Classical topics in projective geometry

Projective geometry

In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...

• Affine space
  Affine space
  In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
  
  
• Projective space
  Projective space
  In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
  
  
• Projective line
  Projective line
  In mathematics, a projective line is a one-dimensional projective space. The projective line over a field K, denoted P1, may be defined as the set of one-dimensional subspaces of the two-dimensional vector space K2 .For the generalisation to the projective line over an associative ring, see...
  
  , cross-ratio
  Cross-ratio
  In geometry, the cross-ratio, also called double ratio and anharmonic ratio, is a special number associated with an ordered quadruple of collinear points, particularly points on a projective line...
  
  
• Projective plane
  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 that do not intersect...
  
  
  • Line at infinity
    Line at infinity
    In geometry and topology, the line at infinity is a line that is added to the real plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line.-Geometric formulation:In...
    
    
  • Complex projective plane
• Complex projective space
  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 a complex projective space label the complex lines...
  
  
• Plane at infinity
  Plane at infinity
  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. The result of the addition is the projective 3-space, P^3...
  
  , hyperplane at infinity
• Projective frame
  Projective frame
  In the mathematical field of projective geometry, a projective frame is an ordered collection of points in projective space which can be used as reference points to describe any other point in that space...
  
  
• Projective transformation
• Fundamental theorem of projective geometry
• Duality (projective geometry)
  Duality (projective geometry)
  A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
  
  
• Real projective plane
  Real projective plane
  In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
  
  
• Real projective space
  Real projective space
  In mathematics, real projective space, or RPn, is the topological space of lines through 0 in Rn+1. It is a compact, smooth manifold of dimension n, and a special case of a Grassmannian.-Construction:...
  
  
• Segre embedding
  Segre embedding
  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...
  
  , multi-way projective space
• Rational normal curve

Algebraic curve

Algebraic curve

In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...

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• Conics, Pascal's theorem
  Pascal's theorem
  In projective geometry, Pascal's theorem states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the Pascal line of that configuration.- Related results :This theorem...
  
  , Brianchon's theorem
  Brianchon's theorem
  In geometry, Brianchon's theorem, named after Charles Julien Brianchon , is as follows. Let ABCDEF be a hexagon formed by six tangent lines of a conic section...
  
  
• Twisted cubic
  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...
  
  
• Elliptic curve
  Elliptic curve
  In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
  
  , cubic curve
  • Elliptic function
    Elliptic function
    In complex analysis, an elliptic function is a function defined on the complex plane that is periodic in two directions and at the same time is meromorphic...
    
    , Jacobi's elliptic functions
    Jacobi's elliptic functions
    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...
    
    , Weierstrass's elliptic functions
    Weierstrass's elliptic functions
    In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass...
    
    
  • Elliptic integral
    Elliptic integral
    In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler...
    
    
  • Complex multiplication
    Complex multiplication
    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 In mathematics, complex multiplication is the...
    
    
  • Weil pairing
    Weil pairing
    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 on the torsion subgroup of E...
    
    
• Hyperelliptic curve
• Klein quartic
  Klein quartic
  In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this genus, namely order 168 orientation-preserving automorphisms, and 336 automorphisms if orientation may be reversed...
  
  
• modular curve
  Modular curve
  In number theory and algebraic geometry, a modular curve Y 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 group of integral 2×2 matrices SL...
  
  
  • modular equation
    Modular equation
    In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli.The most frequent use of the term...
    
    
  • modular function
  • modular group
    Modular group
    In mathematics, the modular group Γ is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics...
    
    
  • Supersingular primes
    Supersingular prime (for an elliptic curve)
    In algebraic number theory, a supersingular prime is a certain type of prime number. If E is an elliptic curve defined over the rational numbers, then a prime p is supersingular for E if the reduction of E modulo p is a supersingular elliptic curve over the residue field Fp.Noam Elkies showed that...
    
    
• Fermat curve
  Fermat curve
  In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates by the Fermat equationX^n + Y^n = Z^n.\ Therefore in terms of the affine plane its equation is...
  
  
• Bézout's theorem
  Bézout's theorem
  Bézout's theorem is a statement in algebraic geometry concerning the number of common points, or intersection points, of two plane algebraic curves. The theorem claims that the number of common points of two such curves X and Y is equal to the product of their degrees...
  
  
• Brill–Noether theory
• Edwards curve
  Edwards curve
  In mathematics, an Edwards curve is a new representation of elliptic curves, discovered by Harold M. Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography...
  
  
• Genus (mathematics)
  Genus (mathematics)
  In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
  
  
• Riemann surface
  Riemann surface
  In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
  
  
• Riemann–Hurwitz formula
• Riemann–Roch theorem
• Abelian integral
• Differential of the first kind
  Differential of the first kind
  In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces and algebraic curves , for everywhere-regular differential 1-forms...
  
  
• Jacobian variety
  Jacobian variety
  In mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
  
  
  • Generalized Jacobian
    Generalized Jacobian
    - In algebraic geometry :In mathematics, there are several notions of generalized Jacobians, which are algebraic groups or complex manifolds that are in some sense analogous to the Jacobian variety of an algebraic curve, or related to the Albanese variety and Picard variety that generalize it to...
    
    
• Hurwitz's automorphisms theorem
• Clifford's theorem
  Clifford's theorem
  In mathematics, Clifford's theorem on special divisors is a result of W. K. Clifford on algebraic curves, showing the constraints on special linear systems on a curve C....
  
  
• Gonality of an algebraic curve
  Gonality of an algebraic curve
  In mathematics, the gonality of an algebraic curve C is defined as the lowest degree of a rational map from C to the projective line, which is not constant...
  
  
• Weil's reciprocity law
• Goppa code

Algebraic surface

Algebraic surface

In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...

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• Enriques-Kodaira classification
  Enriques-Kodaira classification
  In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space...
  
  
• List of algebraic surfaces
• Ruled surface
  Ruled surface
  In geometry, a surface S is ruled if through every point of S there is a straight line that lies on S. The most familiar examples are the plane and the curved surface of a cylinder or cone...
  
  
• Cubic surface
  Cubic surface
  A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...
  
  
• Veronese surface
  Veronese surface
  In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese...
  
  
• Del Pezzo 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...
  
  
• Rational surface
  Rational surface
  In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two...
  
  
• Enriques surface
  Enriques surface
  In mathematics, Enriques surfaces, discovered by , are complex algebraic surfacessuch that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square...
  
  
• K3 surface
  K3 surface
  In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
  
  
• Hodge index theorem
  Hodge index theorem
  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...
  
  
• Elliptic surface
  Elliptic surface
  In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....
  
  
• Surface of general type
  Surface of general type
  In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this...
  
  
• Zariski surface
  Zariski surface
  In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the surface...
  
  

Algebraic geometry: classical approach

• Algebraic variety
  Algebraic variety
  In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
  
  
  • Hypersurface
    Hypersurface
    In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
    
    
  • Quadric
    Quadric
    In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
    
    
  • Dimension of an algebraic variety
    Dimension of an algebraic variety
    In mathematics, the dimension of an algebraic variety V in algebraic geometry is defined, informally speaking, as the number of independent rational functions that exist on V.For example, an algebraic curve has by definition dimension 1...
    
    
  • Hilbert's Nullstellensatz
    Hilbert's Nullstellensatz
    Hilbert's Nullstellensatz is a theorem which establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields...
    
    
  • Complete variety
  • Elimination theory
    Elimination theory
    In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables....
    
    
  • Quasiprojective variety
    Quasiprojective variety
    In mathematics, a quasiprojective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset...
    
    
    • Gröbner basis
      Gröbner basis
      In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R...
      
      
  • Canonical bundle
    Canonical bundle
    In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...
    
    
  • Complete intersection
    Complete intersection
    In mathematics, an algebraic variety V in projective space is a complete intersection if it can be defined by the vanishing of the number of homogeneous polynomials indicated by its codimension...
    
    
  • Serre duality
    Serre duality
    In algebraic geometry, a branch of mathematics, Serre duality is a duality present on non-singular projective algebraic varieties V of dimension n . It shows that a cohomology group Hi is the dual space of another one, Hn−i...
    
    
  • Arithmetic genus
    Arithmetic genus
    In mathematics, the arithmetic genus of an algebraic variety is one of some possible generalizations of the genus of an algebraic curve or Riemann surface.The arithmetic genus of a projective complex manifold...
    
    , geometric genus
    Geometric genus
    In algebraic geometry, the geometric genus is a basic birational invariant pg of algebraic varieties and complex manifolds.-Definition:...
    
    , irregularity
• Tangent space
  Tangent space
  In mathematics, the tangent space of a manifold facilitates the generalization of 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....
  
  , Zariski tangent space
  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...
  
  
• Function field
  Function field
  Function field may refer to:*Function field of an algebraic variety*Function field...
  
  
• Ample vector bundle
• Linear system of divisors
  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....
  
  
• Birational geometry
  Birational geometry
  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 the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
  
  
  • Blowing up
    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 the projectivized tangent space at that point...
    
    
  • Rational variety
    Rational variety
    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...
    
    
  • Unirational variety
• Intersection number
  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 curves, and accounting properly for tangency...
  
  
  • Intersection theory
    Intersection theory
    In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...
    
    
  • Serre's multiplicity conjectures
    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...
    
    
• Albanese variety
  Albanese variety
  In mathematics, the Albanese variety A, named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve, and is the abelian variety generated by a variety V. In other words there is a morphism from the variety V to its Albanese variety A, such that any morphism from V to an...
  
  
• Picard group
• Pluricanonical ring
  Pluricanonical ring
  In mathematics, the pluricanonical ring of an algebraic variety V , or of a complex manifold, is the graded ringR=R \,of sections of powers of the canonical bundle K...
  
  
• Modular form
  Modular form
  In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
  
  
• Moduli space
  Moduli space
  In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects...
  
  
• Modular equation
  Modular equation
  In mathematics, a modular equation is an algebraic equation satisfied by moduli, in the sense of moduli problem. That is, given a number of functions on a moduli space, a modular equation is an equation holding between them, or in other words an identity for moduli.The most frequent use of the term...
  
  
  • J-invariant
    J-invariant
    In mathematics, Klein's j-invariant, regarded as a function of a complex variable τ, is a modular function defined on the upper half-plane of complex numbers.We haveThe modular discriminant \Delta is defined as \Delta=g_2^3-27g_3^2...
    
    
• Algebraic function
  Algebraic function
  In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
  
  
• Algebraic form
• Addition theorem
  Addition theorem
  In mathematics, an addition theorem is a formula such as that for the exponential functionthat expresses, for a particular function f, f in terms of f and f...
  
  
• Invariant theory
  Invariant theory
  Invariant theory is a branch of abstract algebra dealing with actions of groups on algebraic varieties from the point of view of their effect on functions...
  
  
  • Symbolic method of invariant theory
• Geometric invariant theory
  Geometric invariant theory
  In mathematics Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces...
  
  
• Toric geometry
  Toric geometry
  In algebraic geometry, a toric variety or torus embedding is a normal variety containing an algebraic torus as a dense subset, such that the action of the torus on itself extends to the whole variety.-The toric variety of a fan:...
  
  
• Deformation theory
  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 Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach...
  
  
• Singular point
  Mathematical singularity
  In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability...
  
  , non-singular
• Singularity theory
  Singularity theory
  -The notion of singularity:In mathematics, singularity theory is the study of the failure of manifold structure. A loop of string can serve as an example of a one-dimensional manifold, if one neglects its width. What is meant by a singularity can be seen by dropping it on the floor...
  
  
  • Newton polygon
    Newton polygon
    In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields.In the original case, the local field of interest was the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ringover K, where K...
    
    
• Weil conjectures
  Weil conjectures
  In mathematics, the Weil conjectures were some highly-influential proposals by on the generating functions derived from counting the number of points on algebraic varieties over finite fields....
  
  

Complex manifold

Complex manifold

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic....

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• Kähler manifold
  Kähler manifold
  In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
  
  
• Calabi–Yau manifold
• Stein manifold
  Stein manifold
  In mathematics, a Stein manifold in the theory of several complex variables and complex manifolds is a complex submanifold of the vector space of n complex dimensions. The name is for Karl Stein.- Definition :...
  
  
• Hodge theory
  Hodge theory
  In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of the algebraic topology of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised...
  
  
• Hodge cycle
  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...
  
  
• Hodge conjecture
  Hodge conjecture
  The Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of a non-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture says that certain de Rham cohomology classes are algebraic, that is, they...
  
  
• Algebraic geometry and analytic geometry
  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 by the vanishing of analytic functions of several...
  
  
• Mirror symmetry
  Mirror symmetry
  In physics and mathematics, mirror symmetry is a relation that can exist between two Calabi-Yau manifolds. It happens, usually for two such six-dimensional manifolds, that the shapes may look very different geometrically, but nevertheless they are equivalent if they are employed as hidden...
  
  

Algebraic group

Algebraic group

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...

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• Identity component
  Identity component
  In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group...
  
  
• Linear algebraic group
  Linear algebraic group
  In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices that is defined by polynomial equations...
  
  
  • Additive group
    Additive group
    An additive group may refer to:*an abelian group, when it is written using the symbol + for its binary operation*a group scheme representing the underlying-additive-group functor...
    
    
  • Multiplicative group
    Multiplicative group
    In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...
    
    
  • Borel subgroup
    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 group GLn ,...
    
    
  • Parabolic subgroup
  • Radical of an algebraic group
    Radical of an algebraic group
    The radical of an algebraic group is the identity component of its maximal normal solvable subgroup.- External links :*, Encyclopaedia of Mathematics...
    
    
  • Unipotent radical
  • Lie-Kolchin theorem
  • Mumford conjecture
    Mumford conjecture
    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 ≠ 0 in V that is fixed by the action of G, there is a...
    
    
• Abelian variety
  Abelian variety
  In mathematics, particularly in algebraic geometry, complex analysis and 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 functions...
  
  
  • Theta function
• Grassmannian
  Grassmannian
  In mathematics, a Grassmannian is a space which parameterizes all linear subspaces of a vector space V of a given dimension. For example, the Grassmannian Gr is the space of lines through the origin in V, so it is the same as the projective space P. The Grassmanians are compact, topological...
  
  
• Flag manifold
  Flag manifold
  In mathematics, a generalized 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 smooth or complex manifold, called a real or complex flag manifold...
  
  
• Algebraic torus
  Algebraic torus
  In mathematics, an algebraic torus is a type of commutative affine algebraic group. These groups were named by analogy with the theory of tori in Lie group theory...
  
  
• Weil restriction
  Weil restriction
  In mathematics, restriction of scalars is a functor which, for any finite extension of fields L/k and any algebraic variety X over L, produces another variety ResL/kX, defined over k...
  
  
• Differential Galois theory
  Differential Galois theory
  In mathematics, differential Galois theory studies the Galois groups of differential equations.Whereas algebraic Galois theory studies extensions of algebraic fields, differential Galois theory studies extensions of differential fields, i.e. fields that are equipped with a derivation, D. Much of...
  
  

Commutative algebra

Commutative algebra

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

• Prime ideal
  Prime ideal
  In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
  
  
• Valuation (mathematics)
• Regular local ring
  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 ideal m, and suppose a1, ..., an is a minimal set of...
  
  
• Regular sequence (algebra)
  Regular sequence (algebra)
  In commutative algebra, if R is a commutative ring and M an R-module, a nonzero element r in R is called M-regular if r is not a zerodivisor on M, and M/rM is nonzero...
  
  
• Cohen–Macaulay ring
• Gorenstein ring
  Gorenstein ring
  In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition....
  
  
• Koszul complex
  Koszul complex
  In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul...
  
  
• Spectrum of a ring
  Spectrum of a ring
  In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
  
  
• Zariski topology
  Zariski topology
  In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
  
  
• Kähler differential
  Kähler differential
  In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...
  
  
• Generic flatness
  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...
  
  
• Irrelevant ideal
  Irrelevant ideal
  In mathematics, the irrelevant ideal is the ideal of a graded ring consisting of all 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 irrelevant ideal.The terminology arises from the...
  
  

Sheaf theory

• Locally ringed space
• Coherent sheaf
  Coherent sheaf
  In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
  
  
• Invertible sheaf
  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 OX-modules. It is the equivalent in algebraic geometry of the topological notion of a line bundle...
  
  
• Sheaf cohomology
  Sheaf cohomology
  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...
  
  
• Hirzebruch–Riemann–Roch theorem
• Grothendieck–Riemann–Roch theorem
• Coherent duality
  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' theory....
  
  
• Dévissage
  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...
  
  

Schemes

• Affine scheme
• Scheme
  Scheme (mathematics)
  In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
  
  
• Glossary of scheme theory
  Glossary of scheme theory
  This is a glossary of scheme theory. For an introduction to the theory of schemes in algebraic geometry, see affine scheme, projective space, sheaf and scheme. The concern here is to list the fundamental technical definitions and properties of scheme theory...
  
  
• Éléments de géométrie algébrique
  Éléments de géométrie algébrique
  The Éléments de géométrie algébrique by Alexander Grothendieck , or EGA for short, is a rigorous treatise, in French, on algebraic geometry that was published from 1960 through 1967 by the Institut des Hautes Études Scientifiques...
  
  
• Grothendieck's Séminaire de géométrie algébrique
  Grothendieck's Séminaire de géométrie algébrique
  In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris...
  
  
• Flat morphism
  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.,is a flat map for all P in X...
  
  
• Finite morphism
• Quasi-finite morphism
  Quasi-finite morphism
  In algebraic geometry, a branch of mathematics, a morphism f : X → Y of schemes is quasi-finite if it is finite type and satisfies any of the following equivalent conditions:...
  
  
• Group scheme
  Group scheme
  In mathematics, a group scheme is a type of algebro-geometric object 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 have group scheme structure, but group schemes are not...
  
  
• Semistable elliptic curve
  Semistable elliptic curve
  In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field....
  
  
• Grothendieck's relative point of view
  Grothendieck's relative point of view
  Grothendieck's 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

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

• Grothendieck topology
  Grothendieck topology
  In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site.Grothendieck topologies axiomatize the notion...
  
  
• Topos
  Topos
  In mathematics, a topos is a type of category that behaves like the category of sheaves of sets on a topological space...
  
  
• Descent (category theory)
  Descent (category theory)
  In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.A sophisticated...
  
  
  • Grothendieck's Galois theory
    Grothendieck's Galois theory
    In mathematics, Grothendieck's 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...
    
    
• Algebraic stack
• Gerbe
  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 a generalization of principal bundles to the setting of 2-categories...
  
  
• Étale cohomology
  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 conjectures...
  
  
• Motive (mathematics)
• Motivic cohomology
  Motivic cohomology
  Motivic cohomology is a cohomological theory in mathematics, the existence of which was first conjectured by Alexander Grothendieck during the 1960s. At that time, it was conceived as a theory constructed on the basis of the so-called standard conjectures on algebraic cycles, in algebraic geometry...
  
  
• Homotopical algebra
  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...
  
  

Algebraic geometers

• Niels Henrik Abel
  Niels Henrik Abel
  Niels Henrik Abel was a Norwegian mathematician who proved the impossibility of solving the quintic equation in radicals.-Early life:...
  
  
• Carl Gustav Jakob Jacobi
  Carl Gustav Jakob Jacobi
  Carl Gustav Jacob Jacobi was a German mathematician, widely considered to be the most inspiring teacher of his time and is considered one of the greatest mathematicians of his generation.-Biography:...
  
  
• Jakob Steiner
  Jakob Steiner
  Jakob Steiner was a Swiss mathematician who worked primarily in geometry.-Personal and professional life:...
  
  
• Julius Plücker
  Julius Plücker
  Julius Plücker 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 discovery of the electron. He also vastly extended the study of Lamé curves.- Early...
  
  
• Bernhard Riemann
  Bernhard Riemann
  Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
  
  
• William Kingdon Clifford
  William Kingdon Clifford
  William Kingdon Clifford FRS 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 his honour, with interesting applications in contemporary mathematical physics...
  
  
• Italian school of algebraic geometry
  Italian school of algebraic geometry
  In relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more done internationally in birational geometry, particularly on algebraic surfaces. There were in the region of 30 to 40 leading mathematicians who made major...
  
  
  • Guido Castelnuovo
    Guido Castelnuovo
    Guido Castelnuovo was an Italian mathematician. His father, Enrico Castelnuovo, was a novelist and campaigner for the unification of Italy...
    
    
  • Francesco Severi
    Francesco Severi
    Francesco Severi was an Italian mathematician.Severi was born in Arezzo, Italy. He is famous for his contributions to algebraic geometry and the theory of functions of several complex variables. He became the effective leader of the Italian school of algebraic geometry...
    
    
• Solomon Lefschetz
  Solomon Lefschetz
  Solomon Lefschetz was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear ordinary differential equations.-Life:...
  
  
• Oscar Zariski
  Oscar Zariski
  Oscar Zariski was a Russian mathematician and one of the most influential algebraic geometers of the 20th century.-Education:...
  
  
• Erich Kähler
  Erich Kähler
  was a German mathematician with wide-ranging geometrical interests.Kähler was born in Leipzig, and studied there. He received his Ph.D. in 1928 from the University of Leipzig. He held professorial positions in Königsberg, Leipzig, Berlin and Hamburg...
  
  
• W. V. D. Hodge
  W. V. D. Hodge
  William Vallance Douglas Hodge FRS was a Scottish mathematician, specifically a geometer.His discovery of far-reaching topological relations between algebraic geometry and differential geometry—an area now called Hodge theory and pertaining more generally to Kähler manifolds—has been a major...
  
  
• Kunihiko Kodaira
• André Weil
  André Weil
  André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
  
  
• Jean-Pierre Serre
  Jean-Pierre Serre
  Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
  
  
• Alexander Grothendieck
  Alexander Grothendieck
  Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
  
  
• David Mumford
  David Mumford
  David Bryant Mumford is an American mathematician known for distinguished 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 the National Medal of Science...
  
  
• Igor Shafarevich
  Igor Shafarevich
  Igor Rostislavovich Shafarevich is a Soviet and Russian mathematician, founder of a school of algebraic number theory and algebraic geometry in the USSR, and a political writer. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharov's Human Rights...
  
  
• Heisuke Hironaka
  Heisuke Hironaka
  is a Japanese mathematician. After completing his undergraduate studies at Kyoto University, he received his Ph.D. from Harvard while under the direction of Oscar Zariski. He won the Fields Medal in 1970....
  
  
• Shigefumi Mori
  Shigefumi Mori
  -References:*Heisuke Hironaka, Fields Medallists Lectures, Michael F. Atiyah , Daniel Iagolnitzer ; World Scientific Publishing, 2007. ISBN 9810231172...
  
  
• Vladimir Voevodsky
  Vladimir Voevodsky
  Vladimir Voevodsky is a Russian American mathematician. His work in developing a homotopy theory for algebraic varieties and formulating motivic cohomology led to the award of a Fields Medal in 2002.- Biography :...