Bott's Formula

tn_variety(n::Int, points::Vector{Pair{P, Int}}) where P

Return an abstract_variety with a torus action, represented by the fixed points.

tn_grassmannian(k::Int, n::Int; weights = :int)

Return a Grassmannian with a torus action, ... .

tn_flag_variety(dims::Int...; weights = :int)

Return a flag variety with a torus action, ... .

 dim(X::TnVariety)

Return the dimension of X.

Examples

julia> G = tn_grassmannian(2, 5);

julia> dim(G)
6

 fixed_points(X::TnVariety)

Return the fixed points representing X.

Examples

julia> G = tn_grassmannian(2, 5);

julia> fixed_points(G)
10-element Vector{Pair{Vector{Int64}, Int64}}:
 [1, 2] => 1
 [1, 3] => 1
 [2, 3] => 1
 [1, 4] => 1
 [2, 4] => 1
 [3, 4] => 1
 [1, 5] => 1
 [2, 5] => 1
 [3, 5] => 1
 [4, 5] => 1

 tangent_bundle(X::TnVariety)

Return the tangent bundle of X.

Examples

julia> G = tn_grassmannian(2, 5);

julia> tangent_bundle(G)
TnBundle of rank 6 on TnVariety of dim 6

 tautological_bundles(X::TnVariety)

If X has been given tautological bundles, return these bundles.

Examples

julia> G = tn_grassmannian(2, 5);

julia> tautological_bundles(G)
2-element Vector{TnBundle}:
 TnBundle of rank 2 on TnVariety of dim 6
 TnBundle of rank 3 on TnVariety of dim 6

linear_subspaces_on_hypersurface(k::Int, d::Int; bott::Bool = true)

If $n=\frac1{k+1}\binom{d+k}d+k$ is an integer, return the number of $k$-dimensional subspaces on a generic hypersurface of degree $d$ in a projective space of dimension $n$.

lines_on_hypersurface(n::Int; bott::Bool = true)

Return the number of lines on a hypersurface of degree $d = 2*n-3$ in a projective space of dimension $n$.

Examples

julia> linear_subspaces_on_hypersurface(1,3)
27

julia> linear_subspaces_on_hypersurface(2,5)
420760566875

julia> [lines_on_hypersurface(n) for n=2:10]
9-element Vector{QQFieldElem}:
 1
 27
 2875
 698005
 305093061
 210480374951
 210776836330775
 289139638632755625
 520764738758073845321