Examples and Counterexamples in Elementary Algebraic Geometry¶

A curated collection of concrete examples, counterexamples, constructions, and computations in elementary algebraic geometry.

About¶

This project collects worked examples that complement standard textbooks (Hartshorne, Vakil, Stacks Project, etc.), giving learners concrete objects to work with when entering the field.

Overview of scheme morphism properties (chart by Konrad Voelkel)

The collection covers:

Schemes & morphisms: affineness, flatness, smoothness, etale maps, base change
Sheaves & bundles: coherent sheaves, locally free sheaves, normal bundles
Curves: genus, inflection points, automorphisms, Hasse principle violations
Cohomology: local duality, Borel-Moore homology, derived categories, Fourier-Mukai transforms
Computations: Picard/class groups, intersection theory, Hodge theory, weight filtrations
Moduli & deformation: stability, moduli stacks, \(\operatorname{Pic}(\mathscr{M}_{1,1})\), deformation of sheaves

ECAG-Bench¶

ECAG-Bench is an LLM benchmark derived from this collection, testing whether language models can construct concrete algebraic geometry examples satisfying given properties.

Contributing¶

We welcome contributions! See CONTRIBUTING.md for guidelines. Examples marked as Stub are open for community contributions.

License¶

CC BY 4.0 — Waerden, 2018-2026.