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Subsections
We list some simple constructors for simple toric varieties.
There are more general constructors for toric varieties
(either from their fans or their Cox rings) in other sections.
Projective n-space Pn defined over the field k
as a toric variety.
The (weighted) projective space P(Z) defined over the field k
with weights the positive integer sequence Z as a toric variety.
The fake weighted projective space defined over the field k
with weights the positive integer sequence Z and a single
sequence of quotient weights the sequence Q of rational numbers.
The n-dimensional toric variety n≥2 defined over the field k
with weights beign the two sequences of integers (of the same
length n + 2) that comprise
M and linearisation the length 2 integer sequence v.
(This toric variety is the GIT quotient of kn + 2 by
a 2-dimensional torus acting with weights M and linearisation v.
To get a toric variety of the right dimension, v must lie
in the mobile cone implicit in the notation. In practice, this
means that the columns of M must generate a cone with vertex
in a 2-dimensional toric lattice and v must lie in the `very-interior'
of that cone, in the sense that it must lie in the strict
interior of C and in the subcone generated by all columns
of M except the two most extreme.)
We build a Hirzebruch surface as a GIT quotient.
> X<u,v,x,y> := ToricVariety(Rationals(),[[1,1,0,-1],[0,0,1,1]],[1,1]);
> X;
Toric variety of dimension 2
Variables: u, v, x, y
The components of the irrelevant ideal are:
(y, x), (v, u)
The 2 gradings are:
1, 1, 0, -1,
0, 0, 1, 1
The polarisation (1, 1) that we used is forgotten---all that
is left is X.
Using extreme polarisations can give misleading, but true, results.
The same torus action polarised by (0, 1) defines the projective
plane, but not in a very transparent way.
> Y<a,b,c,d> := ToricVariety(Rationals(),[[1,1,0,-1],[0,0,1,1]],[0,1]);
> Y;
Toric variety of dimension 2
Variables: a, b, c, d
The components of the irrelevant ideal are:
(d, c), (c, b, a)
The 2 gradings are:
1, 1, 0, -1,
0, 0, 1, 1
The zero-dimensional point over the field k defined
as a toric variety.
Projective n-space Pn defined over the field k.
FakeProjectiveSpace(k,W,Q) : Fld,SeqEnum,SeqEnum -> TorVar
The (fake) weighted projective space over the field k with weights
the sequence of integers W (and quotient weights the sequence
of sequences of rational numbers, if provided).
The toric variety (defined over the field k)
corresponding to the toric fan F.
The toric fan corresponding to the toric variety X.
The rays of the fan of the toric variety X.
The lattice of weights of the toric variety X; this is
the lattice which supports the toric fan of X.
The monomial lattice of the toric variety X, namely
the toric lattice dual to that containing the fan of X.
The lattice whose elements represent Weil divisors on the toric
variety X; it is dual to ray lattice of X.
The divisor class lattice of the toric variety X.
A sequence of ideals that are the components of the irrelevant ideal
of the toric variety X.
A sequence of sequences of rational numbers describing
the quotients by finite cyclic groups that arise in
the construction of the toric variety X.
The number of sequences the generate the quotient gradings of
the toric variety X.
True if and only if the toric variety X is nonsingular.
True if and only if the toric variety X is Gorenstein.
True if and only if the toric variety X is Q-Gorenstein.
True if and only if the toric variety X is Q-factorial.
True if and only if the toric variety X has (at worst)
terminal singularities.
True if and only if the toric variety X has (at worst)
canonical singularities.
True if and only if the toric variety X is complete.
True if and only if the toric variety X is projective.
True if and only if the anticanonical divisor of the
toric variety X is ample.
True if and only if the toric variety X has exactly
one Z-grading.
True if and only if the toric variety X has exactly
one Z-grading and no quotient gradings.
The affine patch corresponding to i-th cone of fan of
the toric variety X together with the inclusion map.
ToricAffinePatch(X,S) : TorVar,[RngMPolElt] -> TorVar,TorMap
The toric variety, obtained from the toric variety X by
sett the monomials of the sequence S set to be non-zero
(or alternatively the variables of X with indices from the sequence of
integers S set non-zero).
The inclusion map is returned as a second value.
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