Algebraic Geometry¶

← Back to Domain-Specific Abstractions by Domain

57 domain-specific abstractions whose origin domain is Algebraic Geometry.

Algebraic Cycle — A dimension-graded formal integer sum of integral closed subvarieties that can be compared by geometric equivalence and assembled into Chow groups.
Algebraic space — A sheaf on the étale site admitting a representable étale surjection from a scheme, generalizing schemes by allowing étale-local rather than Zariski-local affine charts.
Ample line bundle — A line bundle whose sufficiently high tensor power gives an embedding of a projective variety into projective space, expressing algebro-geometric positivity.
Castelnuovo–Mumford Regularity — Locate the least projective twist whose diagonal higher-cohomology vanishings persist, yielding one integer bound on global generation, Hilbert-function stabilization, and graded syzygy degrees.
Categorical quotient — A universal invariant morphism from an object with group action through which every other invariant morphism factors uniquely.
Circular algebraic curve — A real plane algebraic curve whose highest-degree homogeneous part is divisible by x squared plus y squared, equivalently passing through both circular points at infinity.
Coherent sheaf — A sheaf of modules locally having a finite presentation whose relations are themselves finitely generated, providing a stable algebraic model of geometric data.
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.
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 topology — The compact Hausdorff totally disconnected refinement of the Zariski topology on a scheme or spectrum, generated by making quasi-compact open sets and their complements open.
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.
Descent (Mathematics) — Recover a global mathematical object from compatible local objects by comparing pullbacks on fiber-product overlaps, enforcing a cocycle, and proving the resulting datum effective.
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.
Enriques–Kodaira classification — A birational classification of compact complex surfaces by Kodaira dimension and minimal-model invariants.
Finite morphism — A morphism of schemes that is affine and whose induced coordinate-ring algebra is finite as a module, generalizing maps with algebraically finite fibers.
Formal scheme — A locally ringed space locally modeled by the formal spectrum of an adic topological ring, retaining infinitesimal neighborhoods through completion.
Formally smooth map — A ring map with the infinitesimal lifting property against nilpotent quotient extensions.
Gabriel–Rosenberg Reconstruction Theorem — A categorical reconstruction theorem recovering a suitably separated scheme, including its topology and structure sheaf, from the abelian category of its quasi-coherent sheaves.
Geometric Langlands correspondence — A conjectural categorical correspondence relating local systems for a reductive group on an algebraic curve to sheaf-theoretic objects on the moduli stack of bundles for its Langlands dual group.
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.
Gerbe — A stack locally equivalent to the classifying stack of a group, serving as a degree-two geometric analogue of a principal bundle and encoding obstruction and twisting data.
Grassmannian — A parameter space whose points are the fixed-dimensional linear subspaces of a vector space.
Hurwitz scheme — An algebraic moduli scheme parameterizing branched covers of a fixed target curve, commonly degree-d genus-g covers of the projective line with specified ramification data.
Ideal sheaf — Assign an ideal of functions to every open set compatibly with restriction, so local vanishing conditions glue into a global sheaf and quasi-coherent ideal sheaves determine closed subschemes.
Iitaka dimension — An invariant measuring asymptotic growth of sections of powers of a line bundle, equivalently the dimension of the image of its associated rational maps.
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 finite type — A scheme morphism that is locally induced by finitely generated algebras, expressing algebraic dependence on finitely many generators without requiring module finiteness.
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.
Nash blowing-up — Replace a singular variety by the closure of the graph of its smooth-point tangent-space map, retaining each singular point together with the limiting tangent spaces approached nearby.
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.
Projective bundle — A fiber bundle or scheme morphism locally modeled on projective space, often obtained by projectivizing a vector bundle.
Quasi-finite morphism — A finite-type morphism of schemes whose fibers are zero-dimensional and finite, equivalently one that is locally finite over each image point.
Quaternary cubic — A homogeneous polynomial of degree three in four variables, whose projective zero locus is a cubic surface and whose coefficients carry a classical ring of invariants.
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.
Regular scheme — A locally Noetherian scheme whose every local ring is regular, so local dimension equals the minimal number of generators of its maximal ideal.
Resolution of singularities — The replacement of a singular algebraic variety by a nonsingular variety connected through a proper birational morphism.
Rosati involution — The positive involutive anti-automorphism of the rational endomorphism algebra of a polarized abelian variety obtained by taking the dual endomorphism and conjugating through the polarization.
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.
Segre Class — A graded intersection-theoretic cycle class of a cone or closed embedding that records its projective directions and reduces to the inverse Chern class of the normal bundle for a regular embedding.
Seshadri Constant — A local positivity invariant of a nef line bundle at a point, computed by the least ratio of curve intersection to curve multiplicity, equivalently by the nef threshold on the blowup.
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.
Tacnode — A plane-curve double singularity where two smooth local branches share the same tangent, canonically modeled by y²=x⁴ and carrying higher contact than an ordinary node.
Tautological Ring — The minimal operation-stable system of natural cycle-class subrings on moduli spaces of stable pointed curves, generated through forgetful and gluing morphisms and carrying the standard psi, kappa, lambda, and boundary constructions.
Ternary cubic — A homogeneous polynomial of degree three in three variables, studied through plane cubic curves and invariant theory.
Toric variety — An algebraic variety containing an algebraic torus as a dense open subset whose self-action extends to the entire variety.
Twisted sheaf — A sheaf-like object whose local pieces glue only up to multiplication by a prescribed gerbe or multiplicative two-cocycle, encoding sheaves on a twisted geometric background.
V-topology — A very fine Grothendieck topology in algebraic geometry whose covers are universally subtrusive and can be tested by lifting valuation-ring maps.