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This is a list of algebraic geometry topics, by Wikipedia page.


Classical topics in

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pr ...

*
Affine space In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relat ...
* Projective space *
Projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
,
cross-ratio In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points ''A'', ''B'', ''C'' and ''D'' on a line, the ...
*
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 pairs of lines (namely, parallel lines) that ...
**
Line at infinity In geometry and topology, the line at infinity is a projective line that 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. The l ...
**
Complex projective plane In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordinates :(Z_1,Z_2,Z_3) \in \mathbf^3,\qquad (Z_1 ...
*
Complex projective space In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of ...
* Plane at infinity,
hyperplane at infinity In geometry, any hyperplane ''H'' of a projective space ''P'' may be taken as a hyperplane at infinity. Then the set complement is called an affine space. For instance, if are homogeneous coordinates for ''n''-dimensional projective space, then ...
* Projective frame *
Projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In genera ...
*
Fundamental theorem of projective geometry In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, ...
*
Duality (projective geometry) In geometry, a striking feature of projective planes is the symmetry of the roles played by points and lines in the definitions and theorems, and ( plane) duality is the formalization of this concept. There are two approaches to the subject of d ...
*
Real projective plane In mathematics, the real projective plane is an example of a compact non- orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has ...
*
Real projective space In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction ...
* Segre embedding of a product of projective spaces * Rational normal curve


Algebraic curve In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane ...
s

*
Conic In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a sp ...
s,
Pascal's theorem In projective geometry, Pascal's theorem (also known as the ''hexagrammum mysticum theorem'') states that if six arbitrary points are chosen on a conic (which may be an ellipse, parabola or hyperbola in an appropriate affine plane) and joined ...
, Brianchon's theorem *
Twisted cubic In mathematics, a twisted cubic is a smooth, rational curve ''C'' of degree three in projective 3-space P3. It is a fundamental example of a skew curve. It is essentially unique, up to projective transformation (''the'' twisted cubic, therefore ...
*
Elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
,
cubic curve In mathematics, a cubic plane curve is a plane algebraic curve defined by a cubic equation : applied to homogeneous coordinates for the projective plane; or the inhomogeneous version for the affine space determined by setting in such an ...
**
Elliptic function In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
,
Jacobi's elliptic functions In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of electronic elliptic filters. While tr ...
,
Weierstrass's elliptic functions In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by t ...
**
Elliptic integral In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
**
Complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visibl ...
**
Weil pairing Weil may refer to: Places in Germany *Weil, Bavaria *Weil am Rhein, Baden-Württemberg * Weil der Stadt, Baden-Württemberg *Weil im Schönbuch, Baden-Württemberg Other uses * Weil (river), Hesse, Germany * Weil (surname), including people with ...
*
Hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...
*
Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact space, compact Riemann surface of genus (mathematics), genus with the highest possible order automorphism group for this genus, namely order orientation-preservi ...
*
Modular curve In number theory and algebraic geometry, a modular curve ''Y''(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular ...
** Modular equation **
Modular function In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...
**
Modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fraction ...
** Supersingular primes * Fermat curve *
Bézout's theorem Bézout's theorem is a statement in algebraic geometry concerning the number of common zeros of polynomials in indeterminates. In its original form the theorem states that ''in general'' the number of common zeros equals the product of the deg ...
* Brill–Noether theory *
Genus (mathematics) In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus o ...
*
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ve ...
* Riemann–Hurwitz formula * Riemann–Roch theorem *
Abelian integral In mathematics, an abelian integral, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form :\int_^z R(x,w) \, dx, where R(x,w) is an arbitrary rational function of the two variables x and w, wh ...
*
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 varieties), for everywhere-regular differential ...
*
Jacobian variety In mathematics, the Jacobian variety ''J''(''C'') of a non-singular algebraic curve ''C'' of genus ''g'' is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of ''C'', hence an abelian var ...
** Generalized Jacobian *
Moduli of algebraic curves In algebraic geometry, a moduli space of (algebraic) curves is a geometric space (typically a scheme or an algebraic stack) whose points represent isomorphism classes of algebraic curves. It is thus a special case of a moduli space. Depending ...
* Hurwitz's theorem on automorphisms of a curve * Clifford's theorem on special divisors *
Gonality of an algebraic curve In mathematics, the gonality of an algebraic curve ''C'' is defined as the lowest degree of a nonconstant rational map from ''C'' to the projective line. In more algebraic terms, if ''C'' is defined over the field ''K'' and ''K''(''C'') denotes th ...
*
Weil reciprocity law In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field ''K''(''C'') of an algebraic curve ''C'' over an algebraically closed field ''K''. Given functions ''f'' and ''g'' in ''K''(''C''), i.e. rational fun ...
* Algebraic geometry codes


Algebraic surfaces

* Enriques–Kodaira classification *
List of algebraic surfaces This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira classification. Kodaira dimension −∞ Rational surfaces * Projective plane Qua ...
*
Ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directri ...
*
Cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather tha ...
* Veronese surface *
Del Pezzo surface In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in some sense the opposite of surfaces of genera ...
* Rational surface * Enriques surface *
K3 surface In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected ...
* Hodge index theorem *
Elliptic surface In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed ...
* Surface of general type * Zariski surface


Algebraic geometry: classical approach

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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 equations over the real or complex numbers ...
**
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 , which is embedded in an ambient space of dimension , generally a Eucl ...
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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 geometry. The theory is simplified by workin ...
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Dimension of an algebraic variety In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commuta ...
**
Hilbert's Nullstellensatz In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ...
** Complete variety **
Elimination theory Elimination may refer to: Science and medicine *Elimination reaction, an organic reaction in which two functional groups split to form an organic product *Bodily waste elimination, discharging feces, urine, or foreign substances from the body ...
**
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 in a polynomial ring over a field . A Grö ...
**
Projective variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables ...
** Quasiprojective variety **
Canonical bundle In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, ...
**
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 space ''P'n'', there sho ...
**
Serre duality In algebraic geometry, a branch of mathematics, Serre duality is a duality for the coherent sheaf cohomology of algebraic varieties, proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Ale ...
**
Spaltenstein variety In algebraic geometry, a Spaltenstein variety is a variety given by the fixed point (mathematics), fixed point set of a nilpotent transformation on a flag variety. They were introduced by . In the special case of full flag varieties the Spaltenstei ...
**
Arithmetic genus In mathematics, the arithmetic genus of an algebraic variety is one of a few possible generalizations of the genus of an algebraic curve or Riemann surface. Projective varieties Let ''X'' be a projective scheme of dimension ''r'' over a field '' ...
,
geometric genus In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for comp ...
,
irregularity Irregular, irregulars or irregularity may refer to any of the following: Astronomy * Irregular galaxy * Irregular moon * Irregular variable, a kind of star Language * Irregular inflection, the formation of derived forms such as plurals in ...
*
Tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
, Zariski tangent space *
Function field of an algebraic variety In algebraic geometry, the function field of an algebraic variety ''V'' consists of objects which are interpreted as rational functions on ''V''. In classical algebraic geometry they are ratios of polynomials; in complex algebraic geometry thes ...
*
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 most important notion of positivity is that of an ...
*
Ample vector 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 most important notion of positivity is that of an ...
*
Linear system of divisors In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the fo ...
*
Birational geometry In mathematics, birational geometry is a field of algebraic geometry in which the goal is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets. This amounts to studying mappings that are given by rationa ...
**
Blowing up In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with th ...
**
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 fields of characteri ...
**
Rational variety In mathematics, a rational variety is an algebraic variety, over a given field ''K'', which is birationally equivalent to a projective space of some dimension over ''K''. This means that its function field is isomorphic to :K(U_1, \dots , U_d), ...
** Unirational variety ** Ruled variety **
Kodaira dimension In algebraic geometry, the Kodaira dimension ''κ''(''X'') measures the size of the canonical model of a projective variety ''X''. Igor Shafarevich, in a seminar introduced an important numerical invariant of surfaces with the notation ''� ...
** Canonical ring **
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 as simple as possible. The subject has its or ...
*
Intersection theory In mathematics, intersection theory is one of the main branches of algebraic geometry, where it gives information about the intersection of two subvarieties of a given variety. The theory for varieties is older, with roots in Bézout's theorem o ...
** Intersection number **
Chow ring In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties ( ...
**
Chern class In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Y ...
**
Serre's multiplicity conjectures In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection n ...
* Albanese variety *
Picard group In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a globa ...
*
Modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory ...
*
Moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such ...
* Modular equation **
J-invariant In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holom ...
*
Algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations additi ...
* Algebraic form * Addition theorem *
Invariant theory Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties, such as vector spaces, from the point of view of their effect on functions. Classically, the theory dealt with the question of explicit descri ...
**
Symbolic method of invariant theory In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, Siegfried Heinrich Aronhold, Alfred Clebsch, and Paul Gordan in the 19th century for computing invariant (mathematics), invariants of algebraic ...
*
Geometric invariant theory In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in clas ...
*
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 whole variety. Some authors also require it to be n ...
*
Deformation theory In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesi ...
*
Singular point Singularity or singular point may refer to: Science, technology, and mathematics Mathematics * Mathematical singularity, a point at which a given mathematical object is not defined or not "well-behaved", for example infinite or not differentiab ...
, non-singular *
Singularity theory In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it ...
** Newton polygon *
Weil conjectures In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory. ...


Complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a ...
s

*
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. The concept was first studied by Jan Ar ...
*
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 physics. Particularly in superstring ...
*
Stein manifold In mathematics, in the theory of several complex variables and complex manifolds, a Stein manifold is a complex submanifold of the vector space of ''n'' complex dimensions. They were introduced by and named after . A Stein space is similar to a St ...
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Hodge theory In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every coh ...
*
Hodge cycle In differential geometry, a Hodge cycle or Hodge class is a particular kind of homology class defined on a complex algebraic variety ''V'', or more generally on a Kähler manifold. A homology class ''x'' in a homology group :H_k(V, \Complex) = H ...
*
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 subvarieties. In simple terms, the Hodge conjec ...
*
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 more general analytic spaces defined locally ...
* Mirror symmetry


Algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. ...
s

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Linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n ...
**
Additive group An additive group is a group of which the group operation is to be thought of as ''addition'' in some sense. It is usually abelian, and typically written using the symbol + for its binary operation. This terminology is widely used with structure ...
**
Multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
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Algebraic torus In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Highe ...
**
Reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
**
Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgrou ...
**
Parabolic subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
** Radical of an algebraic group **
Unipotent radical In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipote ...
** Lie-Kolchin theorem **
Haboush's theorem In mathematics Haboush's theorem, often still referred to as 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' ...
(also known as the Mumford conjecture) *
Group scheme In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups ha ...
*
Abelian variety In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functi ...
**
Theta function In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
*
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
*
Flag manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a s ...
* Weil restriction * Differential Galois theory


Contemporary foundations


Commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Promi ...

* Prime ideal *
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 generalizes to commutative algebra the notion of size ...
*
Krull dimension In commutative algebra, the Krull dimension of a commutative ring ''R'', named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generall ...
*
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 ''A'' be a Noetherian local ring with maximal id ...
* Regular sequence * Cohen–Macaulay ring *
Gorenstein ring In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring ''R'' with finite injective dimension as an ''R''-module. There are many equivalent conditions, some of them listed below, often saying that a Gorenstein ring ...
* Koszul complex *
Spectrum of a ring In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with ...
*
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is ...
*
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 standard in commutative algebra and algebra ...
*
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 to Alexander Grothendieck. Generic flatness ...
*
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 irrelevant ideal if its radical contains the irr ...


Sheaf theory In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...

*
Locally ringed space In mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that play roles of restrictions. Precisely, it is a topological space equipped with a sheaf ...
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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 definition of coherent sheaves is made with ref ...
*
Invertible sheaf 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 ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion ...
*
Sheaf cohomology In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions to solving a geometric problem globally whe ...
*
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 questions can be formulated as questions about the exis ...
* Hirzebruch–Riemann–Roch theorem *
Grothendieck–Riemann–Roch theorem In mathematics, specifically in algebraic geometry, the Grothendieck–Riemann–Roch theorem is a far-reaching result on coherent cohomology. It is a generalisation of the Hirzebruch–Riemann–Roch theorem, about complex manifolds, which is ...
*
Coherent duality In mathematics, coherent duality is any of 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 ...
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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 of noetherian induction. It has many applica ...


Schemes

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Affine scheme In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with t ...
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Scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
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Éléments de géométrie algébrique The ''Éléments de géométrie algébrique'' ("Elements of Algebraic Geometry") by Alexander Grothendieck (assisted by Jean Dieudonné), or ''EGA'' for short, is a rigorous treatise, in French, on algebraic geometry that was published (in eig ...
* Grothendieck's Séminaire de géométrie algébrique *
Fiber product of schemes In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction. It has many interpretations and special cases. For example, the fiber product describes how an algebraic variety over one field dete ...
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Flat morphism 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.e., :f_P\colon \mathcal_ ...
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Smooth scheme In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a s ...
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Finite morphism In algebraic geometry, a finite morphism between two affine varieties X, Y is a dense regular map which induces isomorphic inclusion k\left \righthookrightarrow k\left \right/math> between their coordinate rings, such that k\left \right/math> is ...
* Quasi-finite morphism *
Proper morphism In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field ''k'' a complete variety. For example, every projective variety over a fi ...
* Semistable elliptic curve * Grothendieck's relative point of view *
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), refining the Chow variety. The Hilbert scheme is a d ...


Category theory

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Grothendieck topology In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is c ...
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Topos In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notio ...
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Derived category In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction pro ...
* Descent (category theory) ** Grothendieck's Galois theory *
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 specific to algebraic stacks, such as Artin's repr ...
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Gerbe In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analog ...
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Étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectu ...
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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 theories such as singular cohomology, de Rham cohom ...
* Motivic cohomology * A¹ homotopy theory *
Homotopical algebra 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. The ''homotopical'' nomenclature stems from the fact that a com ...


Algebraic geometers

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Niels Henrik Abel Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
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Carl Gustav Jacob Jacobi Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasio ...
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Jakob Steiner Jakob Steiner (18 March 1796 – 1 April 1863) was a Swiss mathematician who worked primarily in geometry. Life Steiner was born in the village of Utzenstorf, Canton of Bern. At 18, he became a pupil of Heinrich Pestalozzi and afterwards s ...
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Julius Plücker Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the di ...
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Arthur Cayley Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician who worked mostly on algebra. He helped found the modern British school of pure mathematics. As a child, Cayley enjoyed solving complex maths problems ...
* Bernhard Riemann *
Max Noether Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the ...
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William Kingdon Clifford William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in ...
* David Hilbert * Italian school of algebraic geometry ** Guido Castelnuovo ** Federigo Enriques **
Francesco Severi Francesco Severi (13 April 1879 – 8 December 1961) was an Italian mathematician. He was the chair of the committee on Fields Medal on 1936, at the first delivery. Severi was born in Arezzo, Italy. He is famous for his contributions to algebr ...
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Solomon Lefschetz Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ...
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Oscar Zariski , birth_date = , birth_place = Kobrin, Russian Empire , death_date = , death_place = Brookline, Massachusetts, United States , nationality = American , field = Mathematics , work_institutions ...
* W. V. D. Hodge *
Sir Michael Atiyah Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded ...
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Kunihiko Kodaira was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japane ...
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André Weil André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. ...
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Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the ...
* Alexander Grothendieck *
Friedrich Hirzebruch Friedrich Ernst Peter Hirzebruch ForMemRS (17 October 1927 – 27 May 2012) was a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation. He has been described a ...
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Igor Shafarevich Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometr ...
* Heisuke Hironaka *
Shreeram S. Abhyankar Shreeram Shankar Abhyankar (22 July 1930 – 2 November 2012) was an Indian American mathematician known for his contributions to algebraic geometry. He, at the time of his death, held the Marshall Distinguished Professor of Mathematics Chai ...
* Pierre Samuel * C.P. Ramanujam *
David Mumford David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded ...
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Michael Artin Michael Artin (; born 28 June 1934) is a German-American mathematician and a professor emeritus in the Massachusetts Institute of Technology mathematics department, known for his contributions to algebraic geometry.Phillip Griffiths Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particula ...
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Pierre Deligne Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
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Yuri Manin Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical lo ...
* Shigefumi Mori * Vladimir Drinfeld *
Vladimir Voevodsky Vladimir Alexandrovich Voevodsky (, russian: Влади́мир Алекса́ндрович Воево́дский; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic var ...
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Claire Voisin Claire Voisin (born 4 March 1962) is a French mathematician known for her work in algebraic geometry. She is a member of the French Academy of Sciences and holds the chair of Algebraic Geometry at the Collège de France. Work She is noted for h ...
* János Kollár * Caucher Birkar {{DEFAULTSORT:Algebraic geometry Mathematics-related lists Outlines of mathematics and logic Wikipedia outlines