Semistable abelian variety

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a semistable abelian variety is an <a href="/facts/Abelian_variety/dGSDVruG">abelian variety</a> defined over a <a href="/facts/Global_field/na458xWs">global</a> or <a href="/facts/Local_field/vqIgPIX3">local field</a>, which is characterized by how it reduces at the primes of the field.
For an abelian variety 
 
 
 
 A
 
 
 {\displaystyle A}
 
 defined over a field 
 
 
 
 F
 
 
 {\displaystyle F}
 
 with <a href="/facts/Ring_of_integers/KEzxxMbh">ring of integers</a> 
 
 
 
 R
 
 
 {\displaystyle R}
 
, consider the <a href="/facts/N%C3%A9ron_model/6x7RKoCe">Néron model</a> of 
 
 
 
 A
 
 
 {\displaystyle A}
 
, which is a 'best possible' model of 
 
 
 
 A
 
 
 {\displaystyle A}
 
 defined over 
 
 
 
 R
 
 
 {\displaystyle R}
 
. This model may be represented as a <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> over 
 
 
 
 
 S
 p
 e
 c
 
 (
 R
 )
 
 
 {\displaystyle \mathrm {Spec} (R)}
 
 (cf. <a href="/facts/Spectrum_of_a_ring/2k2nDE5w">spectrum of a ring</a>) for which the <a href="/facts/Generic_fibre/3qaparob">generic fibre</a> constructed by means of the <a href="/facts/Morphism/Kdpuyz3S">morphism</a>

S
          p
          e
          c
        
        (
        F
        )
        →
        
          S
          p
          e
          c
        
        (
        R
        )
      
    
    {\displaystyle \mathrm {Spec} (F)\to \mathrm {Spec} (R)}

gives back 
 
 
 
 A
 
 
 {\displaystyle A}
 
. The Néron model is a smooth <a href="/facts/Group_scheme/bJW14FE2">group scheme</a>, so we can consider 
 
 
 
 
 A
 
 0
 
 
 
 
 {\displaystyle A^{0}}
 
, the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model. For a <a href="/facts/Residue_field/e6nflEmB">residue field</a> 
 
 
 
 k
 
 
 {\displaystyle k}
 
, 
 
 
 
 
 A
 
 k
 
 
 0
 
 
 
 
 {\displaystyle A_{k}^{0}}
 
 is a <a href="/facts/Group_variety/DDtBCIvh">group variety</a> over 
 
 
 
 k
 
 
 {\displaystyle k}
 
, hence an extension of an abelian variety by a linear group. If this linear group is an <a href="/facts/Algebraic_torus/U5cLv3mU">algebraic torus</a>, so that 
 
 
 
 
 A
 
 k
 
 
 0
 
 
 
 
 {\displaystyle A_{k}^{0}}
 
 is a <a href="/facts/Semiabelian_variety/dGSDVruG">semiabelian variety</a>, then 
 
 
 
 A
 
 
 {\displaystyle A}
 
 has semistable reduction at the prime corresponding to 
 
 
 
 k
 
 
 {\displaystyle k}
 
. If 
 
 
 
 F
 
 
 {\displaystyle F}
 
 is a <a href="/facts/Global_field/na458xWs">global field</a>, then 
 
 
 
 A
 
 
 {\displaystyle A}
 
 is semistable if it has good or semistable reduction at all primes.
The fundamental <a href="/facts/Semistable_reduction_theorem/d6Tf9rtC">semistable reduction theorem</a> of <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a> states that an abelian variety acquires semistable reduction over a finite extension of 
 
 
 
 F
 
 
 {\displaystyle F}
 
.

Semistable abelian variety open-in-new

Semistable abelian variety