Course Identification

Algebraic geometry
20254111

Lecturers and Teaching Assistants

Prof. Vladimir Hinich
N/A

Course Schedule and Location

2025
First Semester
Tuesday, 11:15 - 13:00, Jacob Ziskind Building, Rm 155
Tuesday, 14:15 - 16:00, Jacob Ziskind Building, Rm 155
05/11/2024
28/01/2025

Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; 4.00 points

Comments

The lecture on on December, 24, 2024 will be cancelled.
There will be two more lectures (at the usual hours 11-13 and 14-16) on Monday, February 3 and February 10
An additional session will be held on February, 4, at the usual time.

Prerequisites

Standard Linear Algebra and Algebra courses.

Restrictions

20

Language of Instruction

English

Attendance and participation

Expected and Recommended

Grade Type

Numerical (out of 100)

Grade Breakdown (in %)

30%
70%

Evaluation Type

Take-home exam

Scheduled date 1

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-
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Estimated Weekly Independent Workload (in hours)

5

Syllabus

Algebraic geometry

  1. Introduction: affine varieties, projective varieties, advertisement: Bezout theorem and 27 lines on a cubic surface.
  2. Spaces with functions as a language of algebraic geometry (in this course). Affine varieties. Algebraic varieties. Existence of affine varieties.
  3. Hilbert basis theorem. Nullstellensatz. Noether normalization lemma. Dimension. Irreducible components. Dimension.
  4. Modules. Rings and modules of fractions. Nakayama lemma.
  5. Affine and finite maps. Closed embedding. Projective varieties. Hypersurfaces and Principal ideal theorem.
  6. Tensor product of modules and algebras. Product of algebraic varieties. Algebraic groups.
  7. Separatedness. Properness. Chow lemma.
  8. Tangent space. Smoothness.
  9. Basics of algebraic curves.
  10. Sheaves. Quasicoherent sheaves. Invertible sheaves. Divisors.
  11. Riemann-Roch theorem for curves.

Learning Outcomes

Upon successful completion of the course the student should be able to demonstrate an understanding of basic concepts and the language of algebraic geometry.

 

Reading List

  1. G. Kempf, Algebraic varieties, Cambridge University Press (principal source)
  2. R. Hartshorne, Algebraic geometry, GTM, Springer
  3. M. Atiyah, I. Macdonald, Introduction to commutative algebra, Addison-Wesley.

Website

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