Toric Varieties

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 \({\mathbb{P}}^n\) defined over the field \(k\) as a toric variety.

ToricVariety(k, Z): Fld, [RngIntElt] -> TorVar

The (weighted) projective space \({\mathbb{P}}(Z)\) 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 \(n\ge 2\) 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 \(k^{n+2}\) 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: Hirzebruch Surface as a GIT Quotient (ex-70f3b4)

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 \(X\).

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ToricVariety(k): Fld -> TorVar

The zero-dimensional point over the field \(k\) defined as a toric variety.

ProjectiveSpace(k, n): Fld, RngIntElt -> Prj

ProjectiveSpaceAsToricVariety(k, n): Fld, RngIntElt -> TorVar

WeightedProjectiveSpace(k, n): Fld, RngIntElt -> TorVar

WPS(k, n): Fld, RngIntElt -> TorVar

Projective \(n\)-space \({\mathbb{P}}^n\) defined over the field \(k\).

ProjectiveSpace(k, W): Fld, [RngIntElt] -> Prj

ProjectiveSpaceAsToricVariety(k, W): Fld, [RngIntElt] -> TorVar

WeightedProjectiveSpace(k, W): Fld, [RngIntElt] -> TorVar

WPS(k, W): Fld, [RngIntElt] -> TorVar

FakeProjectiveSpace(k, W, Q): Fld, [RngIntElt], [FldRatElt] -> 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 \(Q\), if provided).

AbsoluteRationalScroll(k, S): Fld, [RngIntElt] -> TorVar

Given a sequence of weights \(S=(s_1,\ldots,s_n)\) of length \(n\), creates the \(n\)-dimensional rational scroll over \(k\) with gradings \((1,1,-s_1,\ldots,-s_n), (0,0,1,\ldots,1).\)

RationalScroll(k, s, A): Fld, RngIntElt, [RngIntElt] -> TorVar

The fibre bundle of projective \(r\)-spaces over projective \(s\)-space, where \(A = [a_0,...,a_r]\) is a sequence of non-negative integers of length at least \(2\), that is \({\rm Proj}(O(a_0)\oplus\ldots\oplus O(a_r))\).

RuledSurface(k, n): Fld, RngIntElt -> TorVar

The rational ruled Hirzebruck surface of degree \(n \geq 0\) over \(k\). This is the rational ruled surface with special section of self-intersection \(-n\). It is the toric variety having the gradings \((1,1,-n,0)\) and \((0,0,1,1)\) on four variables.

RuledSurface(k, a1, a2): Fld, RngIntElt, RngIntElt -> TorVar

The rational ruled surface with gradings \((1,1,-a1,-a2)\) and \((0,0,1,1)\) on four variables. This is the ruled surface \(P(O(-a1)\oplus O(-a2))\) in Hartshorne’s notation (Section 2, Chapter V of [Har77]). It is abstractly isomorphic to the Hirzebruch surface of degree \(|a1-a2|\).

HirzebruchSurface(k, n): Fld, RngIntElt -> TorVar

The same as RuledSurface(k,n) except that the last two variables are swapped, ie the gradings are \((1,1,0,-n)\) and \((0,0,1,1)\).

BigTorus(k, N): Rng, TorLat -> TorVar

The big torus \(T\) associated with a toric lattice \(N\).

BigTorus(X): TorVar -> TorVar, TorMap, TorMap

The big torus \(T\) of the toric variety \(X\), together with its embedding into \(X\) and the rational map from \(X\) to \(T\).

RestrictionToSubtorus(Z): Sch -> Sch, TorMap

The restriction of \(Z\) to the largest subtorus of the ambient containing \(Z\).

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 \(X\).

PicardLattice(X): TorVar -> TorLat

The lattice whose integral elements correspond to Cartier divisors up to linear equivalence and modulo torsion.

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 \(X\).

Gradings(X): Sch -> SeqEnum

A sequence of sequences representing all of the gradings to be applied to the indeterminates of the coordinate ring of the ambient of \(X\).

NumberOfGradings(X): Sch -> RngIntElt

The number of gradings to be applied to the indeterminates of the coordinate ring of the ambient of \(X\).

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 -> RngIntElt

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 \({\mathbb{Q}}\)-Gorenstein.

IsQFactorial(X): TorVar -> BoolElt

Return true if and only if the toric variety \(X\) is \({\mathbb{Q}}\)-factorial.

IsIsolated(X): TorVar -> BoolElt

Return true if and only if the toric variety \(X\) has only isolated singularities.

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.

IsWeakFano(X): TorVar -> BoolElt

Return true if and only if the toric variety \(X\) is a weak Fano variety.

IsFakeWeightedProjectiveSpace(X): TorVar -> BoolElt

Return true if and only if the toric variety \(X\) has exactly one \({\mathbb{Z}}\)-grading.

IsWeightedProjectiveSpace(X): TorVar -> BoolElt

Return true if and only if the toric variety \(X\) has exactly one \({\mathbb{Z}}\)-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.