Math 720: Seminar: Commutative Algebra and Algebraic Geometry
Spring 2017

Instr: Mike O'Sullivan


I will try to keep to this schedule but will update it as needed.

IVA is

Cox, Little, O'Shea Ideals, Varieties, and Algorithms:
An Introduction to Computational Algebraic Geometry and Commutative Algebra

4th Ed., Springer-Verlag, 2015.

> >
Day Topics Reading
Th. 1/19 Short intro to course.
IVA 1.1,2
Videos The polynomial ring in one variable: Main Properties IVA 1.5, H. Ch. 1-5
Tu. 1/24 Ideals in polynomial rings.
Varieties: Solutions of an ideal.
IVA 1. 2,4-5
Videos Rings, ideals, homomorphisms. The Chinese Remainder Theorem. H. Ch. 3, 6
Th. 1/26 Parametrization and implicitization.
Computing using SageMathCloud
IVA 1.3,4,IVA 2.1
Tu. 1/31 Monomial Orderings
Monomial ideals and Dickson's Lemma
IVA 2.2
IVA 2.4
Th. 2/2 Groebner basis and
division in multivariate polynomial rings.
IVA 2. 3,5
Tu. 2/7 The ascending chain condition
Hilbert basis theorem.
IVA 2.4
Tu. 2/9 Problems solved with Groebner bases.
Computations in Sage.
IVA 2.4
Tu. 2/14 Syzygy polynomials
Buchberger's algorithm.
IVA 2.5, 6,7
Th. 2/16 Buchberger's algorithm, proof. IVA 2.6, 7, 9
Tu. 2/21 TEST: Division, Groebner basis.
Systems of representatives for a quotient ring.
IVA Ch 1, 2
Th. 2/23 Elimination and Extension theorems.
Geometrical interpretation.
IVA 3.1,2
Tu. 2/28 Extension theorems, a proof.
Computational examples of elimination and extension.
IVA 3.1,2
Th. 3/2 Parametrized varieties and implicitization. IVA 1.3, 2.8(end), 3.3
Tu. 3/7 Radical ideals and Hilbert's Nullstellensatz.
Zariski closure.
IVA 4.1,2
Th. 3/9 Hilbert's Nullstellensatz.and the algebra geometry dictionary
Radical ideals: computation questions.
IVA 4.1,2,4 (1st pages), 5 (1st pages)
Tu. 3/14 Operations on ideals and geometric analogues. IVA 4.3
Th. 3/16 Computation of the intersection.
Ideal quotients.
IVA 4.3,4
Tu. 3/21 Ideal quotients and saturations.
Primes and irreducibility.
IVA 4.4,5
Th. 3/23 Primes and irreducibility
The closure theorem.
IVA 4.4,5
Tu. 4/4 Review problems. IVA Ch 3, 4.
Tu. 4/6 TEST. IVA Ch 3, 4.
Tu. 4/11 Localization/rings of fractions. Notes.
Tu. 4/13 Localization/rings of fractions: Problem solving. Notes.
Tu. 4/18 Cone varieties, homogeneous ideals, the cone over a variety. VIII.1-4x.
Th. 4/20 Projective geometry. VIII.1-4.
Tu. 4/25 Cone varieties, homogeneous ideals,
Projective varieties: problem solving.
Tu. 4/27 No class: discuss projects with me. .
Tu. 5/2 ?. .
Th. 5/4 No class: discuss projects with me. .
Tu. 5/9 (Final 10:30-12:30) Presentations. .