Intersection theory: More info in docu (#5537)

* Intersection theory: More info in docu

Author: wdecker

docs/oscar_references.bib

@@ -2494,6 +2494,18 @@ @Article{Moe93
   doi           = {10.1007/BF01200146}
 }
 
+@InBook{Mur14,
+  author        = {Jacob Murre},
+  title         = {Chapter Nine. Lectures on Algebraic Cycles and Chow Groups},
+  booktitle     = {Hodge Theory},
+  address       = {Princeton},
+  publisher     = {Princeton University Press},
+  pages         = {410--448},
+  year          = {2014},
+  doi           = {10.1515/9781400851478.410},
+  lastchecked   = {2025-11-07}
+}
+
 @Article{Nik79,
   author        = {Nikulin, V. V.},
   title         = {Integer symmetric bilinear forms and some of their geometric applications},
@@ -2636,6 +2648,15 @@ @MastersThesis{Peg14
   school        = {University of Bremen}
 }
 
+@Article{Pfi07,
+  author        = {Pfister, Gerhard},
+  title         = {On modular computation of standard basis},
+  journal       = {Anal. Stiint. Univ. Ovidius Constanta},
+  volume        = {15},
+  pages         = {129--138},
+  year          = {2007}
+}
+
 @Article{Pol56,
   bibkey        = {Pol56},
   author        = {Pólya, G.},

experimental/IntersectionTheory/docs/src/intro.md

@@ -26,49 +26,67 @@ on a general quintic hypersurface in $\mathbb P^4$, see [ES02](@cite). We quote
 > One way to approach enumerative problems is to find a suitable complete parameter space for the objects that one wants to count, and express the locus of objects satisfying given conditions as a certain zero-cycle on the parameter space. For this method to yield an explicit numerical answer, one needs in particular to be able to evaluate the degree of a given zerodimensional cycle class. This is possible in principle whenever the numerical intersection ring (cycles modulo numerical equivalence) of the parameter space is known, say in terms of generators and relations.
 
 A cycle on a variety $X$ is a finite formal sum of subvarieties of $X$, with integer coefficients. That is, a *cycle* on $X$
-is an element of the free abelian group $Z(X)$ generated by the subvarieties of $X$. This group is graded by dimension:
-$Z_\ast(X) = \bigoplus^{\dim(X)}_{k=0} Z_k(X)$, where $Z_k(X)$ is generated by the cycles of dimension $k$ (called the
-*$k$-cycles* on $X$). Useful cycle theories are obtained by choosing an *adequate equivalence relation* on cycles
-which gives rise to a suitable concept of *moving cycles* and, thus, to a well-defined intersection product on cycle classes
-(see [Sam60](@cite) and the textbooks listed below). As a result, for each such relation $\sim$, the  group $Z_\ast(X)/\sim$
-formed by the cycle classes is turned into a ring (*intersection ring*). In what follows, we grade such a ring by the
-codimension of cycles (to indicate this, with respect to notation, we use an upper $*$ rather than the lower $*$
-used in $Z_\ast(X)$ above). Note that the degree-0  part of such a graded ring (cycle classes of dimension $\dim(X)$), is
-isomorphic to $\mathbb Z$, generated by the class $[X]$of $X$, the *fundamental class* of $X$ (recall that we assume
-that $X$ is irreducible).
+is an element of the free abelian group generated by the subvarieties of $X$. We consider this group with its grading by
+codimension, that is, we work with the graded group $Z^\ast(X) = \bigoplus^{\dim(X)}_{c=0} Z^c(X)$, where
+$Z^c(X)$ consists of the cycles of codimension $c$ on $X$.
 
-The finest adequate equivalence relation is *rational equivalence* which is the  most common relation to work with
-(see the textbooks listed below). The resulting intersection ring of $X$, denoted here by $A^\ast(X)$,  is usually called
-the *Chow ring* of $X$. Note that some of the graded parts of such a  ring may be huge and difficult to handle. For
-example, while $A^0(X)$ is isomorphic to $\mathbb Z$ as pointed out above, it may happen that the group $A^{\dim(X)}$
-is not even finitely generated. For the applications to enumerative geometry
-considered here, it is more convenient to follow the authors of `Schubert` and work with numerical equivalence,
-the coarsest adequate equivalence relation. This means that we only care about the intersection numbers with respect
-to classes in complementary codimension. In many cases of interest in enumerative geometry, this has the additional
-benefit that we can deduce pushforwards related to a morphisms of varieties when only the corresponding pullback
-homomorphism is known (see the section on [Abstract Variety Maps](@ref)).
+A useful cycle theory is obtained by considering an *adequate
+equivalence relation*, given on the cycle groups $Z^\ast(X)$ of all varieties $X$ (see [Sam60](@cite), [Mur14](@cite)). Such a
+relation $\sim$ is compatible with the group structure and  grading on $Z^\ast(X)$. It gives rise to a suitable concept
+of *moving cycles* and, thus, to a well-defined intersection product on cycle classes which makes the quotient group
+$C_{\sim}^\ast(X) := Z^\ast(X)/\sim$ into a graded ring (*intersection ring*). The construction of such rings is functorial
+with respect to morphisms of varieties in the sense that, given a morphism $f:X \rightarrow Y$, there are associated
+pushforward and pullback maps $f_{\ast}$ and $f^{\ast}$ for cycle classes (recall that we work with projective varieties).
+More precisely, for each $d$, building the group $C_d^{\sim}(X)$ of cycle classes of dimension $d$ gives rise to a covariant
+functor from varieties to groups via $f_{\ast}$, while building $C_{\sim}^\ast(X)$ gives rise to a contravariant functor from varieties
+to rings via $f^{\ast}$. Moreover, we have the *projection formula*
 
-For our purposes,  it is also convenient to allow rational coefficients when forming cycles.
-That is, if $N^\ast(X)$ is the numerical intersection ring of $X$, we consider the ring
+$f_\ast(f^\ast(\beta)\cdot \alpha) = f_\ast(\alpha)\cdot \beta \;\text{ for all }\; \alpha\in C_{\sim}^\ast(X), \beta\in C_{\sim}^\ast(Y)$
 
-$N^\ast(X)_{\mathbb Q} = N^\ast(X) \otimes_{\mathbb Z} {\mathbb Q} = \bigoplus^{\dim(X)}_{k=0} N^k(X)_{\mathbb Q}.$
+(here, we extend $f_{\ast}$ by additivity to all cycle classes on $Y$).
 
-The graded pieces $N^k(X)_{\mathbb Q}$ are finite-dimensional $\mathbb Q$-vector spaces. In particular, as
-$\mathbb Q$-vector spaces, both $N^{0}(X)_{\mathbb Q}$ and $N^{\dim(X)}(X)_{\mathbb Q}$ are isomorphic to
-$\mathbb Q$. In fact, we have a well-defined *degree homomorphism* , or *integral*,
 
-$\deg =\int_X : N^{\dim(X)}(X)_{\mathbb Q}  \overset{\cong}\longrightarrow \mathbb Q$
+Since we grade by codimension, the degree-0  part of an intersection ring consists of cycle classes of dimension
+$\dim(X)$. Note that this part is isomorphic to $\mathbb Z$: It is generated by the class of $X$ in $C_{\sim}^\ast(X)$,
+the *fundamental class* of $X$ with respect to $\sim$  (recall that we assume that $X$ is irreducible). On the other
+hand, the degree-$n$  part of an intersection ring consists of classes of 0-cycles, that is, cycles of dimension zero.
+We have a well-defined *degree homomorphism*, or *integral*,
 
-which sends the class of each point of $X$ to 1, and which extends to a homomorphism on all of $N^\ast(X)_{\mathbb Q}$:
+$\deg =\int_X : C_{\sim}^{\dim(X)}(X) \rightarrow \mathbb Z$
 
-$\int_X : N^\ast(X)_{\mathbb Q} \rightarrow \mathbb Q,  \quad c = \sum _{k = 0}^{\dim(X)} c_k\mapsto \int c_{\dim(X)}.$
+which sends the class of a point of $X$ to 1, and which extends to a homomorphism on all of $C_{\sim}^\ast(X)$ :
 
-The dimensions $\beta_k = \dim N^c(X)_{\mathbb Q}$ are called the *Betti-numbers* of $X$.
-Since the pairings
+$\int_X : C_{\sim}^\ast(X)\rightarrow \mathbb Z,  \quad \alpha = \sum _{c = 0}^{\dim(X)} \alpha_c\mapsto \int \alpha_{\dim(X)}.$
 
-$N^k(X)_{\mathbb Q}\times N^{\dim(X)-c}(X)_{\mathbb Q}\rightarrow N^{\dim(X)}(X)_{\mathbb Q} \overset{\cong}\longrightarrow \mathbb Q$
+In particular, we have the intersection pairings
+
+$C_{\sim}^c(X)\times C_{\sim}^{\dim(X)-c}(X)\rightarrow C_{\sim}^{\dim(X)}(X) \overset{\deg}\longrightarrow \mathbb Z.$
+
+The finest adequate equivalence relation is *rational equivalence* which is the  most common relation to work with.
+The resulting intersection ring of $X$ is usually denoted by $A^\ast(X)$ and called
+the *Chow ring* of $X$. Note that such rings are often difficult to handle. For example, while $A^0(X)$ is isomorphic
+to $\mathbb Z$ as pointed out above, it may happen that the group $A^{\dim(X)}$ is not even finitely generated. To
+be unaffected by such problems, we follow the authors of `Schubert` and work with numerical equivalence, the
+coarsest adequate equivalence relation. That is, we only care about the intersection numbers with respect to classes
+in complementary codimension. This is enough for handling most enumerative problems, with the additional benefit
+that we often can compute the pushforward of a cycle along a morphism $f$ of varieties when only the pullback
+homomorphism $ f^\ast$ is known (see [Abstract Variety Maps](@ref)).
+We will write $N^\ast(X)$ for the numerical intersection ring of $X$.
+
+In our implementation of numerical intersection rings, we allow rational coefficients when forming cycles.
+That is, given $X$, we consider the ring
+
+$N^\ast(X)_{\mathbb Q} = N^\ast(X) \otimes_{\mathbb Z} {\mathbb Q} = \bigoplus^{\dim(X)}_{c=0} N^c(X)_{\mathbb Q}$
+and extend notions such as integral or intersection pairing to this situation. The graded pieces $N^c(X)_{\mathbb Q}$ are
+then finite-dimensional $\mathbb Q$-vector spaces,  their dimensions $\beta_c(X) = \dim N^c(X)_{\mathbb Q}$ are
+called the *Betti-numbers* of $X$. Since the intersection pairings
+
+$N^c(X)_{\mathbb Q}\times N^{\dim(X)-c}(X)_{\mathbb Q}\rightarrow N^{\dim(X)}(X)_{\mathbb Q} \overset{\deg}\longrightarrow \mathbb Q$
+
+are nondegenerate by the very definition of numerical equivalence, we have $\beta_c(X) = \beta_{\dim(X)-c}(X)$ for each $c$.
+In particular, $N^0(X)_{\mathbb Q} $ and $N^{\dim(X)}_{\mathbb Q} $ are both 1-dimensional $\mathbb Q$-vector spaces, generated by the fundamental
+class $[X]$ and the class of a point (a class that integrates to 1),  respectively.
 
-are nondegenerate by the definition of numerical equivalence, we have $\beta_k = \beta_{\dim(X)-k}$ for each $k$.
 
 !!! note
     In this chapter, we abuse our notation in that the name *Chow ring* always refers to a ring of type $N^\ast(X)_{\mathbb Q}$.
@@ -85,7 +103,7 @@ the tangent bundle is represented as a collection of data referred to as an *abs
 The main data here is the Chern character of the vector bundle.
 
 In the same spirit, we introduce  *abstract variety maps*. Their key data is the pullback
-morphism between the respective Chow rings.
+homomorphism between the respective Chow rings.
 
 Some comments on this set-up are in order:
 
@@ -96,8 +114,7 @@ $ E^{\prime \prime}$ fit into an exact sequence
 $0\rightarrow E^\prime \rightarrow E \rightarrow E^{\prime \prime} \rightarrow 0$.
 The tensor product turns $K^{\circ}(X)$ into a ring,  the *Grothendieck ring of vector bundles on $X$*.
 By Riemann-Roch, the Chern character defines a ring isomorphism $\text{ch}: K^{\circ}(X)_{\mathbb Q}
-\overset{\simeq}\longrightarrow A^*(X) _{\mathbb Q}$, where $A^*(X)$ is obtained by considering cycles
-modulo rational equivalence. Since rational equivalence is (much) finer than
+\overset{\simeq}\longrightarrow A^*(X) _{\mathbb Q}$. Since rational equivalence is (much) finer than
 numerical equivalence, we obtain a ring epimorphism
 
 $\text{ch}: K^{\circ}(X)_{\mathbb Q}
@@ -111,23 +128,21 @@ $\text{ch}: K^{\circ}(X)_{\mathbb Q}
     vector bundles also to infer information on coherent sheaves, with algebraic operations such as `-` or `*` (tensor product)
     being virtual.
 
-!!! note
-    In many cases, we won't be able to compute the entire Chow ring. To put it in another way, in these cases, the general methods will only
-    yield information on a certain subring of the Chow ring generated by some tautological classes. Then, we are actually working
-    with the class of all varieties sharing the same piece of tautological ring (and, possibly, further data). We illustrate this in
-    [Example: Cubic surfaces](@ref).
-
-!!! note
-    For some nice varieties, numerical equivalence coincides with rational equivalence. For such a variety $X$, the Chow ring
-    coincides with the rational cohomology ring and can be completely computed, so problems as in the previous note disappear.
-	A nice consequence is that the Betti numbers of $X$ introduced above are exactly the (even) Betti numbers
-    of $X$ considered as a compact complex manifold, so we have an equality `sum(betti_numbers(X)) == euler_number(X)`.
-    The class of such varieties $X$ includes projective spaces, Grassmannians, homogeneous spaces for affine algebraic
-    groups (for example, flag varieties), and in general any variety with an affine stratification. Moreover, products,
-    projective bundles, flag bundles, and blowups with center in this class will remain in this class. As Eisenbud and Harris
-	[EH16](@cite) put it: *This class represents a tiny fraction of all varieties, but a large fraction of the set of varieties on which
-	we can effectively carry out intersection theory.* In OSCAR, internally, we use `set_attribute(X, :alg => true)` to declare that
-	$X$  satisfies this property. Accordingly, entering `get_attribute(X, :alg)` reveals whether `:alg` has been set to true or not.
+!!! warning
+    In many cases, there is no algorithm for computing the entire Chow ring: We will only be able to obtain information on a
+    certain subring generated by some tautological classes. We are then actually working with the class of all varieties sharing
+    the same piece of tautological ring (and, possibly, further data). We illustrate this in [Example: Cubic surfaces](@ref).
+
+For some nice varieties, however, numerical equivalence coincides with rational equivalence. For such a variety $X$, the Chow ring
+coincides with the rational cohomology ring and can be completely computed, so problems as above disappear.
+A nice consequence is that the Betti numbers of $X$ introduced above are exactly the (even) Betti numbers
+of $X$ considered as a compact complex manifold, so we have an equality `sum(betti_numbers(X)) == euler_number(X)`.
+The class of such varieties $X$ includes projective spaces, Grassmannians, homogeneous spaces for affine algebraic
+groups (for example, flag varieties), and in general any variety with an affine stratification. Moreover, products,
+projective bundles, flag bundles, and blowups with center in this class will remain in this class. As Eisenbud and Harris
+[EH16](@cite) put it: *This class represents a tiny fraction of all varieties, but a large fraction of the set of varieties on which
+we can effectively carry out intersection theory.* In OSCAR, internally, we use `set_attribute(X, :alg => true)` to declare that
+$X$  satisfies this property. Accordingly, entering `get_attribute(X, :alg)` reveals whether `:alg` has been set to true or not.
 
 General textbooks offering details on theory and algorithms include:
 - [EH16](@cite)

experimental/IntersectionTheory/src/Main.jl

@@ -767,10 +767,10 @@ end
 @doc raw"""
      abstract_variety(n::Int, A::Union{MPolyDecRing, MPolyQuoRing{<:MPolyDecRingElem}})
 
-Return an abstract variety by specifying its dimension `n` and Chow ring `A`.
+Return an abstract variety by specifying its dimension `n` and its Chow ring `A`.
 
 !!! note
-    We allow graded polynomial rings here since for the construction of a new abstract variety, the expert user may find it useful to start from the underlying graded polynomial ring of the Chow ring, and add its defining relations step by step.
+    We allow (graded) polynomial rings here since for the construction of a new abstract variety, the expert user may find it useful to start from the underlying graded polynomial ring of the Chow ring, and add its defining relations step by step.
 
 !!! note
     In addition to the dimension and the Chow ring, further data making up an abstract variety can be set. See the corresponding setter functions in the section [Some Particular Constructions](@ref) of the documentation.