This is an introductory course in algebraic geometry in which we mainly study the computational aspect of algebraic geometry. In this course, macaulay-2 (a mathematical software) will be used for the computation.

Instructor: Amit Kumar Singh

Tutor: Abhiram Subramanian

Books:

1. Computational Algebraic Geometry by Hal Schenck.

2. Ideals, Varieties, and Algorithms by D. Cox, J. Little and D. O'Shea.

3. Computations in algebraic geometry with Macaulay 2 (https://macaulay2.com/Book/ComputationsBook/book/book.pdf)

Topics covered

Lecture 1: (Polynomials and affine spaces) Studied a basic properties of monomials. Defined affine n-spcaces and their basic notions.

Suggested reading:  Cox, Little and O'Shea, Chapter 1, sections 1 and 2.

Lecture note CAG-lec-01.pdf

Lecture 2: (Affine varieties) Defined affine varieties. Studied the bijection between the set of affine varieties in the affine n-space and the set of radical ideals in the polynomial ring in n-variables. 

Suggested reading:  Hartshorne's book titled Algebraic Geometry, [Chapter-I, section 1] 

Lecture note CAG-lec-02.pdf

Lecture 3: (Zariski topology) Defined Zariski topology on the affine varieties. Shows how the Zariski topology is different from the usual metric topology. Studied the irreducibility of topological spaces.

Suggested reading:  [Hartshorne's book titled Algebraic Geometry, Chapter-I (section 1)] and [Cox, Little and O'Shea, Chapter 4].

Homework-01 (due Aug 19, 2024)

Lecture 4: (Projective varieties) Defined the projective varieties and other basic notions related to it. Further studied that there is a bijective inclusion-reversing correspondence between the set of projective varieties X and the set of homogeneous radical ideals of S(Y) not equal to the irrelevant ideal. 

Suggested reading:  [Hartshorne's book titled Algebraic Geometry, Chapter-I (section 2)] and [Cox, Little and O'Shea, Chapter 4].

Lecture note CAG-lec-04.pdf

Lecture 5: (Sheaf of regular functions) 

Suggested reading:  [Hartshorne's book titled Algebraic Geometry, Chapter-I (section 3)] and [Cox, Little and O'Shea, Chapter 4].

Lecture 6: (Practical Session) Learned about the basic notion of Macaulay 2 (https://macaulay2.com/). Computed several examples of radical ideals and primary decompositions and so on with the help of Macaulay.  Notes

Suggested reading:  Hal Schenck, Chapter 1

Lecture 7: (Orderings on the Monomials in k[x1, . . . , xn]) we can reconstruct the monomial x^α = x^α1 ··· x^αn from the n-tuple of exponents α = (α1 , . . . , αn ) ∈ Z_n≥0 . This observation establishes a one-to-one correspondence between the monomials in k[x1,...,xn] and Z_n≥0. Furthermore, any ordering > we establish on the space Zn≥0 will give us an ordering on monomials: if α > β according to this ordering, we will also say that x^α > x^β. 

Suggested reading:  [Cox, Little and O'Shea, Chapter 2, Sections 1, 2].

Lecture 8: (A Division Algorithm in k[x1, . . . , xn]) Studied a division algorithm for polynomials in k[x1, . . . , xn] that extends the algorithm for k[x].

Suggested reading:  [Cox, Little and O'Shea, Chapter 2, Section 3].

Homework-02 (due Sep 2, 2024)

Lecture 9: Studied the monomial ideals and Dickson's lemma. And applications of the Dickson's lemma.

Suggested reading:  [Cox, Little and O'Shea, Chapter 2, Section 4].

Lecture 10: Studied the Hilbert Basis Theorem and Gröbner Bases. And applications of Gröbner Bases, Buchberger's algorithm for modules.

Suggested reading:  [Cox, Little and O'Shea, Chapter 2, Section 5].

Lecture 11: Studied the Hilbert Fuctions and polynomials.

Suggested reading:  Hal Schenck, Chapter 2. 

Lecture note CAG-lec-11.pdf

Lecture 12: (Practical Session)  Computed several examples of the Hilbert functions , Hilbert Series and so on with the help of Macaulay 2 (https://macaulay2.com/). 

Suggested reading:  Hal Schenck, Chapter 2 and do all the exercises with the help of Macaulay 2.

Lecture 13: Studied the free resolution of K[x_1, . . . ,x_n]-modules.

Suggested reading:  Hal Schenck, Chapter 3. 

Lecture note CAG-lec-13.pdf

Lecture 14: Studied the graded free resolution of K[x_1, . . . ,x_n]-modules, and computed the hilbert functions and polynomials of the graded module by using the graded free resolutions.

Suggested reading:  Hal Schenck, Chapter 3. 

Lecture note CAG-lec-14.pdf

Lecture 15: (Practical Session)  Computed several examples of the graded free resolution of K[x_1, . . . ,x_n]-modules, and computed the hilbert functions and polynomials of the graded module by using the graded free resolutions with the help of Macaulay 2 (https://macaulay2.com/). 

Suggested reading:  Hal Schenck, Chapter 3 and do all the exercises with the help of Macaulay 2.

Lecture 16: (Practical Session)  Continued Lecture 15.

Suggested reading:  Hal Schenck, Chapter 3 and 4 and do all the exercises with the help of Macaulay 2.