Introduction to Algebraic Geometry

Week 1: Topics in Commutative Algebra : Integral Extensions, Going-up and Going-down theorems.
Week 2: Topics in Commutative Algebra (Contd.): Valuation Rings, Noether Normalization lemma, Discrete Valuation Rings, Dimension Theory, Affine Algebraic Sets, Weak Nullstellensatz, Strong Nullstellensatz.
Week 3: Regular Morphisms, Abstract Algebraic Sets, Zariski Topology on Affine Spaces, Irreducible Components, Affine and Quasi-affine Varieties, Projective Spaces.
Week 4: Zariski Topology on Projective Spaces, Projective and Quasi-projective Varieties, Regular Functions on Quasi-projective Varieties, Presheaves & Sheaves, Morphisms of Presheaves & Sheaves.
Week 5: Stalks, Sheafification, Push-forward, Inverse Image, Subsheaves, Gluing of Sheaves, Prevarieties, Ring of Regular Functions and Field of Rational Functions on Affine Varieties.
Week 6: Equivalence of Categories between the Category of Affine k-varieties and the Category of Finitely Generated Integral Domains over k, Closed and Open Immersions, Constructible Sets, Chevalley's Theorem, Product of Prevarieties.
Week 7: Products and Coproducts in a Category, Product of Quasi-affine Varieties, Diagonal Morphisms, Varieties, Global Regular Functions on Projective Varieties are Constants.
Week 8: Segre Embeddings, d-uple embedding, Gluing Morphisms, Finite Morphisms.
Week 9: Immersions, Subvarieties, Criterion for Closed Immersions, Graph of a Morphism, Separatedness.
Week 10: Proper Morphisms, Complete Varieties, Zariski Tangent Space, Singular & Non-singular Points.
Week 11: Blow-ups, Smooth Morphisms, Sard's Theorem, Bertini's Theorem.
Week 12: Abstract Non-singular Curves, Introduction to Affine Schemes.

References:
1) Algebraic Geometry by Robin Hartshorne
2) The Red Book of Varieties and Schemes by David Mumford

The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).

Date and Time of Exams: : October 25, 2025 Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.

The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.

Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE

Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
Exam score = 75% of the proctored certification exam score out of 100

Final score = Average assignment score + Exam score

Please note that assignments encompass all types (including quizzes, programming tasks, and essay submissions) available in the specific week.

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75. If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.

Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Madras .It will be e-verifiable at nptel.ac.in/noc.

Only the e-certificate will be made available. Hard copies will not be dispatched.

Once again, thanks for your interest in our online courses and certification. Happy learning.