Families of Spaces

family_of_spaces(coordinate_ring::MPolyRing, grading::Matrix{Int64}, dim::Int)

Return a family of spaces that is (currently) used to build families of F-theory models, defined by using a family of base spaces for an elliptic fibration. It is specified by the following data:

• A polynomial ring. This may be thought of as the coordinate ring of the generic member in this family of spaces.
• A grading for this polynomial ring. This may be thought of as specifying certain line bundles on the generic member in this family of spaces. Of particular interest for F-theory is always the canonical bundle. For this reason, the first row is always thought of as grading the coordinate ring with regard to the canonical bundle. Or put differently, if a variable shares the weight 1 in the first row, then this means that we should think of this coordinate as a section of 1 times the canonical bundle.
• An integer, which specifies the dimension of the generic member within this family of spaces.

Note that the coordinate ring can have strictly more variables than the dimension. This is a desired feature for most, if not all, F-theory literature constructions.

Examples

julia> coord_ring, _ = QQ[:f, :g, :Kbar, :u];

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> family_of_spaces(coord_ring, grading, 3)
Family of spaces of dimension d = 3

family_of_spaces(coordinate_ring::MPolyDecRing{QQFieldElem, QQMPolyRing}, dim::Int)

Return a family of spaces.

Examples

julia> coord_ring, _ = QQ[:f, :g, :Kbar, :u];

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> graded_coord_ring, _ = grade(coord_ring, grading);

julia> family_of_spaces(graded_coord_ring, 3)
Family of spaces of dimension d = 3

coordinate_ring(f::FamilyOfSpaces)

Return the coordinate ring of a generic member of the family of spaces.

Examples

julia> ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> f = family_of_spaces(ring, grading, 3)
Family of spaces of dimension d = 3

julia> coordinate_ring(f)
Multivariate polynomial ring in 4 variables over QQ graded by
  f -> [4 0]
  g -> [6 0]
  Kbar -> [1 0]
  u -> [0 1]

weights(f::FamilyOfSpaces)

Return the grading of the coordinate ring of a generic member of the family of spaces.

Examples

julia> ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> f = family_of_spaces(ring, grading, 3)
Family of spaces of dimension d = 3

julia> weights(f)
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

dim(f::FamilyOfSpaces)

Return the dimension of the generic member of the family of spaces.

Examples

julia> ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> f = family_of_spaces(ring, grading, 3)
Family of spaces of dimension d = 3

julia> dim(f)
3

irrelevant_ideal(f::FamilyOfSpaces)

Return the equivalent of the irrelevant ideal for the generic member of the family of spaces.

Examples

julia> coord_ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> f = family_of_spaces(coord_ring, grading, 3)
Family of spaces of dimension d = 3

julia> irrelevant_ideal(f)
Ideal generated by
  f
  g
  Kbar
  u

ideal_of_linear_relations(f::FamilyOfSpaces)

Return the equivalent of the ideal of linear relations for the generic member of the family of spaces.

Examples

julia> coord_ring, (f, g, Kbar, u) = QQ[:f, :g, :Kbar, :u]
(Multivariate polynomial ring in 4 variables over QQ, QQMPolyRingElem[f, g, Kbar, u])

julia> grading = [4 6 1 0; 0 0 0 1]
2×4 Matrix{Int64}:
 4  6  1  0
 0  0  0  1

julia> f = family_of_spaces(coord_ring, grading, 3)
Family of spaces of dimension d = 3

julia> ideal_of_linear_relations(f)
Ideal generated by
  -5*f + 3*g + 2*Kbar
  -3*f + 2*g