Glossary of algebraic geometry

This is a glossary of algebraic geometry.
See also <a href="/facts/Glossary_of_commutative_algebra/dPjY9Aqe">glossary of commutative algebra</a>, <a href="/facts/Glossary_of_classical_algebraic_geometry/ldUtJh1X">glossary of classical algebraic geometry</a>, and <a href="/facts/Glossary_of_ring_theory/eqENNbJ8">glossary of ring theory</a>. For the number-theoretic applications, see <a href="/facts/Glossary_of_arithmetic_and_Diophantine_geometry/MlmGW0Gm">glossary of arithmetic and Diophantine geometry</a>.
For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme S and a morphism an S-morphism.

η
      
    
    {\displaystyle \eta }

A <a href="/facts/Generic_point/3qaparob">generic point</a>. For example, the point associated to the zero ideal for any integral affine scheme.

F(n), F(D)
1.  If X is a projective scheme with <a href="/facts/Serre%27s_twisting_sheaf/gcvPmjpU">Serre's twisting sheaf</a> 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 1
 )
 
 
 {\displaystyle {\mathcal {O}}_{X}(1)}
 
 and if F is an 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-module, then 
 
 
 
 F
 (
 n
 )
 =
 F
 
 ⊗
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 n
 )
 .
 
 
 {\displaystyle F(n)=F\otimes _{{\mathcal {O}}_{X}}{\mathcal {O}}_{X}(n).}

2.  If D is a Cartier divisor and F is an 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
-module (X arbitrary), then 
 
 
 
 F
 (
 D
 )
 =
 F
 
 ⊗
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 D
 )
 .
 
 
 {\displaystyle F(D)=F\otimes _{{\mathcal {O}}_{X}}{\mathcal {O}}_{X}(D).}
 
 If D is a Weil divisor and F is reflexive, then one replaces F(D) by its reflexive hull (and calls the result still F(D)).

|D|
The <a href="/facts/Complete_linear_system/L8zleDlj">complete linear system</a> of a <a href="/facts/Weil_divisor/V0QDypeo">Weil divisor</a> D on a normal complete variety X over an algebraically closed field k; that is, 
 
 
 
 
 |
 
 D
 
 |
 
 =
 
 P
 
 (
 Γ
 (
 X
 ,
 
 
 
 O
 
 
 
 X
 
 
 (
 D
 )
 )
 )
 
 
 {\displaystyle |D|=\mathbf {P} (\Gamma (X,{\mathcal {O}}_{X}(D)))}
 
. There is a bijection between the set of k-rational points of |D| and the set of effective Weil divisors on X that are linearly equivalent to D.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> The same definition is used if D is a <a href="/facts/Cartier_divisor/V0QDypeo">Cartier divisor</a> on a complete variety over k.

[X/G]
The <a href="/facts/Quotient_stack/lwme8cqw">quotient stack</a> of, say, an algebraic space X by an action of a group scheme G.

X
        
          /
        
        
        
          /
        
        G
      
    
    {\displaystyle X/\!/G}

The <a href="/facts/GIT_quotient/BGS7ASbx">GIT quotient</a> of a scheme X by an <a href="/facts/Action_of_a_group_scheme/jVo9ubJf">action of a group scheme</a> G.

Ln
An ambiguous notation. It usually means an n-th tensor power of L but can also mean the self-intersection number of L. If 
 
 
 
 L
 =
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle L={\mathcal {O}}_{X}}
 
, the structure sheaf on X, then it means the direct sum of n copies of 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
.

O
            
          
          
            X
          
        
        (
        −
        1
        )
      
    
    {\displaystyle {\mathcal {O}}_{X}(-1)}

The <a href="/facts/Tautological_line_bundle/Vc2CmshX">tautological line bundle</a>. It is the dual of <a href="/facts/Serre%27s_twisting_sheaf/gcvPmjpU">Serre's twisting sheaf</a> 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 1
 )
 
 
 {\displaystyle {\mathcal {O}}_{X}(1)}
 
.

O
            
          
          
            X
          
        
        (
        1
        )
      
    
    {\displaystyle {\mathcal {O}}_{X}(1)}

<a href="/facts/Serre%27s_twisting_sheaf/gcvPmjpU">Serre's twisting sheaf</a>. It is the dual of the <a href="/facts/Tautological_line_bundle/Vc2CmshX">tautological line bundle</a> 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 −
 1
 )
 
 
 {\displaystyle {\mathcal {O}}_{X}(-1)}
 
. It is also called the hyperplane bundle.

O
            
          
          
            X
          
        
        (
        D
        )
      
    
    {\displaystyle {\mathcal {O}}_{X}(D)}

1.  If D is an <a href="/facts/Effective_Cartier_divisor/V0QDypeo">effective Cartier divisor</a> on X, then it is the inverse of the ideal sheaf of D.
2.  Most of the times, 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 D
 )
 
 
 {\displaystyle {\mathcal {O}}_{X}(D)}
 
 is the image of D under the natural group homomorphism from the group of Cartier divisors to the Picard group 
 
 
 
 Pic
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {Pic} (X)}
 
 of X, the group of isomorphism classes of line bundles on X.
3.  In general, 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 D
 )
 
 
 {\displaystyle {\mathcal {O}}_{X}(D)}
 
 is the sheaf corresponding to a <a href="/facts/Weil_divisor/V0QDypeo">Weil divisor</a> D (on a <a href="/facts/Normal_scheme/TQ6cB9By">normal scheme</a>). It need not be locally free, only <a href="/facts/Reflexive_sheaf/IvoIt2Gh">reflexive</a>.
4.  If D is a 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
-divisor, then 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 (
 D
 )
 
 
 {\displaystyle {\mathcal {O}}_{X}(D)}
 
 is 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
 of the integral part of D.

Ω
          
            X
          
          
            p
          
        
      
    
    {\displaystyle \Omega _{X}^{p}}

1.  
 
 
 
 
 Ω
 
 X
 
 
 1
 
 
 
 
 {\displaystyle \Omega _{X}^{1}}
 
 is the sheaf of <a href="/facts/K%C3%A4hler_differentials/1aPfygbf">Kähler differentials</a> on X.
2.  
 
 
 
 
 Ω
 
 X
 
 
 p
 
 
 
 
 {\displaystyle \Omega _{X}^{p}}
 
 is the p-th <a href="/facts/Exterior_power/rdN4TvpR">exterior power</a> of 
 
 
 
 
 Ω
 
 X
 
 
 1
 
 
 
 
 {\displaystyle \Omega _{X}^{1}}
 
.

Ω
          
            X
          
          
            p
          
        
        (
        log
        ⁡
        D
        )
      
    
    {\displaystyle \Omega _{X}^{p}(\log D)}

1.  If p is 1, this is the sheaf of <a href="/facts/Logarithmic_K%C3%A4hler_differential/6Hhf0KmM">logarithmic Kähler differentials</a> on X along D (roughly differential forms with simple poles along a divisor D.)
2.  
 
 
 
 
 Ω
 
 X
 
 
 p
 
 
 (
 log
 ⁡
 D
 )
 
 
 {\displaystyle \Omega _{X}^{p}(\log D)}
 
 is the p-th exterior power of 
 
 
 
 
 Ω
 
 X
 
 
 1
 
 
 (
 log
 ⁡
 D
 )
 
 
 {\displaystyle \Omega _{X}^{1}(\log D)}
 
.

P(V)
The notation is ambiguous. Its traditional meaning is the <a href="/facts/Projectivization/KB2shv7I">projectivization</a> of a finite-dimensional k-vector space V; i.e.,

P
        
        (
        V
        )
        =
        Proj
        ⁡
        (
        k
        [
        V
        ]
        )
        =
        Proj
        ⁡
        (
        Sym
        ⁡
        (
        
          V
          
            ∗
          
        
        )
        )
      
    
    {\displaystyle \mathbf {P} (V)=\operatorname {Proj} (k[V])=\operatorname {Proj} (\operatorname {Sym} (V^{*}))}

(the <a href="/facts/Proj_construction/gcvPmjpU">Proj</a> of the <a href="/facts/Ring_of_polynomial_functions/q99CvhPj">ring of polynomial functions</a> k[V]) and its k-points correspond to lines in V. In contrast, Hartshorne and EGA write P(V) for the Proj of the symmetric algebra of V.

Q-factorial
A normal variety is 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  
-factorial if every 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  
-Weil divisor is 
  
    
      
        
          Q
        
      
    
    {\displaystyle \mathbb {Q} }
  
-Cartier.

Spec(R)
The set of all prime ideals in a ring R with Zariski topology; it is called the <a href="/facts/Prime_spectrum/2k2nDE5w">prime spectrum</a> of R.

SpecX(F)
The <a href="/facts/Relative_Spec/2k2nDE5w">relative Spec</a> of the OX-algebra F. It is also denoted by Spec(F) or simply Spec(F).

Specan(R)
The set of all valuations for a ring R with a certain weak topology; it is called the <a href="/facts/Berkovich_spectrum/GKI96eyP">Berkovich spectrum</a> of R.

Glossary of algebraic geometry open-in-new

Glossary of algebraic geometry