Schemes, Varieties & Moduli Spaces

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Abstractions about the objects and machinery of algebraic geometry, including scheme- and stack-theoretic foundations (morphisms, sheaves, formal schemes), varieties classified by geometric type (toric, ruled, spherical), and moduli constructions (Donaldson-Thomas theory, virtual fundamental classes, quotient stacks).

41 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Algebraic geometry code — An error-correcting linear code obtained by evaluating functions or taking residues on rational points of an algebraic curve over a finite field.
• Arakelov theory — An arithmetic geometry that augments schemes over the integers with analytic data at infinite places.
• Borel–Weil–Bott theorem — A theorem realizing irreducible representations of compact or complex semisimple Lie groups as the unique nonzero cohomology of suitable line bundles on flag varieties.
• Canonical ring — The graded ring formed from global sections of all nonnegative tensor powers of a variety's canonical bundle or canonical divisor.
• Complete intersection — A scheme or variety whose defining ideal is locally generated by exactly its codimension number of elements, giving the expected minimal equation count.
• Complex polytope — A regular incidence geometry modeled in complex unitary space that generalizes real regular polytopes through complex reflections and phase-valued incidence.
• Cone (algebraic geometry) — A relative affine scheme obtained as the spectrum of a graded quasi-coherent algebra, carrying the scaling action induced by its grading and admitting an associated projective cone.
• Constructible sheaf — A sheaf that becomes locally constant with finite-type stalks on each piece of a finite stratification of its underlying space.
• Cotangent sheaf — The sheaf of relative Kahler differentials that universally represents derivations for a morphism of schemes or ringed spaces.
• Degeneration (algebraic geometry) — A family of algebraic varieties or schemes whose general fibers specialize to a distinguished, often more singular, fiber, with flatness controlling which invariants are preserved.
• Derived scheme — A scheme-like geometric object whose structure sheaf carries homotopical or differential-graded information encoding higher intersections and deformation data.
• Dimension of an algebraic variety — The intrinsic number of independent parameters of an algebraic variety, equivalently the Krull dimension of its coordinate ring in the affine irreducible case.
• Donaldson–Thomas theory — An enumerative theory assigning virtual counts to moduli spaces of stable sheaves, ideal sheaves, or related objects on Calabi–Yau threefolds.
• Formal scheme — A locally ringed space locally modeled by the formal spectrum of an adic topological ring, retaining infinitesimal neighborhoods through completion.
• Frobenioid — A category equipped with degree, divisor-like, and Frobenius structure that categorifies monoid actions arising from arithmetic line bundles.
• Geometric quotient — A quotient of an algebraic variety by a group action whose fibers are exactly orbits, topology is the quotient topology and regular functions are the invariant functions.
• Grassmannian — A parameter space whose points are the fixed-dimensional linear subspaces of a vector space.
• Inertia stack — A stack whose objects pair an object of an algebraic or differentiable stack with one of its automorphisms, thereby recording isotropy and conjugation data.
• Linear algebraic group — A matrix group defined over a field by polynomial equations in its entries and inverse determinant conditions.
• Morphism of algebraic varieties — A map between algebraic varieties that is locally given by regular polynomial or rational-function expressions without poles.
• Morphism of schemes — A morphism of locally ringed spaces between schemes, combining a continuous map of spectra with a compatible local homomorphism of structure sheaves.
• Motivic integration — An algebraic-geometric integration theory assigning classes in a Grothendieck ring to measurable subsets or functions on arc spaces, retaining geometric information beyond numerical measure.
• Nori motive — A mixed-motive construction obtained from a diagram of algebraic varieties, pairs and cohomology through Nori’s universal abelian category and coalgebra representation.
• Normal scheme — A scheme whose local rings are integrally closed domains at every point.
• Perfect obstruction theory — A morphism from a perfect two-term complex to a space's cotangent complex that correctly captures first-order deformations and obstructions and supports a virtual fundamental class.
• Prestack — A category fibered in groupoids over a site whose isomorphisms satisfy descent, while objects need not yet glue effectively as they do in a stack.
• Principal homogeneous space — A nonempty set or space with a free and transitive action of a group, also called a torsor.
• Pseudo-canonical variety — An algebraic variety whose canonical divisor or class is pseudo-ample under the convention used, placing it in the general-type side of birational classification.
• Quotient space of an algebraic stack — Associate an algebraic stack with its underlying Zariski topological space of points or integral substacks, functorially turning stack morphisms into continuous maps while forgetting stabilizer data.
• Quotient stack — An algebraic stack [X/G] that retains stabilizers and families of objects while representing a group action's quotient.
• Ran space — The topological or algebro-geometric space that organizes all nonempty finite subsets of a base space as a single varying configuration object.
• Representation on coordinate rings — A group action on an affine algebraic variety induces a contragredient linear action on its coordinate ring by precomposing regular functions with the inverse geometric action.
• Resolution of singularities — The replacement of a singular algebraic variety by a nonsingular variety connected through a proper birational morphism.
• Ruled join — The projective variety formed by the union of all lines connecting points of two separately embedded projective subvarieties.
• Ruled variety — An algebraic variety birational to a product with a projective line, so its generic points lie on a rational one-parameter ruling.
• S-equivalence — The equivalence relation identifying semistable vector bundles or sheaves whose Jordan–Hölder graded objects are isomorphic.
• Severi–Brauer variety — An algebraic variety over a field that becomes projective space after extending scalars to an algebraic closure, encoding a central simple algebra and its splitting.
• Sheaf of algebras — A sheaf on a ringed space whose sections form algebras over the structure sheaf compatibly with restriction.
• Spherical variety — An algebraic variety with an action of a reductive group for which a Borel subgroup has an open dense orbit.
• Standard monomial theory — A method constructing explicit bases for coordinate rings and line-bundle sections on flag and Schubert varieties through ordered products satisfying straightening relations.
• Toric variety — An algebraic variety containing an algebraic torus as a dense open subset whose self-action extends to the entire variety.