Intersection theory: Correct one method for the function `abstract_fl… (

oscar-system#3998)

* Intersection theory: Correct one method for the function `abstract_flag_variety` and rename this method to function `abstract_flag_bundle`

docs/oscar_references.bib

@@ -1092,6 +1092,16 @@ @Article{GS79
   doi           = {10.1016/0166-218X(79)90004-0}
 }
 
+@Misc{GSS22,
+  author        = {Daniel R. Grayson and Alexandra Seceleanu and Michael E. Stillman},
+  title         = {Computations in intersection rings of flag bundles},
+  year          = {2022},
+  url           = {https://arxiv.org/abs/1205.4190},
+  eprint        = {1205.4190},
+  archiveprefix = {arXiv},
+  primaryclass  = {math.AG}
+}
+
 @InCollection{GTZ88,
   author        = {Gianni, Patrizia and Trager, Barry and Zacharias, Gail},
   title         = {Gröbner bases and primary decomposition of polynomial ideals},

experimental/IntersectionTheory/docs/src/AbstractVarieties.md

@@ -47,7 +47,7 @@ abstract_hirzebruch_surface(n::Int)
 ```
 
 ```@docs
-abstract_flag_variety(F::AbstractBundle, dims::Int...; symbol::String = "c")
+abstract_flag_bundle(F::AbstractBundle, dims::Int...; symbol::String = "c")
 ```
 
 ```@docs

experimental/IntersectionTheory/docs/src/intro.md

@@ -39,6 +39,10 @@ General textbooks offering details on theory and algorithms include:
 - [EH16](@cite)
 - [Ful98](@cite)
 
+For computations in the Chow rings of abstract flag bundles see 
+- [GSS22](@cite).
+
+
 ## Contact
 
 Please direct questions about this part of OSCAR to the following people:

experimental/IntersectionTheory/src/IntersectionTheory.jl

@@ -11,6 +11,7 @@ import ..Oscar: trivial_line_bundle
 
 export a_hat_genus
 export abstract_bundle
+export abstract_flag_bundle
 export abstract_flag_variety
 export abstract_grassmannian
 export abstract_hirzebruch_surface
@@ -89,6 +90,7 @@ using .IntersectionTheory
 
 export a_hat_genus
 export abstract_bundle
+export abstract_flag_bundle
 export abstract_flag_variety
 export abstract_grassmannian
 export abstract_hirzebruch_surface

experimental/IntersectionTheory/src/Main.jl

@@ -1816,6 +1816,39 @@ AbstractVariety of dim 2
 julia> degree(Z)
 4
 
+```
+
+```jldoctest
+julia> P = abstract_projective_space(4, symbol = "H")
+AbstractVariety of dim 4
+
+julia> F = exterior_power(cotangent_bundle(P), 3)*OO(P,3)
+AbstractBundle of rank 4 on AbstractVariety of dim 4
+
+julia> G = OO(P, 1)+4*OO(P)
+AbstractBundle of rank 5 on AbstractVariety of dim 4
+
+julia> Z = degeneracy_locus(F, G, 3) # rational surface in P4
+AbstractVariety of dim 2
+
+julia> K = canonical_class(Z)
+z - H
+
+julia> integral(K^2)
+-7
+
+julia> H = hyperplane_class(Z)
+H
+
+julia> integral(H^2) # degree of surface
+8
+
+julia> A = K+H
+z
+
+julia> integral(A^2) # degree of first adjoint surface which is a Del Pezzo surface in P5
+5
+
 ```
 """
 function degeneracy_locus(F::AbstractBundle, G::AbstractBundle, k::Int; class::Bool=false)
@@ -1830,12 +1863,12 @@ function degeneracy_locus(F::AbstractBundle, G::AbstractBundle, k::Int; class::B
       return F.parent.ring(0)
     end
   end
-  Gr = (m-k == 1) ? abstract_projective_bundle(F) : abstract_flag_variety(F, m-k)
+  Gr = (m-k == 1) ? abstract_projective_bundle(F) : abstract_flag_bundle(F, m-k)
   S = Gr.bundles[1]
   D = zero_locus_section(dual(S) * G)
   D.struct_map = hom(D, F.parent) # skip the flag abstract_variety
   if isdefined(F.parent, :O1)
-    D.O1 = pullback(D.struct_map, F.parent.O1) #TOCHECK other fields of D
+    D.O1 = pullback(D.struct_map, F.parent.O1)
   end
   set_attribute!(D, :description, "Degeneracy locus of rank $k from $F to $G")
   return D
@@ -1961,6 +1994,12 @@ Quotient
     h -> [1]
   by ideal (h^3, z^2 + 3*z*h + 3*h^2)
 
+julia> gens(PT)[1]
+z
+
+julia> gens(PT)[1] == hyperplane_class(PT)
+true
+
 julia> [chern_class(T, i) for i = 1:2]
 2-element Vector{MPolyQuoRingElem{MPolyDecRingElem{QQFieldElem, QQMPolyRingElem}}}:
  3*h
@@ -2007,7 +2046,7 @@ function abstract_projective_bundle(F::AbstractBundle; symbol::String = "z")
   pback = hom(PR, R1, gens(R1)[2:end])
   pfwd = hom(R1, R, pushfirst!(gens(R), R()))
 
-  # construct the ideal
+  # construct the relations
 
   rels = [sum(pback(chern_class(F, i).f) * z^(r-i) for i in 0:r)]
   if R isa MPolyQuoRing
@@ -2134,9 +2173,9 @@ end
     abstract_flag_variety(dims::Int...; base::Ring = QQ, symbol::String = "c")
     abstract_flag_variety(dims::Vector{Int}; base::Ring = QQ, symbol::String = "c")
 
-Given integers, say, $d_1, \dots, d_{k}, n$ or a vector of such integers, return the abstract 
-flag variety $\mathrm{F}(d_1, \dots, d_{k}; n)$ of nested sequences of subspaces of dimensions 
-$d_1; \dots, d_{k}$ of an $n$-dimensional vector space.
+Given integers, say, $d_1, \dots, d_{k}, n$ with $0 < d_1 < \dots < d_{k} < n$ or a vector of such integers, 
+return the abstract flag variety $\mathrm{F}(d_1, \dots, d_{k}; n)$ of nested sequences of subspaces of 
+dimensions $d_1, \dots, d_{k}$ of an $n$-dimensional vector space.
 
 # Examples
 ```jldoctest
@@ -2204,21 +2243,63 @@ function abs_flag(dims::Vector{Int}; base::Ring=QQ, symbol::String="c")
 end
 
 @doc raw"""
-    abstract_flag_variety(F::AbstractBundle, dims::Int...; symbol::String = "c")
-    abstract_flag_variety(F::AbstractBundle, dims::Vector{Int}; symbol::String = "c")
+    abstract_flag_bundle(F::AbstractBundle, dims::Int...; symbol::String = "c")
+    abstract_flag_bundle(F::AbstractBundle, dims::Vector{Int}; symbol::String = "c")
+
+Given integers, say, $d_1, \dots, d_{k}, n$ with $0 < d_1 < \dots < d_{k} < n$ or a vector of such integers, 
+and given an abstract bundle $F$ of rank $n$, return the abstract flag bundle of nested sequences 
+of subspaces of dimensions $d_1, \dots, d_{k}$ in the fibers of $F$.
+
+!!! note
+    Entering the number $n$ can be omitted since this number can be recovered as the rank of $F$.
+
+# Examples
+```jldoctest
+julia> P = abstract_projective_space(4)
+AbstractVariety of dim 4
+
+julia> F = exterior_power(cotangent_bundle(P),  3)*OO(P,3)
+AbstractBundle of rank 4 on AbstractVariety of dim 4
+
+julia> FB = abstract_flag_bundle(F, 1, 3)
+AbstractVariety of dim 9
+
+julia> CR = chow_ring(FB)
+Quotient
+  of multivariate polynomial ring in 5 variables over QQ graded by
+    c[1, 1] -> [1]
+    c[2, 1] -> [1]
+    c[2, 2] -> [2]
+    c[3, 1] -> [1]
+    h -> [1]
+  by ideal with 5 generators
+
+julia> modulus(CR)
+Ideal generated by
+  h^5
+  -c[1, 1]*c[2, 2]*c[3, 1] + h^4
+  -c[1, 1]*c[2, 1]*c[3, 1] - c[1, 1]*c[2, 2] - c[2, 2]*c[3, 1] - 2*h^3
+  -c[1, 1]*c[2, 1] - c[1, 1]*c[3, 1] - c[2, 1]*c[3, 1] - c[2, 2] + 4*h^2
+  -c[1, 1] - c[2, 1] - c[3, 1] - 3*h
+
+julia> FB.bundles
+3-element Vector{AbstractBundle}:
+ AbstractBundle of rank 1 on AbstractVariety of dim 9
+ AbstractBundle of rank 2 on AbstractVariety of dim 9
+ AbstractBundle of rank 1 on AbstractVariety of dim 9
 
-Given integers, say, $d_1, \dots, d_{k}$ or a vector of such integers, and given an
-abstract bundle $F$, return the abstract flag variety (flag bundle) of nested sequences 
-of subspaces of dimensions $d_1; \dots, d_{k}$ in the fibers of $F$.
+```
 """
-function abstract_flag_variety(F::AbstractBundle, dims::Int...; symbol::String = "c")
-  abstract_flag_variety(F, collect(dims), symbol=symbol)
+function abstract_flag_bundle(F::AbstractBundle, dims::Int...; symbol::String = "c")
+  abstract_flag_bundle(F, collect(dims), symbol=symbol)
 end
 
-function abstract_flag_variety(F::AbstractBundle, dims::Vector{Int}; symbol::String="c")
+function abstract_flag_bundle(F::AbstractBundle, dims::Vector{Int}; symbol::String = "c")
   X, n = F.parent, F.rank
   !(n isa Int) && error("expect rank to be an integer")
-  # compute the ranks and relative dim
+
+  # compute the ranks of successive subqotients and the relative dimension
+
   l = length(dims)
   ranks = pushfirst!([dims[i+1]-dims[i] for i in 1:l-1], dims[1])
   @assert all(r->r>0, ranks) && dims[end] <= n
@@ -2228,40 +2309,43 @@ function abstract_flag_variety(F::AbstractBundle, dims::Vector{Int}; symbol::Str
     l += 1
   end
   d = sum(ranks[i] * sum(dims[end]-dims[i]) for i in 1:l-1)
-  # FIXME ordering
+
   # construct the ring
+
   R = X.ring
   if R isa MPolyQuoRing
     PR = base_ring(R)
   else
     PR = R
   end
-  # syms = vcat([_parse_symbol(symbol, i, 1:r) for (i,r) in enumerate(ranks)]..., string.(gens(R.R)))
   syms = vcat([_parse_symbol(symbol, i, 1:r) for (i,r) in enumerate(ranks)]..., string.(gens(PR)))
-  # ord = prod(ordering_dp(r) for r in ranks) * ordering(X.ring.R.R)
   w = vcat([collect(1:r) for r in ranks]..., gradings(R))
   R1 = graded_polynomial_ring(X.base, syms, w)[1]
-  pback = hom(R, R1, gens(R1)[n+1:end])   # FIXME should always be well-defined
+  pback = hom(PR, R1, gens(R1)[n+1:end])
   pfwd = hom(R1, R, vcat(repeat([R()], n), gens(R)))
 
   # compute the relations
+
   c = pushfirst!([1+sum(gens(R1)[dims[i]+1:dims[i+1]]) for i in 1:l-1], 1+sum(gens(R1)[1:dims[1]]))
-  # XXX cannot mod using graded ring element
-  Rx, x = R1.R["x"]
-  fi = pback(total_chern_class(F))[0:n]
+  Rx, x = R1["x"]
+  fi = pback(total_chern_class(F).f)[0:n]
   f = sum(fi[i+1].f * x^(n-i) for i in 0:n)
   gi = prod(c)[0:n]
   g = sum(gi[i+1].f * x^(n-i) for i in 0:n)
   rels = [R1(coeff(mod(f, g), i)) for i in 0:n-1]
-  if R isa MPolyQuoRing rels = vcat(pback.(R.(gens(R.I))), rels) end
+  if R isa MPolyQuoRing
+    rels = vcat(pback.(gens(R.I)), rels)
+  end
   AFl = quo(R1, ideal(rels))[1]
   c = AFl.(c)
-  Fl = AbstractVariety(X.dim + d, AFl)
+
+  # construct the abstract_variety
 
+  Fl = AbstractVariety(X.dim + d, AFl)  
   Fl.bundles = [AbstractBundle(Fl, r, ci) for (r,ci) in zip(ranks, c)]
   section = prod(top_chern_class(E)^sum(dims[i]) for (i, E) in enumerate(Fl.bundles[2:end]))
   if isdefined(X, :point)
-    Fl.point = pback(X.point) * section
+    Fl.point = pback(X.point.f) * section
   end
   pˣ = Fl.(gens(R1)[n+1:end])
   pₓ = x -> (@warn("possibly wrong ans"); X(pfwd(div(simplify(x).f, simplify(section).f))))
@@ -2276,6 +2360,8 @@ function abstract_flag_variety(F::AbstractBundle, dims::Vector{Int}; symbol::Str
   Fl.struct_map = p
   set_attribute!(Fl, :description => "Relative flag abstract_variety Flag$(tuple(dims...)) for $F")
   set_attribute!(Fl, :section => section)
-  if l == 2 set_attribute!(Fl, :grassmannian => :relative) end
+  if l == 2
+     set_attribute!(Fl, :grassmannian => :relative)
+  end
   return Fl
 end

experimental/IntersectionTheory/test/runtests.jl

@@ -205,7 +205,7 @@ let pushforward = IntersectionTheory.pushforward
 
     # flag bundle
     X, (F,) = abstract_variety(2, [4=>"c"])
-    FlF = abstract_flag_variety(F, 2)
+    FlF = abstract_flag_bundle(F, 2)
     @test dim(FlF) == 6
     @test rank.(tautological_bundles(FlF)) == [2, 2]
     p = FlF.struct_map