Algebraic Geometry MT4214 (IISER Pune)
January-April 2024
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 :
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
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
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
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
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*
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
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.