Algebraic geometry
Graduate · Math
Syllabus focus
Topics typically covered
Click a topic for the full text and related unit practice.
Graduate Algebraic Geometry — scope drawn from open algebraic geometry notes and typical US graduate algebraic geometry syllabi.
Varieties and morphisms
- Affine and projective varieties — Affine and projective varieties; Nullstellensatz
- Regular maps and rational maps — Regular maps and rational maps
- Birational equivalence — Birational equivalence
- Dimension and tangent spaces — Dimension and tangent spaces
- Smoothness — Smoothness
- Bezout's theorem — Bezout's theorem
- Intersection multiplicity (introduction) — Intersection multiplicity (introduction)
- Blow-ups and resolution of singularities (overview) — Blow-ups and resolution of singularities (overview)
Schemes and sheaves
- Sheaves on topological spaces — Sheaves on topological spaces; sheafification
- Affine schemes and Spec of a ring — Affine schemes and Spec of a ring
- Schemes: gluing, morphisms, and closed subschemes — Schemes: gluing, morphisms, and closed subschemes
- Structure sheaf — Structure sheaf
- Locally ringed spaces — Locally ringed spaces
- Fiber products — Fiber products
- Base change (introduction) — Base change (introduction)
Cohomology and curves
- Sheaf cohomology on schemes (Cech — Sheaf cohomology on schemes (Cech
- Derived functors overview) — Derived functors overview)
- Divisors and line bundles on curves — Divisors and line bundles on curves
- Riemann–Roch theorem for curves — Riemann–Roch theorem for curves
- Genus and classification of algebraic curves (introduction) — Genus and classification of algebraic curves (introduction)
- Moduli of elliptic curves (preview) — Moduli of elliptic curves (preview)
Study units
Each unit includes a study guide, worksheets, review, practice test, and answer key. One unit is free; subscribe for the full class.
- Varieties and morphisms
Affine and projective varieties; Nullstellensatz
Coming soon - Schemes and sheaves
Sheaves on topological spaces; sheafification
Coming soon - Cohomology and curves
Sheaf cohomology on schemes (Cech
Coming soon
Notes
Graduate Algebraic Geometry — scope drawn from open algebraic geometry notes and typical US graduate algebraic geometry syllabi. Topic outline: `content/topics/graduate/algebraic_geometry.json`.