Intersect. Theory: Code cleanup, correction, new example

experimental/IntersectionTheory/docs/src/intro.md

@@ -6,12 +6,18 @@ CurrentModule = Oscar
 
 In this chapter, we
 - introduce OSCAR tools which support computations in intersection theory, and
-- give examples which illustrate how intersection theory is used to
-solve problems from enumerative geometry.
+- give examples which illustrate how intersection theory is used to solve problems from enumerative geometry.
 
 !!! note
-    Making use of OSCAR, a first version of what we present here was
-    written by Jeiao Song as a julia package.
+    Making use of OSCAR, a first version of what we present here was written by Jeiao Song as a julia package.
+    This package was "heavily inspired by the Macaulay2 package Schubert2 and the Sage library Chow. Some
+	functionalities from  [the Sage library] Schubert3 are also implemented."
+
+!!! note
+    Schubert2 was written by Daniel R. Grayson, Michael E. Stillman, Stein A. Strømme, David Eisenbud, and Charley Crissman
+    while Chow is due to Manfred Lehn and Christoph Sorger. Schubert3  as well as the Singular library schubert.lib were
+	written by Dang Tuan Hiep. The basis for all this work, including ours, is the Maple package Schubert written
+	by Sheldon Katz and Stein A. Strømme.
 
 Throughout the chapter, the varieties we consider are smooth projective varieties over the complex numbers.
 

experimental/IntersectionTheory/src/Main.jl

@@ -1295,7 +1295,7 @@ function schur_functor(F::AbstractBundle, λ::Partition)
   λ = conjugate(λ)
   X = F.parent
   w = _wedge(sum(λ), chern_character(F))
-  S, ei = polynomial_ring(QQ, "e#" => 1:length(w))
+  S, ei = polynomial_ring(QQ, ["e$i" for i in 1:length(w)])
   e = i -> i < 0 ? S() : ei[i+1]
   M = [e(λ[i]-i+j) for i in 1:length(λ), j in 1:length(λ)]
   sch = det(matrix(S, M)) # Jacobi-Trudi
@@ -1373,18 +1373,19 @@ Return an additive basis of the Chow ring of `X` in codimension `k`.
 basis(X::AbstractVariety, k::Int) = basis(X)[k+1]
 
 @doc raw"""
-    betti(X::AbstractVariety)
+    betti_numbers(X::AbstractVariety)
 
-Return the Betti numbers of the Chow ring of `X`. Note that these are not
-necessarily equal to the usual Betti numbers, i.e., the dimensions of
-(co)homologies.
+Return the Betti numbers of the Chow ring of `X`. 
+
+!!! note
+    The Betti number of `X` in a given degree is the number of elements of `basis(X)` in that degree.
 
 # Examples
 ```jldoctest
 julia> P2xP2 = abstract_projective_space(2, symbol = "k")*abstract_projective_space(2, symbol = "l")
 AbstractVariety of dim 4
 
-julia> betti(P2xP2)
+julia> betti_numbers(P2xP2)
 5-element Vector{Int64}:
  1
  2
@@ -1402,7 +1403,7 @@ julia> basis(P2xP2)
 
 ```
 """
-betti(X::AbstractVariety) = length.(basis(X))
+betti_numbers(X::AbstractVariety) = length.(basis(X))
 
 @doc raw"""
     integral(x::MPolyDecRingElem)