Inter-universal Teichmüller theory
Inter-universal Teichmüller theory (abbreviated as IUT or IUTT) is the name given by mathematician Shinichi Mochizuki to a theory he developed in the 2000s, following his earlier work in arithmetic ge
Hypertoric variety
In mathematics, a hypertoric variety or toric hyperkähler variety is a quaternionic analog of a toric variety constructed by applying the hyper-Kähler quotient construction of N. J. Hitchin, A. Karlhe
Ruled variety
In algebraic geometry, a variety over a field k is ruled if it is birational to the product of the projective line with some variety over k. A variety is uniruled if it is covered by a family of ratio
Biholomorphism
In the mathematical theory of functions of one or more complex variables, and also in complex algebraic geometry, a biholomorphism or biholomorphic function is a bijective holomorphic function whose i
Spectral space
In mathematics, a spectral space is a topological space that is homeomorphic to the spectrum of a commutative ring. It is sometimes also called a coherent space because of the connection to coherent t
Wall-crossing
In algebraic geometry and string theory, the phenomenon of wall-crossing describes the discontinuous change of a certain quantity, such as an integer , an index or a space of BPS state, across a codim
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The de
Spherical variety
In algebraic geometry, given a reductive algebraic group G and a Borel subgroup B, a spherical variety is a G-variety with an open dense B-orbit. It is sometimes also assumed to be normal. Examples ar
Associative algebra
In mathematics, an associative algebra A is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some f
Divisorial scheme
In algebraic geometry, a divisorial scheme is a scheme admitting an "ample family" of line bundles, as opposed to an ample line bundle. In particular, a quasi-projective variety is a divisorial scheme
Drinfeld module
In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, t
Smooth completion
In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve X is a complete smooth algebraic curve which contains X as an open subset. Smooth completio
Stein factorization
In algebraic geometry, the Stein factorization, introduced by Karl Stein for the case of complex spaces, states that a proper morphism can be factorized as a composition of a finite mapping and a prop
Tate twist
In number theory and algebraic geometry, the Tate twist, named after John Tate, is an operation on Galois modules. For example, if K is a field, GK is its absolute Galois group, and ρ : GK → AutQp(V)
Fröberg conjecture
In algebraic geometry, the Fröberg conjecture is a conjecture about the possible Hilbert functions of a set of forms. It is named after Ralf Fröberg, who introduced it in Fröberg . The Fröberg–Iarrobi
Nash blowing-up
In algebraic geometry, Nash blowing-up is a process in which, roughly speaking, each singular point is replaced by all limiting positions of the tangent spaces at the non-singular points. More formall
Torsor (algebraic geometry)
In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torso
Canonical singularity
In mathematics, canonical singularities appear as singularities of the canonical model of a projective variety, and terminal singularities are special cases that appear as singularities of minimal mod
Algebraic geometry of projective spaces
Projective space plays a central role in algebraic geometry. The aim of this article is to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective space
Dwork family
In algebraic geometry, a Dwork family is a one-parameter family of hypersurfaces depending on an integer n, studied by Bernard Dwork. Originally considered by Dwork in the context of local zeta-functi
Grothendieck–Katz p-curvature conjecture
In mathematics, the Grothendieck–Katz p-curvature conjecture is a local-global principle for linear ordinary differential equations, related to differential Galois theory and in a loose sense analogou
Enumerative geometry
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by means of intersection theory.
Motive (algebraic geometry)
In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theorie
Projective line over a ring
In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A with 1, the projective line P(A) over A consists of points identified by
Smooth topology
In algebraic geometry, the smooth topology is a certain Grothendieck topology, which is finer than étale topology. Its main use is to define the cohomology of an algebraic stack with coefficients in,
Residue field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a f
Discrepancy (algebraic geometry)
In algebraic geometry, given a pair (X, D) consisting of a normal variety X and a -divisor D on X (e.g., canonical divisor), the discrepancy of the pair (X, D) measures the degree of the singularity o
Castelnuovo–Mumford regularity
In algebraic geometry, the Castelnuovo–Mumford regularity of a coherent sheaf F over projective space Pn is the smallest integer r such that it is r-regular, meaning that whenever i > 0. The regularit
Nonlinear algebra
Nonlinear algebra is the nonlinear analogue to linear algebra, generalizing notions of spaces and transformations coming from the linear setting. Algebraic geometry is one of the main areas of mathema
Crepant resolution
In algebraic geometry, a crepant resolution of a singularity is a resolution that does not affect the canonical class of the manifold. The term "crepant" was coined by Miles Reid by removing the prefi
Differential of the first kind
In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally, algebraic variet
Moishezon manifold
In mathematics, a Moishezon manifold M is a compact complex manifold such that the field of meromorphic functions on each component M has transcendence degree equal the complex dimension of the compon
Pencil (geometry)
In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane, or the set of circles that pass through two give
Regular local ring
In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let
Syntomic topology
In algebraic geometry, the syntomic topology is a Grothendieck topology introduced by . Mazur defined a morphism to be syntomic if it is flat and locally a complete intersection. The syntomic topology
Quadrisecant
In geometry, a quadrisecant or quadrisecant line of a space curve is a line that passes through four points of the curve. This is the largest possible number of intersections that a generic space curv
Derived stack
In algebraic geometry, a derived stack is, roughly, a stack together with a sheaf of commutative ring spectra. It generalizes a derived scheme. Derived stacks are the "spaces" studied in derived algeb
Primary decomposition
In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed as an intersection, called primary decomposition, of finit
Level structure (algebraic geometry)
In algebraic geometry, a level structure on a space X is an extra structure attached to X that shrinks or eliminates the automorphism group of X, by demanding automorphisms to preserve the level struc
Unibranch local ring
In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared (obtained by quotienting A by its nilradical) is an integral domain, and the integral closure B of Ared is also a
Cone (algebraic geometry)
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec of a quasi-coherent graded OX-algebra R is called the cone or affine cone of R.
GIT quotient
In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme with an action by a group scheme G is the affine scheme , the prime spectrum of the ri
Normal crossing singularity
In algebraic geometry a normal crossing singularity is a singularity similar to a union of coordinate hyperplanes. The term can be confusing because normal crossing singularities are not usually norma
Valuation (algebra)
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generaliz
Sheaf on an algebraic stack
In algebraic geometry, a quasi-coherent sheaf on an algebraic stack is a generalization of a quasi-coherent sheaf on a scheme. The most concrete description is that it is the data consists of, for eac
Algebraic cobordism
In mathematics, algebraic cobordism is an analogue of complex cobordism for smooth quasi-projective schemes over a field. It was introduced by Marc Levine and Fabien Morel . An oriented cohomology the
Local parameter
In the geometry of complex algebraic curves, a local parameter for a curve C at a smooth point P is just a meromorphic function on C that has a simple zero at P. This concept can be generalized to cur
Newton–Okounkov body
In algebraic geometry, a Newton–Okounkov body, also called an Okounkov body, is a convex body in Euclidean space associated to a divisor (or more generally a linear system) on a variety. The convex ge
Affine variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field k is the zero-locus in the affine space kn of some finite family of polynomials of n variables
Kähler differential
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced by Erich Kähler in the 1930s. It was adopted as st
Buchberger's algorithm
In the theory of multivariate polynomials, Buchberger's algorithm is a method for transforming a given set of polynomials into a Gröbner basis, which is another set of polynomials that have the same c
Kodaira–Spencer map
In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point
Zariski–Riemann space
In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a locally ringed space whose points are valuation rings containing k and contained in K. They generalize
Crystalline cohomology
In mathematics, crystalline cohomology is a Weil cohomology theory for schemes X over a base field k. Its values Hn(X/W) are modules over the ring W of Witt vectors over k. It was introduced by Alexan
Numerical algebraic geometry
Numerical algebraic geometry is a field of computational mathematics, particularly computational algebraic geometry, which uses methods from numerical analysis to study and manipulate the solutions of
Cotangent sheaf
In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of -modules that represents (or classifies) S-derivations in the sense: for any -modules F, there is
Generic flatness
In algebraic geometry and commutative algebra, the theorems of generic flatness and generic freeness state that under certain hypotheses, a sheaf of modules on a scheme is flat or free. They are due t
Quippian
In mathematics, a quippian is a degree 5 class 3 contravariant of a plane cubic introduced by Arthur Cayley and discussed by Igor Dolgachev . In the same paper Cayley also introduced another similar i
Néron–Tate height
In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after Andr
Plücker embedding
In mathematics, the Plücker map embeds the Grassmannian , whose elements are k-dimensional subspaces of an n-dimensional vector space V, in a projective space, thereby realizing it as an algebraic var
Zariski tangent space
In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V (and more generally). It does not use differential calculus, bein
Gröbner fan
In computer algebra, the Gröbner fan of an ideal in the ring of polynomials is a concept in the theory of Gröbner bases. It is defined to be a fan consisting of cones that correspond to different mono
Matsusaka's big theorem
In algebraic geometry, given an ample line bundle L on a compact complex manifold X, Matsusaka's big theorem gives an integer m, depending only on the Hilbert polynomial of L, such that the tensor pow
Zariski space
In algebraic geometry, a Zariski space, named for Oscar Zariski, has several different meanings:
* A topological space that is Noetherian (every open set is quasicompact)
* A topological space that
Macdonald polynomials
In mathematics, Macdonald polynomials Pλ(x; t,q) are a family of orthogonal symmetric polynomials in several variables, introduced by Macdonald in 1987. He later introduced a non-symmetric generalizat
Rosati involution
In mathematics, a Rosati involution, named after Carlo Rosati, is an involution of the rational endomorphism ring of an abelian variety induced by a polarization. Let be an abelian variety, let be the
Mirror symmetry conjecture
In mathematics, mirror symmetry is a conjectural relationship between certain Calabi–Yau manifolds and a constructed "mirror manifold". The conjecture allows one to relate the number of rational curve
Normal cone
In algebraic geometry, the normal cone of a subscheme of a scheme is a scheme analogous to the normal bundle or tubular neighborhood in differential geometry.
Flat module
In algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a se
Determinantal variety
In algebraic geometry, determinantal varieties are spaces of matrices with a given upper bound on their ranks. Their significance comes from the fact that many examples in algebraic geometry are of th
Virasoro conjecture
In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro
Cotangent complex
In mathematics, the cotangent complex is a common generalisation of the cotangent sheaf, normal bundle and virtual tangent bundle of a map of geometric spaces such as manifolds or schemes. If is a mor
Geometric quotient
In algebraic geometry, a geometric quotient of an algebraic variety X with the action of an algebraic group G is a morphism of varieties such that (i) For each y in Y, the fiber is an orbit of G.(ii)
Conifold
In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a c
K-stability of Fano varieties
In mathematics, and in particular algebraic geometry, K-stability is an algebro-geometric stability condition for projective algebraic varieties and complex manifolds. K-stability is of particular imp
Derived scheme
In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf either of simplicial commutative rings or of commutative ring spectra on X such that (1) the pair is a
Quadric (algebraic geometry)
In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geome
Correspondence (algebraic geometry)
In algebraic geometry, a correspondence between algebraic varieties V and W is a subset R of V×W, that is closed in the Zariski topology. In set theory, a subset of a Cartesian product of two sets is
K-stability
In mathematics, and especially differential and algebraic geometry, K-stability is an algebro-geometric stability condition, for complex manifolds and complex algebraic varieties. The notion of K-stab
Relative dimension
In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient map , the difference dim V − dim Q is the relative d
Gröbner basis
In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal i
Nakayama's lemma
In mathematics, more specifically abstract algebra and commutative algebra, Nakayama's lemma — also known as the Krull–Azumaya theorem — governs the interaction between the Jacobson radical of a ring
Rational normal scroll
In mathematics, a rational normal scroll is a ruled surface of degree n in projective space of dimension n + 1. Here "rational" means birational to projective space, "scroll" is an old term for ruled
Direction cosine
In analytic geometry, the direction cosines (or directional cosines) of a vector are the cosines of the angles between the vector and the three positive coordinate axes. Equivalently, they are the con
Stable vector bundle
In mathematics, a stable vector bundle is a (holomorphic or algebraic) vector bundle that is stable in the sense of geometric invariant theory. Any holomorphic vector bundle may be built from stable o
Hodge–Arakelov theory
In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and p-adic Hodge theory for elliptic curves carried out in the framework of Arakelov theory. It was introduced by M
Frobenius manifold
In the mathematical field of differential geometry, a Frobenius manifold, introduced by Dubrovin, is a flat Riemannian manifold with a certain compatible multiplicative structure on the tangent space.
Grassmann bundle
In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X: such that the fiber is the Grassmannian of the d-dimensional vector subspaces of .
Mixed Hodge structure
In algebraic geometry, a mixed Hodge structure is an algebraic structure containing information about the cohomology of general algebraic varieties. It is a generalization of a Hodge structure, which
Étale spectrum
In algebraic geometry, a branch of mathematics, the étale spectrum of a commutative ring or an E∞-ring, denoted by Specét or Spét, is an analog of the prime spectrum Spec of a commutative ring that is
Categorical quotient
In algebraic geometry, given a category C, a categorical quotient of an object X with action of a group G is a morphism that (i) is invariant; i.e., where is the given group action and p2 is the proje
P-adic Teichmüller theory
In mathematics, p-adic Teichmüller theory describes the "uniformization" of p-adic curves and their moduli, generalizing the usual Teichmüller theory that describes the uniformization of Riemann surfa
Semiorthogonal decomposition
In mathematics, a semiorthogonal decomposition is a way to divide a triangulated category into simpler pieces. One way to produce a semiorthogonal decomposition is from an exceptional collection, a sp
Parshin's conjecture
In mathematics, more specifically in algebraic geometry, Parshin's conjecture (also referred to as the Beilinson–Parshin conjecture) states that for any smooth projective variety X defined over a fini
Milnor number
In mathematics, and particularly singularity theory, the Milnor number, named after John Milnor, is an invariant of a function germ. If f is a complex-valued holomorphic function germ then the Milnor
Group-stack
In algebraic geometry, a group-stack is an algebraic stack whose categories of points have group structures or even groupoid structures in a compatible way. It generalizes a group scheme, which is a s
Multiplier ideal
In commutative algebra, the multiplier ideal associated to a sheaf of ideals over a complex variety and a real number c consists (locally) of the functions h such that is locally integrable, where the
Purity (algebraic geometry)
In the mathematical field of algebraic geometry, purity is a theme covering a number of results and conjectures, which collectively address the question of proving that "when something happens, it hap
Motivic zeta function
In algebraic geometry, the motivic zeta function of a smooth algebraic variety is the formal power series Here is the -th symmetric power of , i.e., the quotient of by the action of the symmetric grou
Numerical certification
Numerical certification is the process of verifying the correctness of a candidate solution to a system of equations. In (numerical) computational mathematics, such as numerical algebraic geometry, ca
Dualizing sheaf
In algebraic geometry, the dualizing sheaf on a proper scheme X of dimension n over a field k is a coherent sheaf together with a linear functional that induces a natural isomorphism of vector spaces
Geometrically regular ring
In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes are define
Bundle of principal parts
In algebraic geometry, given a line bundle L on a smooth variety X, the bundle of n-th order principal parts of L is a vector bundle of rank that, roughly, parametrizes n-th order Taylor expansions of
Chow variety
In mathematics, particularly in the field of algebraic geometry, a Chow variety is an algebraic variety whose points correspond to effective algebraic cycles of fixed dimension and degree on a given p
Glossary of algebraic geometry
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see gl
Quot scheme
In algebraic geometry, the Quot scheme is a scheme parametrizing locally free sheaves on a projective scheme. More specifically, if X is a projective scheme over a Noetherian scheme S and if F is a co
Convexity (algebraic geometry)
In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces in quantum cohomology. These moduli spaces are s
Regular chain
In computer algebra, a regular chain is a particular kind of triangular set in a multivariate polynomial ring over a field. It enhances the notion of characteristic set.
Combinatorial mirror symmetry
A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for -dimensional convex polyhedra. The most famous examples of the polar duality provide Plat
Fano variety
In algebraic geometry, a Fano variety, introduced by Gino Fano in (Fano , ), is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over
Logarithmic form
In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne.
Relative cycle
In algebraic geometry, a relative cycle is a type of algebraic cycle on a scheme. In particular, let be a scheme of finite type over a Noetherian scheme , so that . Then a relative cycle is a cycle on
Fujita conjecture
In mathematics, Fujita's conjecture is a problem in the theories of algebraic geometry and complex manifolds, unsolved as of 2017. It is named after Takao Fujita, who formulated it in 1985.
Newton polytope
In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial. It can be used to analyze the polynomial's behavior when specific variables are considered relati
Toroidal embedding
In algebraic geometry, a toroidal embedding is an open embedding of algebraic varieties that locally looks like the embedding of the open torus into a toric variety. The notion was introduced by Mumfo
Algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial e
Donaldson–Thomas theory
In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–
Gauss–Manin connection
In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties . The fibers of the vector bundle are the de Rham cohomolog
Algebraic K-theory
Algebraic K-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called K-g
Hasse–Weil zeta function
In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is a meromorphic function on the complex plane defined in terms of the number o
Exceptional divisor
In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map of varieties is a kind of 'large' subvariety of which is 'crushed' by , in a certain definite sense. More stri
Irrelevant ideal
In mathematics, the irrelevant ideal is the ideal of a graded ring generated by the homogeneous elements of degree greater than zero. More generally, a homogeneous ideal of a graded ring is called an
Tangent space to a functor
In algebraic geometry, the tangent space to a functor generalizes the classical construction of a tangent space such as the Zariski tangent space. The construction is based on the following observatio
Witten conjecture
In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper Witten, and generalize
Klein configuration
In geometry, the Klein configuration, studied by Klein, is a geometric configuration related to Kummer surfaces that consists of 60 points and 60 planes, with each point lying on 15 planes and each pl
A¹ homotopy theory
In algebraic geometry and algebraic topology, branches of mathematics, A1 homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, mor
Bloch's higher Chow group
In algebraic geometry, Bloch's higher Chow groups, a generalization of Chow group, is a precursor and a basic example of motivic cohomology (for smooth varieties). It was introduced by Spencer Bloch a
Bivariant theory
In mathematics, a bivariant theory was introduced by Fulton and MacPherson, in order to put a ring structure on the Chow group of a singular variety, the resulting ring called an operational Chow ring
Tango bundle
In algebraic geometry, a Tango bundle is one of the indecomposable vector bundles of rank n − 1 constructed on n-dimensional projective space Pn by
GKM variety
In algebraic geometry, a GKM variety is a complex algebraic variety equipped with a torus action that meets certain conditions. The concept was introduced by Mark Goresky, Robert Kottwitz, and Robert
F-crystal
In algebraic geometry, F-crystals are objects introduced by that capture some of the structure of crystalline cohomology groups. The letter F stands for Frobenius, indicating that F-crystals have an a
Scorza variety
In mathematics, a k-Scorza variety is a smooth projective variety, of maximal dimension among those whose k–1 secant varieties are not the whole of projective space. Scorza varieties were introduced a
Hartshorne ellipse
In mathematics, a Hartshorne ellipse is an ellipse in the unit ball bounded by the 4-sphere S4 such that the ellipse and the circle given by intersection of its plane with S4 satisfy the Poncelet cond
Functor represented by a scheme
In algebraic geometry, a functor represented by a scheme X is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme S is (up to natural biject
Amoeba (mathematics)
In complex analysis, a branch of mathematics, an amoeba is a set associated with a polynomial in one or more complex variables. Amoebas have applications in algebraic geometry, especially tropical geo
Algebraic geometry and analytic geometry
In mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraic varieties, analytic geometry deals with complex manifolds and the
Regular scheme
In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect
Fourier–Deligne transform
In algebraic geometry, the Fourier–Deligne transform, or ℓ-adic Fourier transform, or geometric Fourier transform, is an operation on objects of the derived category of ℓ-adic sheaves over the affine
Cyclic cover
In algebraic topology and algebraic geometry, a cyclic cover or cyclic covering is a covering space for which the set of covering transformations forms a cyclic group. As with cyclic groups, there may
Positive form
In complex geometry, the term positive form refers to several classes of real differential forms of Hodge type (p, p).
Hironaka's example
In geometry, Hironaka's example is a non-Kähler complex manifold that is a deformation of Kähler manifolds found by Heisuke Hironaka . Hironaka's example can be used to show that several other plausib
Hessenberg variety
In geometry, Hessenberg varieties, first studied by Filippo De Mari, Claudio Procesi, and Mark A. Shayman, are a family of subvarieties of the full flag variety which are defined by a Hessenberg funct
Tate vector space
In mathematics, a Tate vector space is a vector space obtained from finite-dimensional vector spaces in a way that makes it possible to extend concepts such as dimension and determinant to an infinite
Hurwitz scheme
In algebraic geometry, the Hurwitz scheme is the scheme parametrizing pairs where C is a smooth curve of genus g and π has degree d.
Section conjecture
In anabelian geometry, a branch of algebraic geometry, the section conjecture gives a conjectural description of the splittings of the group homomorphism , where is a complete smooth curve of genus at
Morphism of schemes
In algebraic geometry, a morphism of schemes generalizes a morphism of algebraic varieties just as a scheme generalizes an algebraic variety. It is, by definition, a morphism in the category of scheme
Injective sheaf
In mathematics, injective sheaves of abelian groups are used to construct the resolutions needed to define sheaf cohomology (and other derived functors, such as sheaf Ext). There is a further group of
Derived noncommutative algebraic geometry
In mathematics, derived noncommutative algebraic geometry, the derived version of noncommutative algebraic geometry, is the geometric study of derived categories and related constructions of triangula
Cohen–Macaulay ring
In mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under mild assumptions, a local ring is
Moduli stack of elliptic curves
In mathematics, the moduli stack of elliptic curves, denoted as or , is an algebraic stack over classifying elliptic curves. Note that it is a special case of the moduli stack of algebraic curves . In
Ample line bundle
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The m
Diagonal morphism (algebraic geometry)
In algebraic geometry, given a morphism of schemes , the diagonal morphism is a morphism determined by the universal property of the fiber product of p and p applied to the identity and the identity .
System of polynomial equations
A system of polynomial equations (sometimes simply a polynomial system) is a set of simultaneous equations f1 = 0, ..., fh = 0 where the fi are polynomials in several variables, say x1, ..., xn, over
Quasi-separated morphism
In algebraic geometry, a morphism of schemes f from X to Y is called quasi-separated if the diagonal map from X to X × YX is quasi-compact (meaning that the inverse image of any quasi-compact open set
Crystal (mathematics)
In mathematics, crystals are Cartesian sections of certain fibered categories. They were introduced by Alexander Grothendieck, who named them crystals because in some sense they are "rigid" and "grow"
Plücker matrix
The Plücker matrix is a special skew-symmetric 4 × 4 matrix, which characterizes a straight line in projective space. The matrix is defined by 6 Plücker coordinates with 4 degrees of freedom. It is na
Standard conjectures on algebraic cycles
In mathematics, the standard conjectures about algebraic cycles are several conjectures describing the relationship of algebraic cycles and Weil cohomology theories. One of the original applications o
Bracket ring
In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d-by-d minors of a generic d-by-n matrix (xij). The bracket ring may be rega
Constructible set (topology)
In topology, constructible sets are a class of subsets of a topological space that have a relatively "simple" structure.They are used particularly in algebraic geometry and related fields. A key resul
Universal algebraic geometry
In algebraic geometry, universal algebraic geometry generalizes the geometry of rings to geometries of arbitrary varieties of algebras, so that every variety of algebras has its own algebraic geometry
Jouanolou's trick
In algebraic geometry, Jouanolou's trick is a theorem that asserts, for an algebraic variety X, the existence of a surjection with affine fibers from an affine variety W to X. The variety W is therefo
Grothendieck's Galois theory
In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the
Normal bundle
In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding (or immersion).
Euler sequence
In mathematics, the Euler sequence is a particular exact sequence of sheaves on n-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an
Regular semi-algebraic system
In computer algebra, a regular semi-algebraic system is a particular kind of triangular system of multivariate polynomials over a real closed field.
Dévissage
In algebraic geometry, dévissage is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind o
Noncommutative algebraic geometry
Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of non-commutative alg
Weibel's conjecture
In mathematics, Weibel's conjecture gives a criterion for vanishing of negative algebraic K-theory groups. The conjecture was proposed by Charles Weibel and proven in full generality by using methods
Nilcurve
In mathematics, a nilcurve is a pointed stable curve over a finite field with an indigenous bundle whose p-curvature is square nilpotent. Nilcurves were introduced by Mochizuki as a central concept in
Nakai conjecture
In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961.It states that if V is a complex algeb
Tate–Shafarevich group
In arithmetic geometry, the Tate–Shafarevich group Ш(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group W
Trope (mathematics)
In geometry, trope is an archaic term for a singular (meaning special) tangent space of a variety, often a quartic surface. The term may have been introduced by Cayley , who defined it as "the recipro
Group-scheme action
In algebraic geometry, an action of a group scheme is a generalization of a group action to a group scheme. Precisely, given a group S-scheme G, a left action of G on an S-scheme X is an S-morphism su
Néron–Severi group
In algebraic geometry, the Néron–Severi group of a variety is the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. Its ra
N! conjecture
In mathematics, the n! conjecture is the conjecture that the dimension of a certain module of is n!. It was made by A. M. Garsia and M. Haiman and later proved by M. Haiman. It implies Macdonald's pos
Group functor
In mathematics, a group functor is a group-valued functor on the category of commutative rings. Although it is typically viewed as a generalization of a group scheme, the notion itself involves no sch
Derived algebraic geometry
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded
Hilbert's arithmetic of ends
In mathematics, specifically in the area of hyperbolic geometry, Hilbert's arithmetic of ends is a method for endowing a geometric set, the set of ideal points or "ends" of a hyperbolic plane, with an
Semistable reduction theorem
In algebraic geometry, the semistable reduction theorem states that, given a proper flat morphism , there exists a morphism (called base change) such that is semistable (i.e., the singularities are mi
Cone of curves
In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety is a combinatorial invariant of importance to the birational geometry of .
Generalized Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be desc
Formal moduli
In mathematics, formal moduli are an aspect of the theory of moduli spaces (of algebraic varieties or vector bundles, for example), closely linked to deformation theory and formal geometry. Roughly sp
Mixed Hodge module
In mathematics, mixed Hodge modules are the culmination of Hodge theory, mixed Hodge structures, intersection cohomology, and the decomposition theorem yielding a coherent framework for discussing var
Morphism of algebraic stacks
In algebraic geometry, given algebraic stacks over a base category C, a morphism of algebraic stacks is a functor such that . More generally, one can also consider a morphism between prestacks; (a sta
Cohomology of a stack
In algebraic geometry, the cohomology of a stack is a generalization of étale cohomology. In a sense, it is a theory that is coarser than the Chow group of a stack. The cohomology of a quotient stack
Cubic form
In mathematics, a cubic form is a homogeneous polynomial of degree 3, and a cubic hypersurface is the zero set of a cubic form. In the case of a cubic form in three variables, the zero set is a cubic
Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commu
Algebraic stack
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques spec
Noether normalization lemma
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field k, and any finitely generated commutative k-algebra
Faithfully flat descent
Faithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such morphisms, that are flat and surjective
Normal scheme
In algebraic geometry, an algebraic variety or scheme X is normal if it is normal at every point, meaning that the local ring at the point is an integrally closed domain. An affine variety X (understo
Schubert calculus
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part o
Algebraic Geometry (book)
Algebraic Geometry is an algebraic geometry textbook written by Robin Hartshorne and published by Springer-Verlag in 1977.
Condensed mathematics
Condensed mathematics is a theory developed by and Peter Scholze which aims to unify various mathematical subfields, including topology, complex geometry, and algebraic geometry.
Formal scheme
In mathematics, specifically in algebraic geometry, a formal scheme is a type of space which includes data about its surroundings. Unlike an ordinary scheme, a formal scheme includes infinitesimal dat
Gordan's lemma
Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways.
* Let be a matrix of integers. Let be the set of non-negative integer solutions of . Then there
Schubert variety
In algebraic geometry, a Schubert variety is a certain subvariety of a Grassmannian, usually with singular points. Like a Grassmannian, it is a kind of moduli space, whose points correspond to certain
Noetherian topological space
In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open
Bott–Samelson resolution
In algebraic geometry, the Bott–Samelson resolution of a Schubert variety is a resolution of singularities. It was introduced by in the context of compact Lie groups. The algebraic formulation is inde
Gorenstein scheme
In algebraic geometry, a Gorenstein scheme is a locally Noetherian scheme whose local rings are all Gorenstein. The canonical line bundle is defined for any Gorenstein scheme over a field, and its pro
Relative effective Cartier divisor
In algebraic geometry, a relative effective Cartier divisor is roughly a family of effective Cartier divisors. Precisely, an effective Cartier divisor in a scheme X over a ring R is a closed subscheme
Frobenioid
In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by
Deligne–Mumford stack
In algebraic geometry, a Deligne–Mumford stack is a stack F such that 1.
* the diagonal morphism is representable, quasi-compact and separated. 2.
* There is a scheme U and étale surjective map (cal
Multi-homogeneous Bézout theorem
In algebra and algebraic geometry, the multi-homogeneous Bézout theorem is a generalization to multi-homogeneous polynomials of Bézout's theorem, which counts the number of isolated common zeros of a
Nagata ring
In commutative algebra, an N-1 ring is an integral domain whose integral closure in its quotient field is a finitely generated -module. It is called a Japanese ring (or an N-2 ring) if for every finit
Relative canonical model
In the mathematical field of algebraic geometry, the relative canonical model of a singular variety of a mathematical object where is a particular canonical variety that maps to , which simplifies the
Period (algebraic geometry)
In algebraic geometry, a period is a number that can be expressed as an integral of an algebraic function over an algebraic domain. Sums and products of periods remain periods, so the periods form a r
Artin's criterion
In mathematics, Artin's criteria are a collection of related necessary and sufficient conditions on deformation functors which prove the representability of these functors as either Algebraic spaces o
Beilinson regulator
In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology: Here, X is a complex smooth projective variety, for exam
Chow group of a stack
In algebraic geometry, the Chow group of a stack is a generalization of the Chow group of a variety or scheme to stacks. For a quotient stack , the Chow group of X is the same as the G-equivariant Cho
Weighted projective space
In algebraic geometry, a weighted projective space P(a0,...,an) is the projective variety Proj(k[x0,...,xn]) associated to the graded ring k[x0,...,xn] where the variable xk has degree ak.
Quotient stack
In algebraic geometry, a quotient stack is a stack that parametrizes equivariant objects. Geometrically, it generalizes a quotient of a scheme or a variety by a group: a quotient variety, say, would b
Complex dimension
In mathematics, complex dimension usually refers to the dimension of a complex manifold or a complex algebraic variety. These are spaces in which the local neighborhoods of points (or of non-singular
H topology
In algebraic geometry, the h topology is a Grothendieck topology introduced by Vladimir Voevodsky to study the homology of schemes. It combines several good properties possessed by its related "sub"to
Flat topology
In mathematics, the flat topology is a Grothendieck topology used in algebraic geometry. It is used to define the theory of flat cohomology; it also plays a fundamental role in the theory of descent (
Main theorem of elimination theory
In algebraic geometry, the main theorem of elimination theory states that every projective scheme is proper. A version of this theorem predates the existence of scheme theory. It can be stated, proved
Equisingularity
In algebraic geometry, an equisingularity is, roughly, a family of singularities that are not non-equivalent and is an important notion in singularity theory. There is no universal definition of equis
Steinerian
In algebraic geometry, a Steinerian of a hypersurface, introduced by Steiner, is the locus of the singular points of its polar quadrics.
Ind-scheme
In algebraic geometry, an ind-scheme is a set-valued functor that can be written (represented) as a direct limit (i.e., inductive limit) of closed embedding of schemes.
Scheme-theoretic intersection
In algebraic geometry, the scheme-theoretic intersection of closed subschemes X, Y of a scheme W is , the fiber product of the closed immersions . It is denoted by . Locally, W is given as for some ri
LLT polynomial
In mathematics, an LLT polynomial is one of a family of symmetric functions introduced by Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon (1997) as q-analogues of products of Schur functions. J.
Toric stack
In algebraic geometry, a toric stack is a stacky generalization of a toric variety. More precisely, a toric stack is obtained by replacing in the construction of a toric variety a step of taking GIT q
Cartan subgroup
In algebraic geometry, a Cartan subgroup of a connected linear algebraic group over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgro
Hyperplane section
In mathematics, a hyperplane section of a subset X of projective space Pn is the intersection of X with some hyperplane H. In other words, we look at the subset XH of those elements x of X that satisf
Quasi-homogeneous polynomial
In algebra, a multivariate polynomial is quasi-homogeneous or weighted homogeneous, if there exist r integers , called weights of the variables, such that the sum is the same for all nonzero terms of
Algebraic space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing togethe
Algebraic cycle
In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology of V that is directly accessible by algeb
Gromov–Witten invariant
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting presc
Conic bundle
In algebraic geometry, a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the form Theoretically, it can be considered as a Severi–Brauer surface, or more pre
Groupoid object
In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and a generalization of a group objects when the mu
Generic property
In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functio
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure.
Ground field
In mathematics, a ground field is a field K fixed at the beginning of the discussion.
General position
In algebraic geometry and computational geometry, general position is a notion of genericity for a set of points, or other geometric objects. It means the general case situation, as opposed to some mo
Severi variety (Hilbert scheme)
In mathematics, a Severi variety is an algebraic variety in a Hilbert scheme that parametrizes curves in projective space with given degree and geometric genus and at most node singularities. Its dime
Geometric class field theory
In mathematics, geometric class field theory is an extension of class field theory to higher-dimensional geometrical objects: much the same way as class field theory describes the abelianization of th
Catenary ring
In mathematics, a commutative ring R is catenary if for any pair of prime ideals p, q, any two strictly increasing chains p=p0 ⊂p1 ... ⊂pn= q of prime ideals are contained in maximal strictly increasi
Descent along torsors
In mathematics, given a G-torsor X → Y and a stack F, the descent along torsors says there is a canonical equivalence between F(Y), the category of Y-points and F(X)G, the category of G-equivariant X-
Moduli stack of principal bundles
In algebraic geometry, given a smooth projective curve X over a finite field and a smooth affine group scheme G over it, the moduli stack of principal bundles over X, denoted by , is an algebraic stac
Descent (mathematics)
In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with so
Morphism of finite type
For a homomorphism A → B of commutative rings, B is called an A-algebra of finite type if B is a finitely generated as an A-algebra. It is much stronger for B to be a finite A-algebra, which means tha
Nisnevich topology
In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹
Tangent cone
In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities.
Trivial cylinder
In geometry and topology, trivial cylinders are certain pseudoholomorphic curves appearing in certain cylindrical manifolds. In Floer homology and its variants, chain complexes or differential graded
Hilbert–Samuel function
In commutative algebra the Hilbert–Samuel function, named after David Hilbert and Pierre Samuel, of a nonzero finitely generated module over a commutative Noetherian local ring and a primary ideal of
Du Bois singularity
In algebraic geometry, Du Bois singularities are singularities of complex varieties studied by . gave the following characterisation of Du Bois singularities. Suppose that is a reduced closed subschem
Comodule over a Hopf algebroid
In mathematics, at the intersection of algebraic topology and algebraic geometry, there is the notion of a Hopf algebroid which encodes the information of a presheaf of groupoids whose object sheaf an
Stack (mathematics)
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to
Topological recursion
In mathematics, topological recursion is a recursive definition of invariants of spectral curves.It has applications in enumerative geometry, random matrix theory, mathematical physics, string theory,
Homotopy associative algebra
In mathematics, an algebra such as has multiplication whose associativity is well-defined on the nose. This means for any real numbers we have . But, there are algebras which are not necessarily assoc
Abundance conjecture
In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program,stating that for every projective variety with Kawamata log terminal
Multiplicative distance
In algebraic geometry, is said to be a multiplicative distance function over a field if it satisfies,
*
* AB is congruent to A'B' iff
* AB < A'B' iff
*
Complex analytic variety
In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety or complex analytic space is a generalization of a complex manifold which allows the presence o
Contraction morphism
In algebraic geometry, a contraction morphism is a surjective projective morphism between normal projective varieties (or projective schemes) such that or, equivalently, the geometric fibers are all c
Affine Grassmannian (manifold)
In mathematics, there are two distinct meanings of the term affine Grassmannian. In one it is the manifold of all k-dimensional affine subspaces of Rn (described on this page), while in the other the
Infinitesimal cohomology
In mathematics, infinitesimal cohomology is a cohomology theory for algebraic varieties introduced by Grothendieck. In characteristic 0 it is essentially the same as crystalline cohomology. In nonzero
Formally smooth map
In algebraic geometry and commutative algebra, a ring homomorphism is called formally smooth (from French: Formellement lisse) if it satisfies the following infinitesimal lifting property: Suppose B i
Adjunction formula
In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is oft
Beilinson–Bernstein localization
In mathematics, especially in representation theory and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag varieties G/B to representations of the Lie algebra a
Glossary of arithmetic and diophantine geometry
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic ge
Excellent ring
In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring that behaves well with respect to the operation of completion, and is called an excellent ring if it is also universally
Absolute irreducibility
In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field. For example, is absolutely irreducible, but while is
Irreducible component
In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algeb
Grothendieck connection
In algebraic geometry and synthetic differential geometry, a Grothendieck connection is a way of viewing connections in terms of descent data from infinitesimal neighbourhoods of the diagonal.
Analytic space
An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study o
Homogeneous polynomial
In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, is a homogeneous polynomial of degree 5, i
Whitney umbrella
In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface plac
Néron model
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model)for an abelian variety AK defined over the field of fractions K of a Dedekind domain R is the "push-forward" of AK from
Bass–Quillen conjecture
In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring . The conjecture is named for Hyman Bass and Daniel Quillen, who formul
Tautological ring
In algebraic geometry, the tautological ring is the subring of the Chow ring of the moduli space of curves generated by tautological classes. These are classes obtained from 1 by pushforward along var
Affine Grassmannian
In mathematics, the affine Grassmannian of an algebraic group G over a field k is an ind-scheme—a colimit of finite-dimensional schemes—which can be thought of as a flag variety for the loop group G(k
Implicit function
In mathematics, an implicit equation is a relation of the form where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is An implicit fun
Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more
Family of curves
In geometry, a family of curves is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is variable. In general, the parameter(s) influence t
Fano fibration
In algebraic geometry, a Fano fibration or Fano fiber space, named after Gino Fano, is a morphism of varieties whose general fiber is a Fano variety (in other words has ample anticanonical bundle) of
Koszul cohomology
In mathematics, the Koszul cohomology groups are groups associated to a projective variety X with a line bundle L. They were introduced by Mark Green , and named after Jean-Louis Koszul as they are cl
Constructible sheaf
In mathematics, a constructible sheaf is a sheaf of abelian groups over some topological space X, such that X is the union of a finite number of locally closed subsets on each of which the sheaf is a
Standard monomial theory
In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive algebraic group by giving an explicit basis o
Combinatorial commutative algebra
Combinatorial commutative algebra is a relatively new, rapidly developing mathematical discipline. As the name implies, it lies at the intersection of two more established fields, commutative algebra
Bass conjecture
In mathematics, especially algebraic geometry, the Bass conjecture says that certain algebraic K-groups are supposed to be finitely generated. The conjecture was proposed by Hyman Bass.
Theorem of absolute purity
In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states: given
* a regular scheme X over some base scheme,
* a clos
Jacobian conjecture
In mathematics, the Jacobian conjecture is a famous unsolved problem concerning polynomials in several variables. It states that if a polynomial function from an n-dimensional space to itself has Jaco
Dimension of a scheme
In algebraic geometry, the dimension of a scheme is a generalization of a dimension of an algebraic variety. Scheme theory emphasizes the relative point of view and, accordingly, the relative dimensio
Max Noether's theorem on curves
In algebraic geometry, Max Noether's theorem on curves is a theorem about curves lying on algebraic surfaces, which are hypersurfaces in P3, or more generally complete intersections. It states that, f
Cox ring
In algebraic geometry, a Cox ring is a sort of universal homogeneous coordinate ring for a projective variety, and is (roughly speaking) a direct sum of the spaces of sections of all isomorphism class
Symmetric variety
In algebraic geometry, a symmetric variety is an algebraic analogue of a symmetric space in differential geometry, given by a quotient G/H of a reductive algebraic group G by the subgroup H fixed by s
Semi-simplicity
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is
Inverse image functor
In mathematics, specifically in algebraic topology and algebraic geometry, an inverse image functor is a contravariant construction of sheaves; here “contravariant” in the sense given a map , the inve
Geometric Langlands correspondence
In mathematics, the geometric Langlands correspondence is a reformulation of the Langlands correspondence obtained by replacing the number fields appearing in the original number theoretic version by
Degeneration (algebraic geometry)
In algebraic geometry, a degeneration (or specialization) is the act of taking a limit of a family of varieties. Precisely, given a morphism of a variety (or a scheme) to a curve C with origin 0 (e.g.
Distribution on a linear algebraic group
In algebraic geometry, given a linear algebraic group G over a field k, a distribution on it is a linear functional satisfying some support condition. A convolution of distributions is again a distrib
Grassmannian
In mathematics, the Grassmannian Gr(k, V) is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V. For example, the Grassmannian Gr(1, V) is the space of l
Picard–Lefschetz theory
In mathematics, Picard–Lefschetz theory studies the topology of a complex manifold by looking at the critical points of a holomorphic function on the manifold. It was introduced by Émile Picard for co
Noetherian scheme
In algebraic geometry, a noetherian scheme is a scheme that admits a finite covering by open affine subsets , noetherian rings. More generally, a scheme is locally noetherian if it is covered by spect
Derived tensor product
In algebra, given a differential graded algebra A over a commutative ring R, the derived tensor product functor is where and are the categories of right A-modules and left A-modules and D refers to th
Steiner's conic problem
In enumerative geometry, Steiner's conic problem is the problem of finding the number of smooth conics tangent to five given conics in the plane in general position. If the problem is considered in th
Frankel conjecture
In the mathematical fields of differential geometry and algebraic geometry, the Frankel conjecture was a problem posed by Theodore Frankel in 1961. It was resolved in 1979 by Shigefumi Mori, and by Yu
Prestack
In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that (when
Hodge conjecture
In mathematics, the Hodge conjecture is a major unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subva
Local uniformization
In algebraic geometry, local uniformization is a weak form of resolution of singularities, stating roughly that a variety can be desingularized near any valuation, or in other words that the Zariski–R
Mori dream space
In algebraic geometry, a Mori Dream Space is a projective variety whose cone of effective divisors has a well-behaved decomposition into certain convex sets called "Mori chambers". showed that Mori dr
Monsky–Washnitzer cohomology
In algebraic geometry, Monsky–Washnitzer cohomology is a p-adic cohomology theory defined for non-singular affine varieties over fields of positive characteristic p introduced by Paul Monsky and Gerar
Formal holomorphic function
In algebraic geometry, a formal holomorphic function along a subvariety V of an algebraic variety W is an algebraic analog of a holomorphic function defined in a neighborhood of V. They are sometimes
P-curvature
In algebraic geometry, p-curvature is an invariant of a connection on a coherent sheaf for schemes of characteristic p > 0. It is a construction similar to a usual curvature, but only exists in finite
Kleiman's theorem
In algebraic geometry, Kleiman's theorem, introduced by , concerns dimension and smoothness of scheme-theoretic intersection after some perturbation of factors in the intersection. Precisely, it state
Rational mapping
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that va
Minimal model program
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is
Stable principal bundle
In mathematics, and especially differential geometry and algebraic geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles.
Moduli stack of vector bundles
In algebraic geometry, the moduli stack of rank-n vector bundles Vectn is the stack parametrizing vector bundles (or locally free sheaves) of rank n over some reasonable spaces. It is a smooth algebra
Valuative criterion
In mathematics, specifically algebraic geometry, the valuative criteria are a collection of results that make it possible to decide whether a morphism of algebraic varieties, or more generally schemes
Deformation (mathematics)
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or a vector
Supersingular variety
In mathematics, a supersingular variety is (usually) a smooth projective variety in nonzero characteristic such that for all n the slopes of the Newton polygon of the nth crystalline cohomology are al
Cayleyan
In algebraic geometry, the Cayleyan is a variety associated to a hypersurface by Arthur Cayley, who named it the pippian in and also called it the Steiner–Hessian.
Harder–Narasimhan stratification
In algebraic geometry and complex geometry, the Harder–Narasimhan stratification is any of a stratification of the moduli stack of principal G-bundles by locally closed substacks in terms of "loci of
Riemann–Roch-type theorem
In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous is the Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation du
Igusa variety
In mathematics, an Igusa curve is (roughly) a coarse moduli space of elliptic curves in characteristic p with a level p Igusa structure, where an Igusa structure on an elliptic curve E is roughly a po
Log structure
In algebraic geometry, a log structure provides an abstract context to study , and in particular the notion of logarithmic differential form and the related Hodge-theoretic concepts. This idea has app
Pseudoholomorphic curve
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauc
Kuga fiber variety
In algebraic geometry, a Kuga fiber variety, introduced by Kuga, is a fiber space whose fibers are abelian varieties and whose base space is an arithmetic quotient of a Hermitian symmetric space.
Codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective a
Finite algebra
In abstract algebra, an -algebra is finite if it is finitely generated as an -module. An -algebra can be thought as a homomorphism of rings , in this case is called a finite morphism if is a finite -a
Arithmetic and geometric Frobenius
In mathematics, the Frobenius endomorphism is defined in any commutative ring R that has characteristic p, where p is a prime number. Namely, the mapping φ that takes r in R to rp is a ring endomorphi
Normally flat ring
In algebraic geometry, a normally flat ring along a proper ideal I is a local ring A such that is flat over for each integer . The notion was introduced by Hironaka in his proof of the resolution of s
Hitchin system
In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies o
Projective bundle
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a Pn-bundle if it is locally a projective n-space; i.e
Witt vector cohomology
In mathematics, Witt vector cohomology was an early p-adic cohomology theory for algebraic varieties introduced by Serre. Serre constructed it by defining a sheaf of truncated Witt rings Wn over a var
Field with one element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted
Unramified morphism
In algebraic geometry, an unramified morphism is a morphism of schemes such that (a) it is locally of finite presentation and (b) for each and , we have that 1.
* The residue field is a separable alg
Secant variety
In algebraic geometry, the secant variety , or the variety of chords, of a projective variety is the Zariski closure of the union of all secant lines (chords) to V in : (for , the line is the tangent
Quantum cohomology
In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two v
Siegel modular variety
In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular variet
Quotient space of an algebraic stack
In algebraic geometry, the quotient space of an algebraic stack F, denoted by |F|, is a topological space which as a set is the set of all integral substacks of F and which then is given a "Zariski to
Shriek map
In category theory, a branch of mathematics, certain unusual functors are denoted and with the exclamation mark used to indicate that they are exceptional in some way. They are thus accordingly someti
Rees algebra
In commutative algebra, the Rees algebra of an ideal I in a commutative ring R is defined to be The extended Rees algebra of I (which some authors refer to as the Rees algebra of I) is defined as This
Shimura variety
In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebra
Bridgeland stability condition
In mathematics, and especially algebraic geometry, a Bridgeland stability condition, defined by Tom Bridgeland, is an algebro-geometric stability condition defined on elements of a triangulated catego
Étale topology
In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also de
Pseudo algebraically closed field
In mathematics, a field is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced by James Ax in 1967.
Cartier isomorphism
In algebraic geometry, the Cartier isomorphism is a certain isomorphism between the cohomology sheaves of the de Rham complex of a smooth algebraic variety over a field of positive characteristic, and
Localized Chern class
In algebraic geometry, a localized Chern class is a variant of a Chern class, that is defined for a chain complex of vector bundles as opposed to a single vector bundle. It was originally introduced i
Sum of residues formula
In mathematics, the residue formula says that the sum of the residues of a meromorphic differential form on a smooth proper algebraic curve vanishes.
V-topology
In mathematics, especially in algebraic geometry, the v-topology (also known as the universally subtrusive topology) is a Grothendieck topology whose covers are characterized by lifting maps from valu
Suita conjecture
In mathematics, the Suita conjecture is a conjecture related to the theory of the Riemann surface, the boundary behavior of conformal maps, the theory of Bergman kernel, and the theory of the L2 exten
Kummer configuration
In geometry, the Kummer configuration, named for Ernst Kummer, is a geometric configuration of 16 points and 16 planes such that each point lies on 6 of the planes and each plane contains 6 of the poi
Adequate equivalence relation
In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties used to obtain a well-working theory of s
Stability (algebraic geometry)
In mathematics, and especially algebraic geometry, stability is a notion which characterises when a geometric object, for example a point, an algebraic variety, a vector bundle, or a sheaf, has some d
Generic point
In algebraic geometry, a generic point P of an algebraic variety X is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost
Fondements de la Géometrie Algébrique
Fondements de la Géometrie Algébrique (FGA) is a book that collected together seminar notes of Alexander Grothendieck. It is an important source for his pioneering work on scheme theory, which laid fo
Canonical ring
In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is the graded ring of sections of powers of the canonical bundle K. Its nth graded
Logarithmic pair
In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were studied by .
Frobenius splitting
In mathematics, a Frobenius splitting, introduced by Mehta and Ramanathan, is a splitting of the injective morphism OX→F*OX from a structure sheaf OX of a characteristic p > 0 variety X to its image F
Berkovich space
In mathematics, a Berkovich space, introduced by Berkovich, is a version of an analytic space over a non-Archimedean field (e.g. p-adic field), refining Tate's notion of a rigid analytic space.
Erdős–Diophantine graph
An Erdős–Diophantine graph is an object in the mathematical subject of Diophantine equations consisting of a set of integer points at integer distances in the plane that cannot be extended by any addi
Hilbert's fifteenth problem
Hilbert's fifteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. The problem is to put Schubert's enumerative calculus on a rigorous founda
Ruled join
In algebraic geometry, given irreducible subvarieties V, W of a projective space Pn, the ruled join of V and W is the union of all lines from V to W in P2n+1, where V, W are embedded into P2n+1 so tha
ℓ-adic sheaf
In algebraic geometry, an ℓ-adic sheaf on a Noetherian scheme X is an inverse system consisting of -modules in the étale topology and inducing . Bhatt–Scholze's gives an alternative approach.
Esquisse d'un Programme
"Esquisse d'un Programme" (Sketch of a Programme) is a famous proposal for long-term mathematical research made by the German-born, French mathematician Alexander Grothendieck in 1984. He pursued the
Triangular decomposition
In computer algebra, a triangular decomposition of a polynomial system S is a set of simpler polynomial systems S1, ..., Se such that a point is a solution of S if and only if it is a solution of one
Horrocks bundle
In algebraic geometry, Horrocks bundles are certain indecomposable rank 3 vector bundles (locally free sheaves) on 5-dimensional projective space, found by .
Resolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, a non-singular variety W with a proper birational map W→V. For varieties over
Complex geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the stud
Elimination theory
In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating some variables between polynomials of several variables, in order to s
Motivic integration
Motivic integration is a notion in algebraic geometry that was introduced by Maxim Kontsevich in 1995 and was developed by Jan Denef and François Loeser. Since its introduction it has proved to be qui
Chordal variety
In algebraic geometry, a chordal variety of a variety is the union of all the chords (lines meeting 2 points), including the limiting cases of tangent lines.
Italian school of algebraic geometry
In relation to the history of mathematics, the Italian school of algebraic geometry refers to mathematicians and their work in birational geometry, particularly on algebraic surfaces, centered around
Tate topology
In mathematics, the Tate topology is a Grothendieck topology of the space of maximal ideals of a k-affinoid algebra, whose open sets are the admissible open subsets and whose coverings are the admissi
Last geometric statement of Jacobi
In differential geometry the last geometric statement of Jacobi is a conjecture named after Carl Gustav Jacob Jacobi. According to this conjecture: Every caustic from any point on an ellipsoid other t
Foundations of Algebraic Geometry
Foundations of Algebraic Geometry is a book by André Weil that develops algebraic geometry over fields of any characteristic. In particular it gives a careful treatment of intersection theory by defin
Motivic L-function
In mathematics, motivic L-functions are a generalization of Hasse–Weil L-functions to general motives over global fields. The local L-factor at a finite place v is similarly given by the characteristi
Chow group
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by Claude Chevalley) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The
Projective variety
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space over k that is the zero-locus of some finite family of homogeneous polynomials o
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension n − 1, which is embedded in a
Hodge–Tate module
In mathematics, a Hodge–Tate module is an analogue of a Hodge structure over p-adic fields. Serre introduced and named Hodge–Tate structures using the results of Tate on p-divisible groups.
Finite morphism
In algebraic geometry, a finite morphism between two affine varieties is a dense regular map which induces isomorphic inclusion between their coordinate rings, such that is integral over . This defini
Cramer's paradox
In mathematics, Cramer's paradox or the Cramer–Euler paradox is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitr
Gibbons–Hawking space
In mathematical physics, a Gibbons–Hawking space, named after Gary Gibbons and Stephen Hawking, is essentially a hyperkähler manifold with an extra U(1) symmetry. (In general, Gibbons–Hawking metrics
Torus action
In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus T is called a T-variety. In different
Baily–Borel compactification
In mathematics, the Baily–Borel compactification is a compactification of a quotient of a Hermitian symmetric space by an arithmetic group, introduced by Walter L. Baily and Armand Borel .
Calabi–Yau manifold
In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical phys
Galois geometry
Galois geometry (so named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or G
Alternating algebra
In mathematics, an alternating algebra is a Z-graded algebra for which xy = (−1)deg(x)deg(y)yx for all nonzero homogeneous elements x and y (i.e. it is an anticommutative algebra) and has the further
Flag bundle
In algebraic geometry, the flag bundle of a flag of vector bundles on an algebraic scheme X is the algebraic scheme over X: such that is a flag of vector spaces such that is a vector subspace of of di
Chevalley restriction theorem
In the mathematical theory of Lie groups, the Chevalley restriction theorem describes functions on a Lie algebra which are invariant under the action of a Lie group in terms of functions on a Cartan s
Coherent sheaf cohomology
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questio
Flat function
In mathematics, especially real analysis, a flat function is a smooth function all of whose derivatives vanish at a given point . The flat functions are, in some sense, the antitheses of the analytic
Flip (mathematics)
In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative canonical ring. In dimension 3 flips are used to
Mirror symmetry (string theory)
In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds lo
Gopakumar–Vafa invariant
In theoretical physics, Rajesh Gopakumar and Cumrun Vafa introduced in a series of papers new topological invariants, called Gopakumar–Vafa invariants, that represent the number of BPS states on a Cal
Fake projective space
In mathematics, a fake projective space is a complex algebraic variety that has the same Betti numbers as some projective space, but is not isomorphic to it. There are exactly 50 fake projective plane
List of topologies on the category of schemes
The most fundamental item of study in modern algebraic geometry is the category of schemes. This category admits many different Grothendieck topologies, each of which is well-suited for a different pu
Wu's method of characteristic set
Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept
General elephant
In algebraic geometry, general elephant is an idiosyncratic name for a general element of the anticanonical system of a variety, introduced by . For 3-folds the general elephant problem (or conjecture
Imaginary line (mathematics)
In complex geometry, an imaginary line is a straight line that only contains one real point. It can be proven that this point is the intersection point with the conjugated line. It is a special case o
K-groups of a field
In mathematics, especially in algebraic K-theory, the algebraic K-group of a field is important to compute. For a finite field, the complete calculation was given by Daniel Quillen.
Pursuing Stacks
Pursuing Stacks (French: À la Poursuite des Champs) is an influential 1983 mathematical manuscript by Alexander Grothendieck. It consists of a 12-page letter to Daniel Quillen followed by about 600 pa
Étale homotopy type
In mathematics, especially in algebraic geometry, the étale homotopy type is an analogue of the homotopy type of topological spaces for algebraic varieties. Roughly speaking, for a variety or scheme X
Cohomological descent
In algebraic geometry, a cohomological descent is, roughly, a "derived" version of a fully faithful descent in the classical descent theory. This point is made precise by the below: the following are
Complete intersection
In mathematics, an algebraic variety V in projective space is a complete intersection if the ideal of V is generated by exactly codim V elements. That is, if V has dimension m and lies in projective s
Fujiki class C
In algebraic geometry, a complex manifold is called Fujiki class C if it is to a compact Kähler manifold. This notion was defined by .
Dessin d'enfant
In mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute Galois group of the rational number
Griffiths group
In mathematics, more specifically in algebraic geometry, the Griffiths group of a projective complex manifold X measures the difference between homological equivalence and algebraic equivalence, which
Hilbert scheme
In algebraic geometry, a branch of mathematics, a Hilbert scheme is a scheme that is the parameter space for the closed subschemes of some projective space (or a more general projective scheme), refin
Arakelov theory
In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Suslin homology
In mathematics, the Suslin homology is a homology theory attached to algebraic varieties. It was proposed by Suslin in 1987, and developed by Suslin and Voevodsky. It is sometimes called singular homo
Bloch's formula
In algebraic K-theory, a branch of mathematics, Bloch's formula, introduced by Spencer Bloch for , states that the Chow group of a smooth variety X over a field is isomorphic to the cohomology of X wi
Projective plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pa
Glossary of classical algebraic geometry
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David Hilbert and the Italian school of algebraic geo
Toric variety
In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whol
Kobayashi metric
In mathematics and especially complex geometry, the Kobayashi metric is a pseudometric intrinsically associated to any complex manifold. It was introduced by Shoshichi Kobayashi in 1967. Kobayashi hyp
Hilbert series and Hilbert polynomial
In commutative algebra, the Hilbert function, the Hilbert polynomial, and the Hilbert series of a graded commutative algebra finitely generated over a field are three strongly related notions which me