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Generalizations of codimension-1 subvarieties of algebraic varieties
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common
Divisor_(algebraic_geometry)
Concept in algebraic geometry
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_of_divisors
Relation between genus, degree, and dimension of function spaces over surfaces
important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic
Riemann–Roch_theorem
Topics referred to by the same term
divides evenly another integer Divisor (ring theory), a generalization of the preceding concept Divisor (algebraic geometry), a generalization of codimension
Divisor_(disambiguation)
Generalization of vector bundles
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric
Coherent_sheaf
Branch of algebraic geometry
arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around
Arithmetic_geometry
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety A over the complex numbers (and principally
Theta_divisor
Algebraic variety of dimension two
intrinsic geometry of algebraic surfaces is a central topic in algebraic geometry. The theory is much more complicated than for algebraic curves (one-dimensional
Algebraic_surface
V, and is called the Kodaira dimension. Hartshorne, Robin (1975). Algebraic Geometry, Arcata 1974. p. 7. Birkar, Caucher; Cascini, Paolo; Hacon, Christopher
Canonical_ring
Greatest common divisor of polynomials
In algebra, the greatest common divisor (frequently abbreviated GCD or gcd) of two polynomials is a polynomial, of the highest possible degree, which is
Polynomial greatest common divisor
Polynomial_greatest_common_divisor
Mathematical linear code
Algebraic geometry codes, often abbreviated AG codes, are a type of linear code that generalize Reed–Solomon codes. The Russian mathematician V. D. Goppa
Algebraic_geometry_code
Family of geometric objects with a common property
In geometry, a pencil is a family of geometric objects with a common property, for example the set of lines that pass through a given point in a plane
Pencil_(geometry)
theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory,
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry
Glossary of arithmetic and diophantine geometry
Glossary_of_arithmetic_and_diophantine_geometry
and Lazarsfeld (2009). In algebraic geometry, the multiplier ideal of an effective Q {\displaystyle \mathbb {Q} } -divisor measures singularities coming
Multiplier_ideal
Mathematical concept in algebraic geometry
the algebraic relation y 2 = x 5 + 1 {\displaystyle y^{2}=x^{5}+1} . Algebraic function field Cartier divisor Hartshorne, Robin (1977), Algebraic Geometry
Function field of an algebraic variety
Function_field_of_an_algebraic_variety
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
called divisors. The earliest work on algebraic cycles focused on the case of divisors, particularly divisors on algebraic curves. Divisors on algebraic curves
Algebraic_cycle
the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly
Hodge_index_theorem
space plays a central role in algebraic geometry. This article aims to define the notion in terms of abstract algebraic geometry and to describe some basic
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
Structure in algebraic geometry
In algebraic geometry, a motive (or sometimes motif, following French usage) is an abstract object introduced by Alexander Grothendieck in the 1960s as
Motive_(algebraic_geometry)
Commutative algebra studies commutative rings, their ideals, and modules over such rings
ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings
List of commutative algebra topics
List_of_commutative_algebra_topics
Mathematical theorem
if s > 0. Topological Methods in Algebraic Geometry by Friedrich Hirzebruch ISBN 3-540-58663-6 Zariski, Oscar (1995), Algebraic surfaces, Classics in Mathematics
Riemann–Roch theorem for surfaces
Riemann–Roch_theorem_for_surfaces
In mathematics, specifically algebraic geometry, an exceptional divisor for a regular map f : X → Y {\displaystyle f:X\rightarrow Y} of varieties is a
Exceptional_divisor
Concept in algebraic geometry
In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line
Nef_line_bundle
Skeletonized version of algebraic geometry
optimizing departure times for a network of trains. Tropical geometry is a variant of algebraic geometry in which polynomial graphs resemble piecewise linear
Tropical_geometry
Type of sheaf of modules
is the equivalent in algebraic geometry of the topological notion of a line bundle. Due to their interactions with Cartier divisors, they play a central
Invertible_sheaf
In algebraic geometry, given a pair (X, D) consisting of a normal variety X and a Q {\displaystyle \mathbb {Q} } -divisor D on X (e.g., canonical divisor)
Discrepancy (algebraic geometry)
Discrepancy_(algebraic_geometry)
Unsolved problem in geometry
unsolved problem in algebraic geometry and complex geometry that relates the algebraic topology of a non-singular complex algebraic variety to its subvarieties
Hodge_conjecture
special divisors is a result of William K. Clifford (1878) on algebraic curves, showing the constraints on special linear systems on a curve C. A divisor on
Clifford's theorem on special divisors
Clifford's_theorem_on_special_divisors
On the Euler characteristic of a holomorphic vector bundle on a compact complex manifold
is an algebraic surface (Noether's formula). Weil's Riemann–Roch theorem for vector bundles on curves, and the Riemann–Roch theorem for algebraic surfaces
Hirzebruch–Riemann–Roch theorem
Hirzebruch–Riemann–Roch_theorem
Concept in algebraic geometry
V {\displaystyle V} , and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor − K {\displaystyle K} with K {\displaystyle
Canonical_bundle
Singularities of algebraic varieties
In algebraic geometry, a normal crossing singularity looks locally like a union of coordinate hyperplanes. There are two variants of the concept, a divisor
Normal_crossing_singularity
Vector bundle existing over a Grassmannian
Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible
Tautological_bundle
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds
Picard_group
Set with operations obeying given axioms
In mathematics, an algebraic structure or algebraic system consists of a nonempty set A (called the underlying set, carrier set or domain), a collection
Algebraic_structure
Algebraic structure with addition and multiplication
influenced by problems and ideas of algebraic number theory and algebraic geometry. In turn, commutative algebra is a fundamental tool in these branches
Ring_(mathematics)
Study of complex manifolds and several complex variables
transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Complex geometry sits at the intersection
Complex_geometry
Effort to birationally classify algebraic varieties
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational
Minimal_model_program
Branch of number theory
Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields
Algebraic_number_theory
Concept in algebraic geometry
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others
Ample_line_bundle
Mathematical object studied in the field of algebraic geometry
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as
Algebraic_variety
In algebraic geometry, a complete algebraic curve is an algebraic curve that is complete as an algebraic variety. A projective curve, a dimension-one
Complete_algebraic_curve
Algebraic structure designed for geometry
number of geometries, including affine geometry, projective geometry, symplectic geometry, and orthogonal geometry. In physics, geometric algebras have been
Geometric_algebra
Group in algebraic geometry
In algebraic geometry, the Néron–Severi group of a variety is the group of divisors modulo algebraic equivalence; in other words it is the group of components
Néron–Severi_group
Projective variety that is also an algebraic group
particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is
Abelian_variety
French mathematician (1928–2014)
of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory
Alexander_Grothendieck
Surgery operation in minimal model program
In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative
Flip_(algebraic_geometry)
Branch of pure mathematics
(for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in
Number_theory
Natural number
a triangular number. 28 also appears in the Padovan sequence. In algebraic geometry, 28 is the number of bitangents to a general plane quartic curve.
28_(number)
In algebraic geometry, a logarithmic pair consists of a variety, together with a divisor along which one allows mild logarithmic singularities. They were
Logarithmic_pair
In algebraic geometry, a branch of mathematics, an adequate equivalence relation is an equivalence relation on algebraic cycles of smooth projective varieties
Adequate_equivalence_relation
Sort of mathematical expression
In algebra, an algebraic fraction is a fraction whose numerator and denominator are algebraic expressions. Two examples of algebraic fractions are 3 x
Algebraic_fraction
emergence of abstract algebra. This approach explored the axiomatic basis of arbitrary algebraic operations. The invention of new algebraic systems based on
History_of_algebra
of geometry. Fundamentally, it studies algebraic varieties. Algebraic graph theory a branch of graph theory in which methods are taken from algebra and
Glossary of areas of mathematics
Glossary_of_areas_of_mathematics
In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic
Theta_characteristic
In mathematics, especially in algebraic K-theory, the algebraic K-group of a field is important to compute. For a finite field, the complete calculation
K-groups_of_a_field
Generalized notion of counting curve intersections
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves
Intersection_number
algebraic geometry, a relative effective Cartier divisor is roughly a family of effective Cartier divisors. Precisely, an effective Cartier divisor in
Relative effective Cartier divisor
Relative_effective_Cartier_divisor
In algebraic geometry, a divisorial scheme is a scheme admitting an ample family of line bundles, as opposed to an ample line bundle. In particular, a
Divisorial_scheme
Meromorphic differential form
In algebraic geometry and the theory of complex manifolds, a logarithmic differential form is a differential form with poles of a certain kind. The concept
Logarithmic_form
Construction in algebraic geometry
effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map. In complex algebraic geometry, the Jacobian
Abel–Jacobi_map
Algebraic geometry theorem
In algebraic geometry, the theorem of Bertini is an existence and genericity theorem for smooth connected hyperplane sections for smooth projective varieties
Theorem_of_Bertini
Class of mathematical expression
In mathematics, division by zero, division where the divisor (denominator) is zero, is a problematic special case. Using fraction notation, the general
Division_by_zero
Field of mathematics using techniques from combinatorics and commutative algebra
Concini, David Eisenbud, and Claudio Procesi. Algebraic combinatorics Polyhedral combinatorics Zero-divisor graph A foundational paper on Stanley–Reisner
Combinatorial commutative algebra
Combinatorial_commutative_algebra
Commutative ring with no zero divisors other than zero
translates, in algebraic geometry, into the fact that the coordinate ring of an affine algebraic set is an integral domain if and only if the algebraic set is
Integral_domain
Curve defined as zeros of polynomials
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
Algebraic_curve
states that this is equivalent to a divisor in the class being the sum of an ample divisor and an effective divisor. Bombieri–Lang conjecture Lang, Serge
Pseudo-canonical_variety
Polynomial equation whose integer solutions are sought
equations define algebraic curves, algebraic surfaces, or, more generally, algebraic sets, their study is a part of algebraic geometry that is called Diophantine
Diophantine_equation
Generalization of algebraic variety
commutative algebra can be viewed as an algebraic approach to affine algebraic varieties. However, many arguments in algebraic geometry work better for
Scheme_(mathematics)
In algebraic geometry, a degeneration (or specialization) is the act of taking a limit of a family of varieties. Precisely, given a morphism π : X → C
Degeneration (algebraic geometry)
Degeneration_(algebraic_geometry)
Algebraic manipulation of "true" and "false"
connection between his algebra and logic was later put on firm ground in the setting of algebraic logic, which also studies the algebraic systems of many other
Boolean_algebra
Algebra with a graded anticommutativity property on multiplication
2 is not a zero divisor is alternating. Alternating multilinear map Exterior algebra Graded-symmetric algebra Supercommutative algebra Nicolas Bourbaki
Alternating_algebra
Mathematical theory
Atsushi; Yamaki, Kazuhiko (2002), "Introduction to Arakelov geometry", Algebraic geometry in East Asia (Kyoto, 2001), River Edge, NJ: World Sci. Publ
Arakelov_theory
Tool to track locally defined data attached to the open sets of a topological space
possibly singular algebraic varieties). 1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: schemes
Sheaf_(mathematics)
In algebraic geometry, convexity is a restrictive technical condition for algebraic varieties originally introduced to analyze Kontsevich moduli spaces
Convexity (algebraic geometry)
Convexity_(algebraic_geometry)
Mathematical construct in computer algebra
and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind
Gröbner_basis
Number divisible only by 1 and itself
important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the
Prime_number
Conjecture in algebraic geometry
specifically arithmetic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms
Tate_conjecture
Construction in algebraic geometry
is a construction in algebraic geometry considered by Solomon Lefschetz, used to analyse the algebraic topology of an algebraic variety V {\displaystyle
Lefschetz_pencil
In algebraic geometry, a log structure provides an abstract context to study semistable schemes, and in particular the notion of logarithmic differential
Log_structure
Mathematical treatise by Euclid
theorem, Thales' theorem, the Euclidean algorithm for greatest common divisors, Euclid's theorem that there are infinitely many prime numbers, and the
Euclid's_Elements
Soviet and Russian mathematician
relation between algebraic geometry and codes, utilizing the Riemann-Roch theorem. Today these codes are called algebraic geometry codes. In 1981 he
Valery_Goppa
German mathematician
field of research was commutative algebra, and analytic and algebraic geometry, in particular derivations, divisor class group, and resultants. Storch
Uwe_Storch
Type of geometric transformation
along a smooth divisor. Infinitely near point – Concept in algebraic geometry Resolution of singularities – Concept in algebraic geometry Fulton, William
Blowing_up
Generalization of a vector bundle
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme X, the relative Spec C = Spec X R {\displaystyle C=\operatorname
Cone_(algebraic_geometry)
of general type Zariski surface Algebraic variety Hypersurface Quadric (algebraic geometry) Dimension of an algebraic variety Hilbert's Nullstellensatz
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Algebraic variety in a projective space
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in
Projective_variety
Concept in algebraic geometry
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety V has a resolution, which is a non-singular variety
Resolution_of_singularities
Group of Italian mathematicians who studied birational geometry (c. 1885–1935)
detailed geometry of the theta-divisor). The classification of algebraic surfaces was a bold and successful attempt to repeat the division of algebraic curves
Italian school of algebraic geometry
Italian_school_of_algebraic_geometry
Mathematical concept
of polynomials, and is thus a projective algebraic variety. See (Griffiths & Harris 1994) In algebraic geometry, complex projective space can be equipped
Complex_projective_space
(algebraic surfaces) Proper base change theorem (algebraic geometry) Puiseux's theorem (algebraic geometry) Ramanujam vanishing theorem (algebraic geometry)
List_of_theorems
About polynomials in several variables
publicized by Abhyankar as an example of a difficult question in algebraic geometry that can be understood using little beyond a knowledge of calculus
Jacobian_conjecture
Geometric concept of a 2D space with "points at infinity" adjoined
example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP2
Projective_plane
Sheaf cohomology on the étale site
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients
Étale_cohomology
In algebraic geometry, a closed immersion i : X ↪ Y {\displaystyle i:X\hookrightarrow Y} of schemes is a regular embedding of codimension r if each point
Regular_embedding
Branch of algebraic geometry concerned with counting solutions
In mathematics, enumerative geometry is the branch of algebraic geometry concerned with counting numbers of solutions to geometric questions, mainly by
Enumerative_geometry
The terminology of algebraic geometry changed drastically during the twentieth century, with the introduction of the general methods, initiated by David
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Number of intersection points of algebraic curves and hypersurfaces
In algebraic geometry, Bézout's theorem is a statement concerning the number of common zeros of n polynomials in n indeterminates. In its original form
Bézout's_theorem
Algebraic structure
integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples
Principal_ideal_domain
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DIVISOR ALGEBRAIC-GEOMETRY
DIVISOR ALGEBRAIC-GEOMETRY
DIVISOR ALGEBRAIC-GEOMETRY
DIVISOR ALGEBRAIC-GEOMETRY
DIVISOR ALGEBRAIC-GEOMETRY
DIVISOR ALGEBRAIC-GEOMETRY
DIVISOR ALGEBRAIC-GEOMETRY
DIVISOR ALGEBRAIC-GEOMETRY
DIVISOR ALGEBRAIC-GEOMETRY
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