Topics in algebraic geometry
(reading course)
(2023, Jan–May)

Instructors: Mrinal Kumar and Ramprasad Saptharishi

Schedule: Monday, Wednesday at 14:00 — 15:30 (IST)

Problem sets

  • Problem set 1: [PDF] (Due date: 2023-03-05)

Lecture-wise plan

Lecture 1 (Wednesday, 2023-01-18)
Broad goals and plan for the course, introduction to ideals and varieties, a proof of (Weak) Hilbert’s Nullstellensatz.
Lecture 2: (Monday, 2023-01-23)
Strong form of Nullstellensatz, introduction to the Zariski topology. Operations on rings – quotients and localisations, viewing points as maximal ideals.
Lecture 3 (Wednesday, 2023-01-25)
The algebraic geometry dictionary, coordinate rings and local rings. Statement of Bezout’s theorem and proof of Pascal’s Mystic Hexagram Theorem.
Lecture 4 (Monday, 2023-01-30)
(presented by Varun) Polynomial rings are Noetherian (Hilbert Basis Theorem)
Lecture 5 (Wednesday, 2023-02-01)
(presented by Varun) Algebraic subsets of the plane
Lecture 6 (Monday, 2023-02-06)
(presented by Varun) Modules and finiteness conditions
Lecture 7 (Wednesday, 2023-02-08)
(presented by Varun) Proof of Zariski’s lemma
Lecture 8 (Monday, 2023-02-13)
(presented by Anirban) maps between varieties and between coordinate rings
Lecture 9 (Wednesday, 2023-02-15)
(presented by Anirban) coordinate changes, rational functions and local rings
Lecture 10 (Wednesday, 2023-02-22)
(presented by Anirban) more on local rings and discrete valuation rings
Lecture 11 (Monday, 2023-02-27)
(presented by Anirban) Properties of finite varieties
Lecture 12 (Wednesday, 2023-03-01)
(presented by Anirban) Finishing the proof that coordinate ring of a finite variety is a product direct product of quotients of local rings. (presented by Ramprasad) visualising exact sequences (from Ravi Vakil’s exposition)
Lecture 13 (Monday, 2023-03-06)
(presented by Ramprasad) Singular points on plane curves, definition of tangent lines and multiplicities via local rings
Lecture 14 (Wednesday, 2023-03-08)
(presented by Ramprasad) More on multiplicities defined via local rings, towards intersection numbers
Lecture 15 (Monday, 2023-03-13)
(presented by Mrinal) A brief introduction to algebraic-geometric codes
Lecture 16 (Tuesday, 2023-04-11)
(presented by Ramprasad) Function fields, valuation rings, and places
Lecture 17 (Wednesday, 2023-04-12)
(presented by Ramprasad) Divisors, Reimann-Roch spaces and definition of algebraic-geometric codes
Lecture 18 (Monday, 2023-04-17)
(presented by Ramprasad) Relationship between degree of a divisor and dimension of the Reimann-Roch space, genus of a curve, rate-distance analysis of AG codes
Lecture 19 (Monday, 2023-04-24)
(presented by Ramprasad) Decoding algorithms for AG codes