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101 relations: Abelian variety, Absolutely irreducible, Affine space, Alexander Grothendieck, Algebra, Algebraic curve, Algebraic geometry, Algebraic geometry of projective spaces, Algebraic manifold, Algebraic space, Algebraic surface, Algebraically closed field, Analytic manifold, André Weil, Birational geometry, Characteristic class, Chern class, Claude Chevalley, Closed immersion, Closed set, Coherent sheaf, Complete variety, Complex number, Complex plane, Complex-analytic variety, Constructible set (topology), Dimension of an algebraic variety, Elliptic curve, Exterior algebra, Factorization of polynomials, Fiber product of schemes, Foundations of Algebraic Geometry, Function (mathematics), Function field of an algebraic variety, Fundamental theorem of algebra, Generic property, Genus–degree formula, Geometry, Graph isomorphism, Grassmannian, Gröbner basis, Hilbert's Nullstellensatz, Homogeneous coordinate ring, Homogeneous coordinates, Homogeneous polynomial, Hypersurface, Ideal (ring theory), Identity function, If and only if, Injective function, ..., Integral domain, Jean-Pierre Serre, Line (geometry), Manifold, Masayoshi Nagata, Moduli of algebraic curves, Monic polynomial, Monomial order, Morphism of algebraic varieties, Motive (algebraic geometry), Nash functions, Natural number, Natural topology, Nilpotent, Nilradical of a ring, Plücker embedding, Polynomial ring, Prime ideal, Projective line, Projective space, Projective variety, Quasi-projective variety, Quotient ring, Real number, Riemann sphere, Ring theory, Ringed space, Scheme (mathematics), Segre embedding, Semialgebraic set, Sheaf (mathematics), Sheaf cohomology, Singular point of an algebraic variety, Smoothness, Solution set, Spectrum of a ring, Springer Science+Business Media, Stack (mathematics), Subset, System of polynomial equations, Tautological bundle, Twisted cubic, Unit circle, Valuation (algebra), Variety, Vector bundle, Vector space, Veronese surface, Zariski topology, Zariski–Riemann space, Zero of a function. Expand index (51 more) »
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 functions.
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Absolutely irreducible
In mathematics, a multivariate polynomial defined over the rational numbers is absolutely irreducible if it is irreducible over the complex field.
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Affine space
In mathematics, an affine space is a geometric structure that generalizes 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 related to parallelism and ratio of lengths for parallel line segments.
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Alexander Grothendieck
Alexander Grothendieck (28 March 1928 – 13 November 2014) was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry.
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Algebra
Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.
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Algebraic curve
In mathematics, a plane real algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Algebraic geometry of projective spaces
Projective space plays a central role in algebraic geometry.
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Algebraic manifold
__notoc__ In mathematics, an algebraic manifold is an algebraic variety which is also a manifold.
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Algebraic space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by for use in deformation theory.
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Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two.
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Algebraically closed field
In abstract algebra, an algebraically closed field F contains a root for every non-constant polynomial in F, the ring of polynomials in the variable x with coefficients in F.
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Analytic manifold
In mathematics, an analytic manifold is a topological manifold with analytic transition maps.
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André Weil
André Weil (6 May 1906 – 6 August 1998) was an influential French mathematician of the 20th century, known for his foundational work in number theory, algebraic geometry.
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Birational geometry
In mathematics, birational geometry is a field of algebraic geometry the goal of which is to determine when two algebraic varieties are isomorphic outside lower-dimensional subsets.
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Characteristic class
In mathematics, a characteristic class is a way of associating to each principal bundle X a cohomology class of X. The cohomology class measures the extent the bundle is "twisted" — and whether it possesses sections.
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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.
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Claude Chevalley
Claude Chevalley (11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory, and the theory of algebraic groups.
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Closed immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes f: Z \to X that identifies Z as a closed subset of X such that regular functions on Z can be extended locally to X. The latter condition can be formalized by saying that f^\#:\mathcal_X\rightarrow f_\ast\mathcal_Z is surjective.
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Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set.
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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.
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Complete variety
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism is a closed map, i.e. maps closed sets onto closed sets.
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Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
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Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
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Complex-analytic variety
In mathematics, specifically complex geometry, a complex-analytic variety is defined locally as the set of common zeros of finitely many analytic functions.
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Constructible set (topology)
In topology, a constructible set in a topological space is a finite union of locally closed sets.
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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.
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Elliptic curve
In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.
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Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.
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Factorization of polynomials
In mathematics and computer algebra, factorization of polynomials or polynomial factorization is the process of expressing a polynomial with coefficients in a given field or in the integers as the product of irreducible factors with coefficients in the same domain.
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Fiber product of schemes
In mathematics, specifically in algebraic geometry, the fiber product of schemes is a fundamental construction.
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Foundations of Algebraic Geometry
Foundations of Algebraic Geometry is a book by that develops algebraic geometry over fields of any characteristic.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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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 these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
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Fundamental theorem of algebra
The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root.
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Generic property
In mathematics, properties that hold for "typical" examples are called generic properties.
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Genus–degree formula
In classical algebraic geometry, the genus–degree formula relates the degree d of an irreducible plane curve C with its arithmetic genus g via the formula: Here "plane curve" means that C is a closed curve in the projective plane \mathbb^2.
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Geometry
Geometry (from the γεωμετρία; geo- "earth", -metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space.
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Graph isomorphism
In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H such that any two vertices u and v of G are adjacent in G if and only if ƒ(u) and ƒ(v) are adjacent in H. This kind of bijection is commonly described as "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection.
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Grassmannian
In mathematics, the Grassmannian is a space which parametrizes all -dimensional linear subspaces of the n-dimensional vector space.
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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.
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Hilbert's Nullstellensatz
Hilbert's Nullstellensatz (German for "theorem of zeros," or more literally, "zero-locus-theorem"—see Satz) is a theorem that establishes a fundamental relationship between geometry and algebra.
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Homogeneous coordinate ring
In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring where I is the homogeneous ideal defining V, K is the algebraically closed field over which V is defined, and is the polynomial ring in N + 1 variables Xi.
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Homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry.
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Homogeneous polynomial
In mathematics, a homogeneous polynomial is a polynomial whose nonzero terms all have the same degree.
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Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface.
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Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.
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Identity function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Injective function
In mathematics, an injective function or injection or one-to-one function is a function that preserves distinctness: it never maps distinct elements of its domain to the same element of its codomain.
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Integral domain
In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
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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.
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Line (geometry)
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.
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Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
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Masayoshi Nagata
Masayoshi Nagata (Japanese: 永田 雅宜 Nagata Masayoshi; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra.
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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.
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Monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1.
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Monomial order
In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e.,.
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Morphism of algebraic varieties
In algebraic geometry, a morphism between algebraic varieties is a function between the varieties that is given locally by polynomials.
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Motive (algebraic geometry)
In algebraic geometry, a motive (or sometimes motif, following French usage) denotes 'some essential part of an algebraic variety'.
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Nash functions
In real algebraic geometry, a Nash function on an open semialgebraic subset U ⊂ Rn is an analytic function f: U → R satisfying a nontrivial polynomial equation P(x,f(x)).
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Natural number
In mathematics, the natural numbers are those used for counting (as in "there are six coins on the table") and ordering (as in "this is the third largest city in the country").
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Natural topology
In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question.
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Nilpotent
In mathematics, an element, x, of a ring, R, is called nilpotent if there exists some positive integer, n, such that xn.
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Nilradical of a ring
In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements of the ring.
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Plücker embedding
In mathematics, the Plücker embedding is a method of realizing the Grassmannian Gr_k(V) of all k-dimensional subspaces of an n-dimensional vector space V as a subvariety of a projective space.
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Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
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Prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers.
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Projective line
In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity.
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Projective space
In mathematics, a projective space can be thought of as the set of lines through the origin of a vector space V. The cases when and are the real projective line and the real projective plane, respectively, where R denotes the field of real numbers, R2 denotes ordered pairs of real numbers, and R3 denotes ordered triplets of real numbers.
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Projective variety
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective ''n''-space Pn over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety.
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Quasi-projective variety
In mathematics, a quasi-projective 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.
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Quotient ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Riemann sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane, the complex plane plus a point at infinity.
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Ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers.
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Ringed space
In mathematics, a ringed space can be equivalently thought of as either Ringed spaces appear in analysis as well as complex algebraic geometry and scheme theory of algebraic geometry.
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Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x.
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Segre embedding
In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety.
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Semialgebraic set
In mathematics, a semialgebraic set is a subset S of Rn for some real closed field R (for example R could be the field of real numbers) defined by a finite sequence of polynomial equations (of the form P(x_1,...,x_n).
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Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
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Sheaf cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space.
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Singular point of an algebraic variety
In the mathematical field of algebraic geometry, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined.
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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
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Solution set
In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities.
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Spectrum of a ring
In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by \operatorname(R), is the set of all prime ideals of R. It is commonly augmented with the Zariski topology and with a structure sheaf, turning it into a locally ringed space.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Stack (mathematics)
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets.
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Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
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System of polynomial equations
A system of polynomial equations is a set of simultaneous equations f1.
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Tautological bundle
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: the fiber of the bundle over a vector space V (a point in the Grassmannian) is V itself.
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Twisted cubic
In mathematics, a twisted cubic is a smooth, rational curve C of degree three in projective 3-space P3.
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Unit circle
In mathematics, a unit circle is a circle with a radius of one.
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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.
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Variety
Variety may refer to.
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Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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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.
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Zariski topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology on algebraic varieties, introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring a topological space, called the spectrum of the ring.
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Zariski–Riemann space
In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring k of a field K is a locally ringed space whose points are valuation rings containing k and contained in K. They generalize the Riemann surface of a complex curve.
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Zero of a function
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
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Abstract algebraic variety, Abstract variety, Affine curve, Algebraic set, Algebraic subvariety, Algebraic varieties, Complex variety, Differential algebraic variety, Projective algebraic set, Subvarieties, Subvariety (mathematics).
References
[1] https://en.wikipedia.org/wiki/Algebraic_variety
