Introduction to Algebraic Geometry (Schedule)

David Hilbert
projective variety
ruled surface
Zariski topology
Date Topics Read Links
2023.01.09 Course overview, Affine space §1.1
2023.01.11 Affine subvarieties, Zariski topology §1.2 notes01
2023.01.12 Parametrizations, Unirationality §1.3 email
2023.01.16 Ideals, Vanishing ideals §1.4
2023.01.18 Monomial ideals, Monomial orders §2.2 notes02
2023.01.19 Polynomial division, Division algorithm §2.3 problems1
2023.01.23 Hilbert basis, Noetherian rings §2.4
2023.01.25 Projects, Remainders §2.5 notes03
2023.01.26 S-polynomials, Buchberger criteria §2.6 research
2023.01.30 Buchberger algorithm, reduced Gröbner bases §2.7
2023.02.01 Macaulay2 basics, More Macaulay2 §2.8 notes04
2023.02.02 Elimination, Projections §3.1 problems2
2023.02.06 Implicitization, Classic examples §3.2
2023.02.08 Rational Implicitization, Toric ideals §3.3 notes05
2023.02.09 Common roots, Resultants §3.4 outline
2023.02.13 Resultants and roots, Properties of resultants §3.5
2023.02.15 Resultants and remainders, Resultants with many variables §3.6 notes06
2023.02.16 Extension theorem (statement and proof) problems3
Winter break
2023.02.27 Nullstellensatz, Radical ideals §4.1
2023.03.01 Maximal ideals, Prime ideals §4.2 notes07
2023.03.02 Sums and products, Intersections §4.3 draft
2023.03.06 Closure theorem, Colon ideals §4.4
2023.03.08 Irreducible decompositions, Primary ideals §4.5 notes08
2023.03.09 Primary decompositions, Associated primes §4.6 problems4
2023.03.13 Coordinate rings, Morphisms of affine subvarieties §5.4
2023.03.15 Homomorphisms of coordinate rings, Dominant morphisms §5.5 notes09
2023.03.16 Projective space (as a set and as a variety) §8.1 feedback
2023.03.20 Projective subvarieties, Homogenization §8.2
2023.03.22 Projective closure, Projective nullstellensatz §8.3 notes10
2023.03.23 Saturation, Projective dictionary §8.4 problems5
2023.03.27 Projective elimination, Completeness §8.5
2023.03.29 Hilbert functions, Hilbert polynomials §9.3 notes11
2023.03.30 Multiplicities, Finite affine subvarieties §9.2
2023.04.03 Finite projective subvarieties, Bézout theorem §8.7 notes12
2023.04.05 Mock presentation, Making presentations §8.6 paper
2023.04.06 Review problems6
2023.04.10 Extra office hour 12:30–13:20
2023.04.13 Extra office hour 13:00– video
2023.04.16 Exam 14:00–17:00 Gym 5