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Toric Varieties

Subsections

Constructors for Toric Varieties

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.

ToricVariety(k,n) : Fld,RngIntElt -> TorVar
Projective n-space Pn defined over the field k as a toric variety.
ToricVariety(k,Z) : Fld,[RngIntElt] -> TorVar
The (weighted) projective space P(Z) defined over the field k with weights the positive integer sequence Z as a toric variety.
ToricVariety(k,Z,Q) : Fld,[RngIntElt],[FldRatElt] -> TorVar
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.
ToricVariety(k,M,v) : Fld,[[RngIntElt]],[RngIntElt] -> TorVar
The n-dimensional toric variety n≥2 defined over the field k with weights begin 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.)

Example Toric_toric-cox-example2 (H118E8)

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.
ToricVariety(k) : Fld -> TorVar
The zero-dimensional point over the field k defined as a toric variety.
ProjectiveSpace(k,n) : Fld,RngIntElt -> Prj
Projective n-space Pn defined over the field k.
ProjectiveSpace(k,W) : Fld,SeqEnum -> Prj
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).

Toric Varieties and Their Fans

ToricVariety(k,F) : Fld,TorFan -> TorVar
The toric variety (defined over the field k) corresponding to the toric fan F.
Fan(X) : TorVar -> TorLat
The toric fan corresponding to the toric variety X.
Rays(X) : TorVar -> SeqEnum
The rays of the fan of the toric variety X.
OneParameterSubgroupsLattice(X) : TorVar -> TorLat
The lattice of weights of the toric variety X; this is the lattice which supports the toric fan of X.
MonomialLattice(X) : TorVar -> TorLat
The monomial lattice of the toric variety X, namely the toric lattice dual to that containing the fan of X.
CoxMonomialLattice(X) : TorVar -> TorLat
The lattice whose elements represent Weil divisors on the toric variety X; it is dual to ray lattice of X.
DivisorClassLattice(X) : TorVar -> TorLat
The divisor class lattice of the toric variety X.
IrrelevantIdeal(X) : TorVar -> SeqEnum
A sequence of ideals that are the components of the irrelevant ideal of the toric variety X.
QuotientGradings(X) : TorVar -> SeqEnum
A sequence of sequences of rational numbers describing the quotients by finite cyclic groups that arise in the construction of the toric variety X.
NumberOfQuotientGradings(X) : TorVar -> SeqEnum
The number of sequences the generate the quotient gradings of the toric variety X.

Properties of Toric Varieties

IsSingular(X) : TorVar -> BoolElt
Return false if and only if the toric variety X is nonsingular.
IsNonsingular(X) : TorVar -> BoolElt
Return true if and only if the toric variety X is nonsingular.
IsGorenstein(X) : TorVar -> BoolElt
Return true if and only if the toric variety X is Gorenstein.
IsQGorenstein(X) : TorVar -> BoolElt
Return true if and only if the toric variety X is Q-Gorenstein.
IsQFactorial(X) : TorVar -> BoolElt
Return true if and only if the toric variety X is Q-factorial.
IsTerminal(X) : TorVar -> BoolElt
Return true if and only if the toric variety X has (at worst) terminal singularities.
IsCanonical(X) : TorVar -> BoolElt
Return true if and only if the toric variety X has (at worst) canonical singularities.
IsComplete(X) : TorVar -> BoolElt
Return true if and only if the toric variety X is complete.
IsProjective(X) : TorVar -> BoolElt
Return true if and only if the toric variety X is projective.
IsFano(X) : TorVar -> BoolElt
Return true if and only if the anticanonical divisor of the toric variety X is ample.
IsFakeWeightedProjectiveSpace(X) : TorVar -> BoolElt
Return true if and only if the toric variety X has exactly one Z-grading.
IsWeightedProjectiveSpace(X) : TorVar -> BoolElt
Return true if and only if the toric variety X has exactly one Z-grading and no quotient gradings.

Affine Patches on Toric Varieties

ToricAffinePatch(X,i) : TorVar,RngIntElt -> TorVar,TorMap
The affine patch corresponding to i-th cone of fan of the toric variety X together with the inclusion map.
ToricAffinePatch(X,S) : TorVar,[RngIntElt] -> TorVar,TorMap
ToricAffinePatch(X,S) : TorVar,[RngMPolElt] -> TorVar,TorMap
The toric variety, obtained from the toric variety X by set 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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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012