Introduction to Algebraic Geometry

• Instructors : Dilip P. Patil and Chandan Saha   Lecture time : M,Th 17:00-18:30   Venue : CSA 117


• Objective : This course is open to all students/faculty who are interested in learning the basics of Algebraic Geometry. The main aims of this course are to make students think, stimulate them into active learning and show them the excitement of doing mathematics on their own, enthuse them for more advanced topics with confidence, assist them to realize their potential, nurture their mathematical talent and appreciate the deep effects on the application world.
  
  Starting with a short recapitulation of the basic algebraic preliminaries on Rings and Ideals (with proofs), some classical as well as modern material will be covered. The focus of the lectures will be on motivating examples, concepts and results which emerged from the two main sources Algebraic Number Theory and Algebraic Geometry. The emphasis will be given to motivate the development of important concepts using as many examples as possible. These examples will range from routine to fairly sophisticated theoretical ones. Special efforts will be made to encourage the participants to ask questions, raise doubts and seek clarifications in the class-room.


• Prerequisites : Efforts will be made to keep the required prerequisites as low as possible. However, some exposure to Linear Algebra/Algebra/Topology will be helpful.


• Syllabus (tentative):
  • Algebraic Preliminaries :
    • Examples of Rings. Ideals, quotient rings, Prime and Maximal ideals, Chinese remainder theorem.
    • Polynomial rings, Zeros of polynomials, Resultants and Discriminants, Gröbner basis, Hilbert basis theorem. Euclidean rings, Principal ideal domains and Factorial rings. Factorisation in rings.
    • Elementary symmetric functions and Fundamental theorem on symmetric functions. Proof of Fundamental theorem of Algebra.
    • Noetherian rings and Modules, Graded rings and modules, Formal power series rings.
    • Local rings, Nakayamma-lemma. Localisation, Primary decomposition, Integral extensions.
    • Finite and Algebraic extensions. Algebraic closure, Algebraically closed fields. Galois Extensions. The fundamental theorem of Galois theory. Transcendental extensions.
    • Finite algebras over a field — Finite dimensional division algebras.
    • Finite type algebras over a field — Noether's Normalisation Lemma.
  
  
  • Basic Algebraic Geometry :
    • Affine varieties, Algebra-Geometry Dictionary : Various forms of Hilbert's Nullstellensatz, Ideal- Variety correspondence, Irreducible varieties and Prime ideals, Decomposition of a variety into irreducibles.
    • The prime spectrum of a ring — Affine schemes.
    • Projective algebraic geometry — Projective varieties, Bezout's theorem.
    • The Dimension of a variety — The Hilbert function and the Dimension, Elementary properties of Dimension, Dimension and algebraic independence, Dimension and non-singularity.


• References :
  • Commutative Algebra :
    1. Artin, M. , Algebra, Prentice-Hall, 1994.
    2. Atiyah, M. F. and Macdonald, I. G., Introduction to Commutative Algebra, Addison-Wesley, 1969.
    3. Jacobson, N., Basic Algebra, Vols. I & II, Hindustan Pub. Co., 1984.
    4. Serre, J.-P. , Local Algebra (Translated from French), Springer Monographs in Mathematics, Springer-Verlag, 2000.
    5. Zariski, O. and Samuel, P., Commutative Algebra, Vols. I & II, Van Nostrand, 1958 and 1960.
  • Algebraic Geometry :
    1. Cox, D., Little, J. and O'Shea, D. , Ideals, Varieties and Algorithms, Undergraduate Texts in Mathematics, Springer-Verlag, 1996.
    2. Cox, D., Little, J. and O'Shea, D. , Using Algebraic Geometry, Graduate Texts in Mathematics, Volume 185, Springer-Verlag, 1998.
    3. Fulton, W., Algebraic Curves, Benjamin, 1969.
    4. Patil, D. P. and Storch, U. : Introduction to Algebraic Geometry and Commutative Algebra, Cambridge University press India Pvt. Ltd. 2012.
    5. Shafarevich, I. R. , Basic Algebraic Geometry, Springer-Verlag, 1974.
  
  
• Lecture notes : (Written by Dilip P. Patil)
  • Polynomials in one variable
  • Polynomials in several variables