BigObject VersalComponent [polymake wiki]

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====== BigObject VersalComponent ======
//from application [[..:fulton|fulton]]//\\
\\
 A component of the versal deformation of a ''[[..:fulton:CyclicQuotient |CyclicQuotient]]'' singularity.

===== Properties =====

==== Algebraic Geometry ====
 Properties relevant for the algebro-geometric side of the components of  the versal deformation of CQS.
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{{anchor:total_space:}}
  ?  **''TOTAL_SPACE''**
  ::
    ? Type:
    :''[[..:ideal:Ideal |Ideal]]''


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==== Combinatorics ====
 These properties capture combinatorial information of the object.  Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.
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{{anchor:alpha_lattice:}}
  ?  **''ALPHA_LATTICE''**
  :: Lattice of alphas as in below article.
  > Jan Arthur Christophersen: On the components and discriminant of the versal base space of cyclic quotient singularities.
    ? Type:
    :''[[..:common#Map |Map]]<[[..:common#Vector |Vector]]<[[..:common#Integer |Integer]]>,[[..:common#Vector |Vector]]<[[..:common#Integer |Integer]]%%>>%%''


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{{anchor:continued_fraction:}}
  ?  **''CONTINUED_FRACTION''**
  :: The continued fraction equivalent to zero describing the versal component. See
  > Jan Arthur Christophersen: On the components and discriminant of the versal base space of cyclic quotient singularities.
  .. and
  > Jan Stevens: On the versal deformation of cyclic quotient singularities.
    ? Type:
    :''[[..:common#Vector |Vector]]<[[..:common#Integer |Integer]]>''


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{{anchor:cqs_matrix:}}
  ?  **''CQS_MATRIX''**
  :: The CQS matrix as described in Def 6.10 of the below article.
  > Theo de Jong, Duco van Straten: Deformation theory of sandwiched singularities.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Rational |Rational]],[[..:common#NonSymmetric |NonSymmetric]]>''


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{{anchor:dimension:}}
  ?  **''DIMENSION''**
  :: Dimension of this component, i.e., number of parameters.
    ? Type:
    :''[[..:common#Integer |Integer]]''


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{{anchor:n_gon_triangulation:}}
  ?  **''N_GON_TRIANGULATION''**
  :: Triangulation of the N-gon corresponding to this component. See
  > Jan Stevens: On the versal deformation of cyclic quotient singularities.
    ? Type:
    :''[[..:graph:Graph |Graph]]<[[..:common#Undirected |Undirected]]>''


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{{anchor:pyramid_lattice:}}
  ?  **''PYRAMID_LATTICE''**
  :: FIXME
    ? Type:
    :''[[..:common#Map |Map]]<[[..:common#Vector |Vector]]<[[..:common#Integer |Integer]]>,[[..:common#Vector |Vector]]<[[..:common#Integer |Integer]]%%>>%%''


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{{anchor:p_resolution:}}
  ?  **''P_RESOLUTION''**
  :: The P-resolution fan corresponding to this component. See
  > Klaus Altmann: P-Resolutions of Cyclic Quotients from the Toric Viewpoint.
    ? Type:
    :''[[..:fan:PolyhedralFan |PolyhedralFan]]<[[..:common#Rational |Rational]]>''


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{{anchor:triangles:}}
  ?  **''TRIANGLES''**
  :: The triangles in the N-gon triangulation corresponding to this component. See
  > Jan Stevens: On the versal deformation of cyclic quotient singularities.
    ? Type:
    :''[[..:common#Matrix |Matrix]]<[[..:common#Integer |Integer]],[[..:common#NonSymmetric |NonSymmetric]]>''


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