CurrentModule = Oscar
Algebraic cycles are formal linear sum of irreducible subvarieies over the integers. Perse, algebraic cycles do not admit a well-defined intersection product.
To see this, think of intersecting a non-trivial algebraic
cycle C with itself. Of course, in set theory we can
intersect C with itself and the result is again C.
However, for a well-defined intersection theory, we would
ask that the self-intersection of C is an algebraic
cycle of strictly smaller dimension.
In theory, this is resolved by saying that the
self-intersection of C is given by intersecting C with
a distinct algebraic cycle D which is obtained by moving
C a little bit. The general phrase for this is to "move
C in general position".
This leads to a famous notion of equivalence among algebraic cycles, the so-called rational equivalence. The set of equivalence classes of algebraic cycles together with the intersection product then furnishes the Chow ring of the variety in question.
For complete and simplicial toric varieties, many things are known about the Chow ring and algebraic cycles (cf. section 12.5 in CLS11:
- By therorem 12.5.3 of CLS11, there is an isomorphism among the Chow ring and the cohomology ring. Note that the cohomology ring is naturally graded (cf. last paragraph on page 593 in CLS11). However, the Chow ring is usually considered as a non-graded ring. To match this general convention, and in particular the implementation of the Chow ring for matroids in OSCAR, the toric Chow ring is constructed as a non-graded ring.
- By therorem 12.5.3 of CLS11, the Chow ring is isomorphic to the quotient of the non-graded Cox ring and a certain ideal. Specifically, the ideal in question is the sum of the ideal of linear relations and the Stanley-Reisner ideal.
- It is worth noting that the ideal of linear relations is not
homogeneous with respect to the class group grading of the Cox ring.
In order to construct the cohomology ring, one can introduce a
$\mathbb{Z}$ -grading on the Cox ring such that the ideal of linear relations and the Stanley-Reißner ideal are homogeneous. - Finally, by lemma 12.5.1 of CLS11, generators of the rational equivalence classes of algebraic cycles are one-to-one to the cones in the fan of the toric variety.
rational_equivalence_class(v::NormalToricVarietyType, p::MPolyQuoRingElem)
rational_equivalence_class(v::NormalToricVarietyType, coefficients::Vector{T}) where {T <: IntegerUnion}
rational_equivalence_class(d::ToricDivisor)
rational_equivalence_class(c::ToricDivisorClass)
rational_equivalence_class(l::ToricLineBundle)
rational_equivalence_class(cc::CohomologyClass)
rational_equivalence_class(sv::ClosedSubvarietyOfToricVariety)
Algebraic cycles can be added and subtracted via the usual + and -
operators. Moreover, multiplication by scalars from the left is supported
for scalars which are integers or of type ZZRingElem.
Note that one can easily define the Chow ring also a formal linear sums of
irreducible subvarieties with coefficients being rational numbers. We
support this more general ring and therefore also allow for left
multiplication with scalars of type QQFieldElem.
The intersection product of algebraic cycles is implemented via *.
This makes sense, since algebraic cycles on toric varieties are
elements of the Chow ring, which in turn is (a certain) quotient of
the Cox ring. Hence, internally, an algebraic cycle can be thought
of as a polynomial in this ring and the intersection product
corresponds to the product of two (equivalence classes of) polynomials.
An algebraic cycle can be intersected n- with itself via ^n,
where n can be an integer of of type ZZRingElem.
A closed subvarieties defines in a natural way a rational equivalence class (cf. [Special constructors](@ref toric_special_constructors)). This allows to compute intersection products among closed subvarieties and rational equivalence classes in the Chow ring.
toric_variety(ac::RationalEquivalenceClass)
polynomial(ac::RationalEquivalenceClass)
polynomial(ring::MPolyQuoRing, ac::RationalEquivalenceClass)
In order to see a geometric interpretation of rational equivalence classes of algebraic cycles most efficiently, it is best to replace self-intersections by transverse complete intersections. Indeed, within the regime of simplicial, complete toric varieties this is always possible. However, this involves a choice. Consequently, the following methods will pick a special choice and return values for that particular choice of representative of the rational equivalence class in question.
representative(ac::RationalEquivalenceClass)
It can be rather convenient to investigate such a representative in order to understand the geometric meaning of a rational equivalence class. For this purpose, we support the following methods.
coefficients(ac::RationalEquivalenceClass)
components(ac::RationalEquivalenceClass)
cohomology_class(ac::RationalEquivalenceClass)
One can check if a rational equivalence class of algebraic cycles
is trivial via is_trivial. Equality can be tested with ==.
chow_ring(v::NormalToricVarietyType)
gens_of_rational_equivalence_classes(v::NormalToricVarietyType)
map_gens_of_chow_ring_to_cox_ring(v::NormalToricVarietyType)