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The Cox ring of a toric variety X is a polynomial ring whose variables
are in bijection with the 1-skeleton of the fan of X together with
three sequences of additional data:
- (1)
- a sequence of the components of an ideal called the irrelevant ideal;
- (2)
- a sequence of integral lattice points determining weights of
Gm actions, called the Z weights;
- (3)
- a sequence of rational lattice points determining weights of finite
cyclic group actions, called the quotient weights.
When the Cox ring of a toric variety is displayed in Magma,
all nontrivial data is also printed, but any sequences that
are empty are omitted.
Cox rings provide a powerful way to construct toric varieties: under
some mild conditions, specification of data of this nature
determines a toric variety.
Cox [Cox95] associates a ring, now called the Cox ring,
to a toric variety, and Magma allows exactly the same construction.
The Cox ring of the toric variety X.
The Cox ring of the toric variety defined over field k by the fan F.
We build the weighted projective space P 2(1, 2, 3).
> P<x,y,z> := ProjectiveSpace(Rationals(),[1,2,3]);
The Cox ring of P 2(1, 2, 3) is the usual homogeneous coordinate ring,
graded by the weights 1, 2, 3 of the space---that is, x has weight 1,
y has weight 2 and z has weight 3.
> CoxRing(P);
Cox ring with underlying Graded Polynomial ring of rank 3 over Rational Field
Order: Graded Reverse Lexicographical
Variables: x, y, z
Variable weights: [1, 2, 3]
The irrelevant ideal is:
(x, y, z)
The grading is:
1, 2, 3
The irrelevant ideal is the usual one for projective spaces: it
decrees that (0, 0, 0) is not a point of P 2 since it lies
in the locus defined by the irrelevant ideal.
We build a toric variety X 2 whose fan resembles that of P 2.
> F2 := Fan([[1,2],[-2,-1],[1,-1]], [[1,2],[1,3],[2,3]]);
> X2<u,v,w> := ToricVariety(Rationals(),F2);
However, X is not isomorphic to P 2. The small catch is that
the 1-skeleton of the fan F 2 that we defined (in other words, those
three vectors (1, 2), ( - 2, - 1) and (1, - 1)) does not generate
the lattice N, but only a sublattice.
So X will be the quotient of P 2 by some finite group action.
> CoxRing(X2);
Cox ring with underlying Polynomial ring of rank 3 over Rational Field
Order: Lexicographical
Variables: u, v, w
The irrelevant ideal is:
(w, v, u)
The quotient grading is:
1/3( 0, 2, 1 )
The integer grading is:
1, 1, 1
The returned data are very similar to those for the Cox ring of P 2.
The difference is in the third piece of data: a sequence
containing the single element 1/3(0, 2, 1).
This indicates that X is the quotient of P 2 by the
action of Z/3 given by
ε: (u, v, w) |-> (u, ε2 v, ε w).
where ε is a cube-root of unity.
The introduction to this section describes Cox rings in abstract
terms. It is possible to define them as polynomial rings plus
additional data without naming a toric variety or a fan in the
first place.
The Cox ring with polynomial ring R of rank n (that is, having
n variables) and additional data as follows:
B is a sequence of ideals (or of sequences of elements of R,
each of which will be interpreted as the generators of ideals);
Z is a sequence of sequences of integers, each one of length n;
Q is a sequence of sequences of rationals, each one of length n.
The sequence B is regarded as the components of the irrelevant
ideal, and Z and Q are the Z weights and quotient weights
respectively.
Return true if and only if the two Cox rings C1 and C2 have
the same underlying polynomial rings and are
defined by the same combinatorial data.
CoefficientRing(C) : RngCox -> Fld
The coefficient field of the Cox ring C.
The underlying polynomial ring of the Cox ring C.
The rank of the underlying polynomial ring of the Cox ring C,
that is, the number of polynomial variables of C.
The irrelevant ideal of the Cox ring C.
A sequence containing the components of the irrelevant ideal of
the Cox ring C.
A sequence of sequences, each containing the generators of the
components of the irrelevant ideal of the Cox ring C.
The Z gradings of the Cox ring C, that is, a sequence
of sequences of integers.
The number of Z gradings of the Cox ring C.
The quotient gradings of the Cox ring C, that is, a sequence of
sequences of rational numbers.
The number of quotient gradings of the Cox ring C.
The i-th indeterminate for the underlying polynomial ring
of the Cox ring C.
Procedure to change the printed names of the indeterminates of
the Cox ring C. The ith indeterminate will be given name
the ith element of the sequence S of strings (which has
length at most the number of indeterminates of C).
This does not change the names of the indeterminates for
calling---this must be done with an explicit assignment or
with the angle bracket notation when defining the Cox ring
in the first place.
The ith variable of the underlying polynomial ring of the Cox
ring C.
It is simple either to recover a toric variety from a
Cox ring (if one exists at all) or the fan and associated
lattice machinery corresponding to the toric variety.
The algorithm is straightforward: use the Z weights
to determine the rays of a fan and then the irrelevant
ideal to construct the rest of the cone structure of the
fan. The quotient weights may then require the lattice
to be extended to a larger lattice but containing the
same fan.
With this point of view, the Cox ring can be regarded
as the primary collection of data of a toric variety.
The toric variety whose Cox ring is C. It is not checked whether the Cox
data defines a toric variety; if you are unsure, you should ask for the fan.
Sometimes, rather than defining a fan, it is easier to construct
a Cox ring first and build a toric variety from that.
> R<x1,x2,x3,y1,y2,y3,y4> := PolynomialRing(Rationals(),7);
> I := [ ideal<R|x1,x2,x3>, ideal<R|y1,y2,y3,y4> ];
> Z := [ [1,1,1,0,-3,-5,-5], [0,0,0,1,1,1,1] ];
> Q := [];
> C := CoxRing(R,I,Z,Q);
> C;
Cox ring C with underlying Polynomial ring of rank 7 over Rational Field
Order: Lexicographical
Variables: x1, x2, x3, y1, y2, y3, y4
The components of the irrelevant ideal are:
(y4, y3, y2, y1), (x3, x2, x1)
The 2 gradings are:
1, 1, 1, 0, -3, -5, -5,
0, 0, 0, 1, 1, 1, 1
> X := ToricVariety(C);
> X;
Toric variety of dimension 5
Variables: x1, x2, x3, y1, y2, y3, y4
The components of the irrelevant ideal are:
(y4, y3, y2, y1), (x3, x2, x1)
The 2 gradings are:
1, 1, 1, 0, -3, -5, -5,
0, 0, 0, 1, 1, 1, 1
Now Magma can compute the fan of X if we really want it.
> Fan(X);
Fan with 7 rays:
( 1, 0, 0, 0, 0),
( 0, 1, 0, 0, 0),
( 0, 0, 1, 0, 0),
( 1, 1, 1, 2, 0),
(-3, -3, -3, -5, 0),
( 0, 0, 0, 0, 1),
( 2, 2, 2, 3, -1)
and 12 cones
The fan associated to the Cox ring is C; an error is reported
if there is no such fan.
The Cox monomial lattice of the Cox ring C.
A basis of the rational function field of the toric variety X expressed in terms of rational monomials in the Cox ring.
A basis of the rational function field of the toric variety X expressed in terms of lattice points in the Cox monomials lattice.
The divisor class lattice of the Cox ring C.
The monomial lattice of the Cox ring C.
The one-parameter subgroups lattice of the Cox ring C.
The ray lattice of the Cox ring C.
The divisor class group of the Cox ring C.
The map from the ray lattice of the Cox ring C to the ambient
lattice of its fan.
WeilToClassLatticesMap(C) : RngCox -> Map
Comparison maps between toric lattices related to the Cox ring
C of a toric variety.
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