Theoretical / proof-based
Algebraic geometry · Graduate · Math
Learning objectives from the Algebraic geometry syllabus, grouped by unit. Click an objective for study materials.
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)
Learning objectives
Click an objective for study materials.
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)
Definitions and structure
- 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
Proofs and reasoning
- 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)
- Sheaves on topological spaces — Sheaves on topological spaces; sheafification
- Affine schemes and Spec of a ring — Affine schemes and Spec of a ring
Abstraction and generalization
- 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)
Modeling and computation
- Apply affine and projective varieties — Apply affine and projective varieties; nullstellensatz in engineering contexts
- Apply regular maps and rational maps in engineering contexts — Apply regular maps and rational maps in engineering contexts
- Apply birational equivalence in engineering contexts — Apply birational equivalence in engineering contexts
- Apply dimension and tangent spaces in engineering contexts — Apply dimension and tangent spaces in engineering contexts
- Apply smoothness in engineering contexts — Apply smoothness in engineering contexts
Data and technology
- Use software to explore algebraic geometry problems numerically — Use software to explore algebraic geometry problems numerically
- Interpret computational results against analytic predictions — Interpret computational results against analytic predictions
- Build spreadsheets or scripts for routine calculations — Build spreadsheets or scripts for routine calculations
- Visualize functions, fields, or datasets tied to course topics — Visualize functions, fields, or datasets tied to course topics
- Connect course methods to lab, industry, or research workflows — Connect course methods to lab, industry, or research workflows
Problem-solving practice
- 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)
- Sheaves on topological spaces — Sheaves on topological spaces; sheafification
- Affine schemes and Spec of a ring — Affine schemes and Spec of a ring
Multi-Unit Problems
Course-level sets that combine skills across study units (coming soon).
Browse Multi-Unit ProblemsWhat each unit includes
Open a unit below for full materials. Typical resources:
- Study guide
- Exam Strategy
- Common Mistakes
- Worksheets
- Word problems
- Mixed Practice
- Multi-Unit Problems
- Review
- Practice test
- Answer key
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
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