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{{short description|Theorem about orthocenter and polars in circle geometry}} '''Brokard's theorem''' is a theorem in [[projective geometry]].<ref>{{cite book ...

...etry]], '''Reider's theorem''' gives conditions for a [[line bundle]] on a projective surface to be [[very ample]]. ...Nef line bundle|nef]] [[divisor (algebraic geometry)|divisor]] on a smooth projective surface ''X''. Denote by ''K''_''X'' the [[canonical divisor]] of ...

...'''Castelnuovo's contraction theorem'''<!-- Who's Castelnuovo? --> is used in the [[Enriques–Kodaira classification|classification theory]] of [[algebrai ...a [[morphism of varieties|morphism]] from <math>X</math> to another smooth projective surface <math>Y</math> such that the curve <math>C</math> has been contract ...

...e [[invertible sheaf]] with [[Iitaka dimension]] at least 2 on a [[complex projective manifold]], then [[Category:Theorems in algebraic geometry]] ...

...nsions require characteristic 0.<ref>{{Springer|id=B/b015770|title=Bertini theorems}}</ref><ref>Hartshorne, Ch. III.10.</ref> ...lly closed field, embedded in a [[algebraic geometry of projective spaces|projective space]] <math>\mathbf P^n</math>. ...

{{short description|Collinearity of the midpoints of parallel chords in a conic}} ...a [[Conic section|conic]]. It states that the midpoints of parallel chords in a conic are located on a common line. ...

{{Short description|Criterion for vector stability in algebraic geometry}} ...ion for the [[Geometric invariant theory#Stability|stability]] of a vector in a [[group representation|representation]] of a complex [[reductive group]]. ...

...; thus, it is the analog for Möbius planes of [[Desargues' Theorem]] for [[projective plane]]s. ...<math>Q_{ij}:=\{A_i,B_i,A_j,B_j\}, \ i<j, </math> are concyclic (contained in a cycle) on at least four cycles <math>c_{ij}</math>, then the sixth quadru ...

...[[finitely generated module|finitely generated]] ''R''-module of finite [[projective dimension]], then: Here pd stands for the projective dimension of a module, and depth for the [[depth (ring theory)|depth]] of a ...

...link=Igor Shafarevich |first=I.R. |last=Shafarevich |title=Basic Algebraic Geometry |url=https://books.google.com/books?id=m6nwCAAAQBAJ |date=2012 |publisher=S In the case of a single equation, this problem is solved by the [[fundamental ...

{{short description|Surface in three-dimensional space}} In three-dimensional space, a '''regulus''' ''R'' is a set of [[skew lines]], ...

{{Short description|Theorem in algebraic geometry}} In [[algebraic geometry]], the '''Bogomolov–Sommese vanishing theorem''' is a result related to the ...

...tive space]] with the property that the line through any two of the points in the subset also passes through at least one other point of the subset. ...ne containing the pair and that every line contains at least three points. In this more general form they are also called '''Sylvester–Gallai designs'''. ...

{{Short description|Algebraic geometry theorem}} In [[algebraic geometry]], the '''Reiss relation''', introduced by {{harvs|txt|last=Reiss|authorlin ...

...the volume of the intersection of two geodesic balls |journal=Differential Geometry and Its Applications}}</ref> It is named after [[Americans|American]] [[mat ...|quaternionic hyperbolic space]]s <math>\mathbb{HH}^n</math>, the [[Cayley projective plane]] <math>\mathbb{C}ayP^2</math>, and the [[Cayley plane|Cayley hyperbo ...

...ngent, <math>s_1,...s_n</math> secants, <math>n</math> is the order of the projective plane (number of points on a line -1)]] In [[projective geometry]], '''Segre's theorem''', named after the Italian mathematician [[Beniamino ...

{{short description|Theorem in algebraic geometry that builds a homotopy equivalent affine variety}} ...Jouanolou's original statement of the theorem required that ''X'' be quasi-projective over an affine scheme, but this has since been considerably weakened. ...

...f a torus ''T'' is called a '''<nowiki />''T''-variety'''. In differential geometry, one considers an action of a real or complex torus on a manifold (or an [[ A [[normal scheme|normal]] algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a [[toric variety]] (for e ...

...</ref> and proved by [[Montserrat Teixidor i Bigas]] and [[Barbara Russo]] in 1999. ...th projective [[algebraic curve|curve]] of [[Genus (mathematics)#Algebraic geometry|genus]] greater or equal to 2. For [[generic property|generic]] vector bund ...

{{short description|Formula relating pairs of elements in a division ring}} {{for|Hua's identity in [[Jordan algebra]]s|Hua's identity (Jordan algebra)}} ...