geometry, AMS (2018)

D. Eisenbud, J. Harris, The geometry of schemes, Springer Graduate Texts in Mathematics 197 (2000).

4. Fulton W. Algebraic curves: An introduction to algebraic geometry. W. A. Benjamin, Inc., NY, 2nd ed. (2008)

A. Gathmann. Algebraic geometry. Book online, (2014).

R. Hartshorne, Algebraic geometry, Springer Graduate Texts in Mathematics 52 (1977).

I. Shafarevich, Basic algebraic geometry, Springer Study Edition (1977).

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