Intersect. Theory: Code cleanup, correction, new example (#4180)

* Intersect. Theory: Code cleanup, correction, new example

Co-authored-by: Lars Göttgens <lars.goettgens@rwth-aachen.de>

docs/src/AlgebraicGeometry/Surfaces/AdjunctionProcess.md

@@ -40,8 +40,20 @@ exceptional cases. In particular, if $X$ has non-negative Kodaira dimension, th
 !!! note
  The surfaces in the examples below are taken from the OSCAR data base of nongeneral type surfaces in $\mathbb P^4$. To ease subsequent computations, the surfaces in the data base where constructed over finite fields. Note, however, that the recipes used in the constructions also work in characteristic zero. So all computations can be confirmed in characteristic zero, although this may be time consuming.
 
+## Adjunction Process
+
+What we describe here goes back to joint work of Wolfram Decker and Frank-Olaf Schreyer. See [DES93](@cite), [DS00](@cite).
+
 ```@docs
 adjunction_process(X::AbsProjectiveVariety, steps::Int=0)
 ```
 
+## Contact
+
+Please direct questions about this part of OSCAR to the following people:
+* [Wolfram Decker](https://math.rptu.de/en/wgs/agag/people/head/decker).
+
+You can ask questions in the [OSCAR Slack](https://www.oscar-system.org/community/#slack).
+
+Alternatively, you can [raise an issue on github](https://www.oscar-system.org/community/#how-to-report-issues).
 

docs/src/AlgebraicGeometry/Surfaces/ParametrizationSurfaces.md

@@ -4,6 +4,19 @@ CurrentModule = Oscar
 
 # Rational Parametrization of Rational Surfaces
 
+What we present here relies on the function `adjunction_process` discussed in the previous section.
+
+## Parametrization
+
 ```@docs
 parametrization(X::AbsProjectiveVariety)
 ```
+
+## Contact
+
+Please direct questions about this part of OSCAR to the following people:
+* [Wolfram Decker](https://math.rptu.de/en/wgs/agag/people/head/decker).
+
+You can ask questions in the [OSCAR Slack](https://www.oscar-system.org/community/#slack).
+
+Alternatively, you can [raise an issue on github](https://www.oscar-system.org/community/#how-to-report-issues).

docs/src/InvariantTheory/intro.md

@@ -5,7 +5,7 @@ CurrentModule = Oscar
 # [Introduction](@id invariant_theory)
 
 The invariant theory part of OSCAR provides functionality for computing polynomial invariants
-of group actions, focusing on finite and linearly reductive groups, respectively.
+of group actions, focusing on finite groups, tori, and linearly reductive groups, respectively.
 
 The basic setting in this context consists of a group $G$, a field $K$, a vector space
 $V$ over $K$ of finite dimension $n,$ and a representation $\rho: G \to \text{GL}(V)$ of $G$ on $V$.

experimental/IntersectionTheory/docs/src/intro.md

@@ -6,20 +6,26 @@ 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.
 
 !!! note
  The Chow ring of a variety `X` is the group of cycles of `X` modulo an equivalence relation,
  together with the intersection pairing which defines the multiplication of the ring. Here,
  in contrast to most textbooks, we consider numerical equivalence classes of cycles rather than
- rational equivalence classes.
+ rational equivalence classes.
 
 Our approach is abstract in the sense that we do not work with concrete varieties; that is,
 our varieties are not given by equations. Instead, we represent a variety by specifying its

experimental/IntersectionTheory/src/IntersectionTheory.jl

@@ -3,7 +3,7 @@ using ..Oscar
 
 import Base: +, -, *, ^, ==, div, zero, one, parent
 import ..Oscar: AffAlgHom, Ring, MPolyDecRingElem, symmetric_power, exterior_power, pullback, canonical_bundle, graph, euler_characteristic, pullback
-import ..Oscar: basis, betti, chow_ring, codomain, degree, det, dim, domain, dual, gens, hilbert_polynomial, hom, integral, rank, signature, partitions
+import ..Oscar: basis, betti_numbers, chow_ring, codomain, degree, det, dim, domain, dual, gens, hilbert_polynomial, hom, integral, rank, signature, partitions
 import ..Oscar.AbstractAlgebra: combinations
 import ..Oscar.AbstractAlgebra.Generic: FunctionalMap
 import ..Oscar: pullback, pushforward, base, OO, product, compose
@@ -22,7 +22,7 @@ export abstract_projective_bundle
 export abstract_projective_space
 export abstract_variety
 export base
-export betti
+export betti_numbers
 export blowup
 export blowup_points
 export bundles
@@ -100,7 +100,7 @@ export abstract_projective_bundle
 export abstract_projective_space
 export abstract_variety
 export base
-export betti
+export betti_numbers
 export blowup
 export blowup_points
 export bundles

experimental/IntersectionTheory/src/Main.jl

@@ -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)