1 — Affine Varieties & Coordinate Rings
Definitions, prime ideals, Zariski topology, Hilbert’s Nullstellensatz, and coordinate rings as function rings.
Affine Variety
Nullstellensatz
Content for roughly two semesters: first, classical algebraic geometry (Hartshorne, Chapter I; Gathmann, Chapters 1–10), then the gateway to schemes (Hartshorne, Chapter II), guided by Ellingsrud–Ottem’s wonderfully intuitive Introduction to Schemes.
Definitions, prime ideals, Zariski topology, Hilbert’s Nullstellensatz, and coordinate rings as function rings.
Projective space via gluing affine patches; homogeneous ideals; projective Nullstellensatz; projective closure.
Zariski open/closed sets, irreducibility, constructible sets, and how topology converses with algebra.
Morphisms of varieties, regular maps, local rings, and a surface-level glance at sheaves.
Rational maps, function fields, dominant maps, birational equivalence, categorical perspectives.
Motivation for blow-ups, examples, resolving singularities, and taming ill-defined maps.
Tangent spaces, regular local rings, non-singular points, and parallels to smooth manifolds.
Chains of irreducible closed sets, Krull dimension, and transcendence degree.
Product varieties, fibre products, parameter spaces, and a first look at moduli.
Why schemes? Limits of classical varieties, gluing spectra, and a teaser of the functorial viewpoint.
Formal motivation, gluing affine schemes, and the guiding philosophy.
Prime spectra, the structure sheaf, basic open sets, and localisation.