Instructor: Dr Anupam Singh

Audience: 4th year BS-MS, MSc students at IISER Pune

Schedule: Monday 4-5 PM, Tuesday 2 PM, Thursday 4-5 PM, Main building 41.

Evaluation: Test I - 20%, mid-sem - 30%, Test II (or mini-projects) - 20%, end-sem - 30%

Prerequisite: Rings and Modules etc

Goal of the course: We aim to introduce the basics of algebraic geometry through an interplay of geometric objects like curves, and surfaces and algebraic objects like rings, ideals, fields and solutions of polynomial equations. This course will be useful for all mathematics students and possibly useful for physics students.

Proposed Content: Revision of the basic commutative algebra required:Noetherian rings, Radicals, Hilbert Nullstellensatz, Noether normalisation, coordinate ring etc.

Affine Algebraic Varieties, Zariski Topology, Morphisms,  Dimension, Projective Varieties, morphisms, Quasi-Projective Varieties, Regular Functions, Examples: Curves, Veronese maps, Segre maps, Grassmannians,  Degree, statement of Bezout theorem.

Smoothness, Tangent space, Rational Maps, Blowing up a point, Birational Equivalence, Maps to projective space.

Introduction to algebraic groups.

Textbooks and reference material :

• An Invitation to Algebraic Geometry by Karen E Smith, Universitext

• Algebraic Geometry, A First Course by Joe Harris

• Basic Algebraic Geometry, by Shafarevich 

• Algebraic Geometry, by Hartshorne

For commutative algebra background:

• Commutative algebra: Atiyah & Macdonald

• Algebra: Dummit & Foote (Chapter 15)

Weekly topics covered :

Assignment 1 

01/01/2024: What is algebraic geometry? Solving equations...

02/01/2024: Rings, Ideals, Prime and Maximal ideals, examples.

04/01/2024: Tutorial

08/01/2024: Nil radical, Jacobson radical and radical of an ideal

09/01/2024: Noetherian ring, Hilbert basis theorem.

11/01/2024: Tutorial

Assignment 2

15/01/2024: Integral extension, integral closure, examples

16/01/2024: Module, Algebra, co-ordinate ring, more examples ring of integers etc

18/01/2014: Tutorial

22/02/2024: Algebraic independence, transcendental field extensions, Noether Normalisation Lemma

23/02/2024: Hilbert's Nullstellensatz-I, affine algebraic sets

25/02/2024: Tutorial

Assignment 3

29/01/2024: Correspondence between zero-sets and radical ideals, Hilbert's Nullstellensatz - II

30/01/2024: Morphisms between algebraic sets

01/02/2024: Tutorial

05/02/2024: Test 1

06/02/2024: Morphism vs algebra homomorphism of the coordinate ring

08/02/2024: Tutorial

Assignment 4

12/02/2024: Irreducible algebraic sets and variety

13/02/2024: Product of affine algebraic sets

15/02/2024: Tutorial 

Mid-sem exam 19-26 February 2024

27/02/2024 Projective space, projective algebraic sets

29/02/2024 Tutorial

04/03/24 Homogeneous ideal, Projective nullstellensatz

05/03/24 Projective closure of affine sets, conics

07/03/24 Tutorial

Assignment 5

11/03/2024: morphisms of projective sets, projective equivalence, Aut(P^n)

12/03/2024: Quasi-projective sets

14/03/2024: Tutorial

18/03/2024: Regular functions, regular functions on affine

19/03/2024: Veronese maps, Segre embedding, Grassmannians

21/03/2024: Tutorial

25/03/2024: Holiday

26/03/2024: Tangent space, non-singular varieties

28/03/2024: Tutorial

29/03/2024: (Extra class at 10 AM - 12PM): Localisation, tangent space <---> M/M^2* 

Assignment 6

01/04/2014: Derivations, point-derivations and tangent space

02/04/2014: Test 

04/04/2024: Tutorial 

08/04/2024: No class 

09/04/2024: Holiday

11/04/2024: Holiday

Assignment 7

15/04/2024: Rational maps, Birational equivalence, variety birational equivalent to hypersurface

16/04/2024: Where we go from here! Elliptic Curves, Curves, Algebraic Groups etc

18/04/2024: Tutorial 

End Sem exam 22/04/2024 at 3 PM.