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This is a list of algebraic geometry topics, by Wikipedia page. Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ...
Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ...
In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ...
In mathematics, a projective space is a fundamental construction from any vector space. ...
In mathematics, the projective line is a fundamental example of an algebraic curve. ...
In mathematics, the cross-ratio cr( w, x, y, z ) of an ordered quadruple of complex numbers (which may be real numbers) is Cross-ratios are preserved by linear fractional transformations, i. ...
In mathematics, a projective plane consists of a set of lines and a set of points with the following properties: Given any two distinct points, there is exactly one line incident with both of them. ...
In geometry and topology, the line at infinity is a line which is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. ...
In mathematics, the complex projective plane, usually denoted CP2, is the two-dimensional complex projective space. ...
In mathematics, complex projective space, or CPn, is the projective space of (complex) lines in Cn+1. ...
In projective geometry, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. ...
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In mathematics, a projective frame in projective geometry is an (n + 2)-tuple of points in general position in the space from which a projective space has been projected, one can take the first n + 1 points to form a basis, and the last to be the sum of the...
A projective transformation is a transformation used in projective geometry: it is the composition of a pair of perspective projections. ...
In mathematics, the fundamental theorem of projective geometry states that if Pn is a projective space and F and F′ are frames of Pn, then there exists a unique projective transformation sending F to F′. In case n = 1 this comes down to saying that given two ordered triples of...
Duality in the projective plane refers to the interchangeability between points and lines which preserves incidence properties. ...
In mathematics, the real projective plane is a two-dimensional manifold, that is, a surface, that has basic applications to geometry, but which cannot be embedded in our usual three-dimensional space. ...
In mathematics, real projective space, or RPn is the projective space of lines in Rn+1. ...
In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety. ...
In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety. ...
In algebraic geometry, an algebraic curve is an algebraic variety of dimension equal to 1. ...
In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. ...
Pascals theorem states that if an arbitrary hexagon is inscribed in any conic section, and opposite pairs of sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration. ...
Let ABCDEF be a hexagon formed by six tangent lines of a conic section. ...
In mathematics, an elliptic curve is a non-singular projective algebraic curve of genus 1 over a field K, together with a distinguished point defined over K. A more accessible (though less accurate) definition is that an elliptic curve is a plane curve defined by an equation of the form...
In mathematics, a cubic curve is a plane curve C defined by a cubic equation F(X,Y,Z) = 0 applied to homogeneous coordinates [X:Y:Z] for the projective plane; or the inhomogeneous version for the affine space determined by setting Z = 1 in such an equation. ...
In complex analysis, an elliptic function is, roughly speaking , a function defined on the complex plane which is periodic in two directions. ...
In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ...
In mathematics, Weierstrass introduced some particular elliptic functions that have become the basis for the most standard notations used. ...
In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler. ...
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at...
In mathematics, the Weil pairing is a construction of roots of unity by means of functions on an elliptic curve E, in such a way as to constitute a pairing (bilinear form, though with multiplicative notation) on the torsion subgroup of E. The name is for André Weil, who gave...
In algebraic geometry, a hyperelliptic curve (over the complex numbers) is an algebraic curve given by an equation of the form where f(x) is a polynomial of degree n > 4 with n distinct roots. ...
The Klein quartic x3y + y3z + z3x = 0, named after Felix Klein, is a Riemann surface, and a curve of genus 3 over the complex numbers C. The Klein quartic has automorphism group isomorphic to the projective special linear group G = PSL(2,7). ...
In mathematics, a modular curve is a Riemann surface, or corresponding algebraic curve, constructed as HΓ where H is the upper half-plane in the complex numbers, and Γ is a Fuchsian group acting on H, with Γ a subgroup of the modular group of integral 2×2 matrices. ...
In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. ...
In mathematics, modular functions are certain kinds of mathematical functions mapping complex numbers to complex numbers. ...
In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...
In mathematics, a supersingular prime is a certain kind of prime number. ...
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (X:Y:Z) by the Fermat equation Xn + Yn = Zn. ...
This article refers to Bézouts theorem in algebraic geometry. ...
In mathematics, the Brill-Noether theory in algebraic geometry is the theory of special divisors on generic algebraic curves. ...
In mathematics, the genus has few different meanings Topology The genus of a connected, oriented surface is an integer representing the maximum number of cuttings along closed simple curves without rendering the resultant manifold disconnected. ...
In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. ...
In mathematics, the Riemann-Hurwitz formula describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. ...
In mathematics, specifically in complex analysis and algebraic geometry, the Riemann-Roch theorem is an important tool in the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. ...
In mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a 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 geometry), for everywhere-regular differential 1-forms. ...
For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ...
In mathematics, Hurwitzs automorphisms theorem bounds the group of automorphisms, via conformal mappings, of a compact Riemann surface of genus g > 1, telling us that the order of the group of such automorphisms is bounded by 84(g − 1). ...
In mathematics, an algebraic surface is an algebraic variety of dimension two. ...
In geometry, a surface is ruled if through every point of there is a straight line that lies on . ...
A cubic surface is a projective variety studied in algebraic geometry. ...
In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space. ...
In mathematics, a del Pezzo surface is a complex two-dimensional Fano variety, i. ...
A K3 manifold is a hyperkähler manifold of real dimension 4, i. ...
In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanned by such curves (up to linear equivalence) has a one-dimensional subspace on which it is positive...
Algebraic geometry: classical approach In classical algebraic geometry (and to some extent also in modern algebraic geometry), the main objects of study are algebraic varieties. ...
In mathematics, a hypersurface is some kind of submanifold. ...
Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ...
In algebraic geometry, the dimension of an algebraic variety V is defined, informally speaking, as the number of independent rational functions that exist on V. So, for example, an algebraic curve has by definition dimension 1. ...
Hilberts Nullstellensatz ( German: theorem of zeros) is a theorem in algebraic geometry that relates varieties and ideals in polynomial rings over algebraically closed fields. ...
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y is a closed map, i. ...
In algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables. ...
In mathematics, a quasiprojective variety in algebraic geometry is, in non-intrinsic terms, a Zariski-open subset of a projective variety. ...
In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis G (named after Wolfgang Gröbner) is a particular kind of generating subset of an ideal I in a polynomial ring R. One can view it as a multivariate, non-linear generalization of: the Euclidean algorithm...
In mathematics, the canonical bundle of a non-singular algebraic variety of dimension is the line bundle which is the th exterior power of the cotangent bundle on . ...
In algebraic geometry, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n (and in greater generality) for vector bundles and the more general coherent sheaves. ...
In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalisations of the genus of an algebraic curve. ...
An irregular is a short name for something that does not follow the expected pattern. ...
The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...
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). ...
In algebraic geometry, the function field of an irreducible algebraic variety is the field of fractions of the ring of regular functions. ...
In mathematics, in algebraic geometry or the theory of complex manifolds, a very ample line bundle is one with enough sections to set up an embedding of its base variety or manifold into projective space. ...
In mathematics, the concept of a linear system of divisors arose first in the form of a linear system of algebraic curves in the projective plane. ...
In mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. ...
In mathematics, blowing up is a type of geometric modification, particularly applied in algebraic geometry, where it is essential in birational geometry. ...
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K. This is a question on its function field: is it up to isomorphism the field of all rational functions for some set of indeterminates...
In mathematics, a rational variety is an algebraic variety, over a given field K, which is birationally equivalent to projective space of some dimension over K. This is a question on its function field: is it up to isomorphism the field of all rational functions for some set of indeterminates...
In mathematics, the concept of intersection number arose in algebraic geometry, where two curves intersecting at a point may be considered to meet twice if they are tangent there. ...
In mathematics, Serres multiplicity conjectures are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. ...
In mathematics, the Albanese variety is a construction of algebraic geometry, which for an algebraic variety V solves a universal problem for morphisms of V into abelian varieties. ...
In mathematics, the Picard group of a ringed space is the group of isomorphism classes of invertible sheaves on , with the group operation being tensor product. ...
In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is a graded commutative ring that is made up of the sections of powers of the canonical bundle K. More precisely, it is the graded ring R such that for n...
A modular form is an analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. ...
In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. ...
In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. ...
In mathematics, Kleins j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half plane of complex numbers with positive imaginary part. ...
In mathematics, an algebraic function of indeterminates X1, X2, ..., Xn, is a function F that satisfies some non-trivial equation P(F, X1, X2, ..., Xn) = 0, with P a polynomial in n + 1 variables over a given field K. That is, F is an implicit function that solves an algebraic...
In the mathematics of the nineteenth century, an important role was played by the algebraic forms that generalise quadratic forms to degrees 3 and more, also known as quantics. ...
In mathematics, an addition theorem is a formula such as that for the exponential function ex + y = ex·ey that expresses, for a particular function f, f(x + y) in terms of f(x) and f(y). ...
In mathematics, invariant theory refers to the study of invariant algebraic forms (equivalently, symmetric tensors) for the action of linear transformations. ...
In mathematics, the symbolic method in invariant theory is a highly formal algorithm developed in the 19th century for computing form invariants — invariants of algebraic forms. ...
In mathematics, geometric invariant theory in algebraic geometry is (technically complex) development building on nineteenth century invariant theory. ...
In mathematics and theoretical physics, toric geometry is a set of methods in algebraic geometry in which complex manifolds are visualized as fiber bundles with multi-dimensional tori as fibers. ...
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 vector of small quantities. ...
In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ...
In mathematics, a singular point of an algebraic variety V is a point P that is special (so, singular), in the geometric sense that V is not locally flat there. ...
For non-mathematical singularity theories, see singularity. ...
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields. ...
In mathematics, the Weil conjectures, which had become theorems by 1975, were some highly-influential proposals from the late 1940s by Andre Weil on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields. ...
In differential geometry, a complex manifold is a manifold such that every neighborhood looks like the complex n-space. ...
In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ...
In mathematics, a Calabi-Yau manifold is a compact K hler manifold with a vanishing first Chern class. ...
In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a closed, complex submanifold of the vector space of n complex dimensions. ...
In mathematics, Hodge theory is the study of the consequences for the algebraic topology of a smooth manifold M of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M. It was developed by W. V. D. Hodge in the 1930s as an extension...
In mathematics, a Hodge cycle is a particular kind of homology class defined on a complex algebraic variety V, or more generally on a Kähler manifold. ...
The Hodge conjecture is a major unsolved problem of algebraic geometry. ...
In mathematics, algebraic geometry and analytic geometry are closely related subjects, where analytic geometry is the theory of complex manifolds and the more general analytic spaces defined locally by the vanishing of analytic functions of several complex variables. ...
In physics and mathematics, mirror symmetry is a surprising relation that can exist between two Calabi-Yau manifolds. ...
In algebraic geometry, an algebraic group is a group that is an algebraic variety, such that the multiplication and inverse are given by regular functions on the variety. ...
In mathematics, the identity component of a topological group G is the connected component C that contains the identity element e. ...
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. ...
In mathematics, an abelian group is a commutative group, i. ...
In mathematics, multiplicative group in group theory may mean any group G written in multiplicative notation (rather than additive notation for an abelian group) for its binary operation or in particular the multiplicative group of a field F, namely F{0} under multiplication, written F* or Fx. ...
In mathematics, a Borel subgroup (named after Armand Borel) of an algebraic group G is a maximal solvable subgroup. ...
In mathematics, a Borel subgroup (named after Armand Borel) of an algebraic group G is a maximal solvable subgroup. ...
In mathematics, an algebraic group G contains a unique maximal normal solvable subgroup; and this subgroup is closed. ...
In mathematics, the unipotent radical of an algebraic group G is the set of unipotent elements in the radical of G. Categories: Algebraic groups ...
In mathematics, the Lie-Kolchin theorem is a theorem in the representation theory of linear algebraic groups. ...
In mathematics, the Mumford conjecture states that for any semisimple algebraic group G, over a field K, and for any linear representation ρ of G on a K-vector space V, given v in V that is fixed by the action of G, there is a G-invariant F on...
In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ...
In mathematics, theta functions are special functions of several complex variables. ...
In mathematics, a Grassmannian is the space of all k-dimensional subspaces of an n-dimensional vector space V, often denoted Gk(V) or simply Gk,n. ...
In mathematics, a flag is an increasing sequence of subspaces of a vector space. ...
In mathematics, an algebraic torus is a particular kind of algebraic group, that becomes of the simple form Π GL1 of a direct product of finitely many copies of the multiplicative group GL1, once the underlying field is extended to an algebraically closed field. ...
In mathematics, specifically the theory of algebraic groups, Weil restriction is a functor allowing one to pass from an algebraic group G over a field L to another one, RG, over a subfield K. The idea is that the group of points G(L) of G over L should be...
Motivation and Basic Idea In mathematics, the antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. ...
Contemporary foundations Main article glossary of scheme theory This is a glossary of scheme theory. ...
In abstract algebra, commutative algebra is the field of study of commutative rings, their ideals, modules and algebras. ...
In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...
Model Theory In logic and model theory, a valuation is a map from the set of variables of a first-order language to the universe of some interpretation of that language. ...
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 exactly the same as its Krull dimension. ...
In commutative algebra, if R is a commutative ring and M an R-module, a R-regular sequence on M is a d-tuple of (non-zero non-unit) elements r1, r2, ..., rd from R such that for each i, ri is not a zerodivisor on the quotient R-module...
In mathematics, a Cohen-Macaulay ring is a commutative noetherian local ring with Krull dimension equal to its depth. ...
In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. ...
In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul. ...
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...
In mathematics, the Zariski topology is a structure basic to algebraic geometry, especially since 1950. ...
In mathematics, the Kähler differentials are a universal construction Ω1S/R associated to a ring homomorphism of commutative rings, φ:R → S, that provides an analogue of the construction of differential forms (1-forms). ...
In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain...
In mathematics, a locally ringed space (or local ringed space) is, intuitively speaking, a space together with, for each of its open sets, a commutative ring the elements of which are thought of as functions defined on that open set. ...
In mathematics, especially in algebraic geometry and the theory of complex manifolds, a coherent sheaf F on a locally ringed space X is a sheaf isomorphic with the cokernel of a morphism of OX_modules OXm → OXn. ...
In mathematics, an invertible sheaf is a coherent sheaf S on a ringed space X, for which there is an inverse T with respect to tensor product of OX-modules. ...
In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric...
In mathematics, specifically in algebraic geometry, the Grothendieck-Riemann-Roch theorem is a far-reaching result on coherent cohomology. ...
Coherent duality in mathematics refers to a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the local theory. ...
Schemes In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec(R), is defined to be the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space. ...
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ...
This is a glossary of scheme theory. ...
The Éléments de géométrie algébrique (Elements of Algebraic Geometry) by Alexander Grothendieck (assisted by Jean Dieudonné), or EGA for short, are an unfinished 1500-page treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut...
In mathematics, Alexander Grothendiecks Séminaire de géométrie algébrique was a unique phenomenon of research and publication outside of the main mathematical journals, reporting on work done starting from 1960 and centred on the IHÉS near Paris (the official title was the seminar of Bois...
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i. ...
In mathematics, a group scheme is a group object (some would prefer to say just group) in the category of schemes. ...
In mathematics, a semistable elliptic curve in diophantine geometry is an elliptic curve that has bad reduction only of multiplicative type. ...
Grothendiecks relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of objects explicitly depending on parameters, as the basic field of study, rather than a single such object. ...
Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ...
In mathematics, a Grothendieck topology is a structure defined on an arbitrary category C which allows the definition of sheaves on C, and with that the definition of general cohomology theories. ...
Sheaves were introduced into mathematics in the 1940s, and a major theme since then has been to study a space by studying sheaves on that space. ...
In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of gluing in topology. ...
In mathematics, Grothendiecks Galois theory is a highly 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 setting of algebraic geometry. ...
In mathematics, an algebraic stack in algebraic geometry is a special case of the concept of a stack, which is useful for working on moduli questions. ...
In mathematics, a gerbe is a construct in homological algebra. ...
In mathematics, the étale cohomology theory of algebraic geometry is a refined construction of homological algebra, introduced in order to attack the Weil conjectures. ...
In algebraic geometry the idea of a motive intuitively refers to some essential part of an algebraic variety. Mathematically, the theory of motives is then the conjectural universal cohomology theory for such objects. ...
Motivic cohomology is a homological theory in mathematics, the existence of which was first conjectured by Grothendieck during the 1960s. ...
In mathematics, homotopical algebra is a collection of concepts comprising the nonabelian aspects of homological algebra as well as possibly the abelian aspects as special cases. ...
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