<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE Book SYSTEM "gapdoc.dtd">
<Book Name="toric"> <!-- REQUIRED -->
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<TitlePage>
<!-- REQUIRED -->
<Title>
<Package>Toric</Package>
</Title>
<Subtitle> <!-- OPTIONAL -->
A &GAP;4 Package for computing with toric varieties
</Subtitle>
<Version>Version 1.4</Version>
<!-- OPTIONAL -->
<Author>
David Joyner<Br/> <!-- REQUIRED -->
<Address>
Mathematics Department,<Br/>
U. S. Naval Academy,<Br/>
Annapolis, MD,<Br/>
21402 USA.
</Address>
<Email> wdjoyner@gmail.com</Email>
<Homepage>http://www.opensourcemath.org/toric/</Homepage>
</Author>
<Date>February 26, 2008</Date>
<!-- OPTIONAL -->
<!-- end of title page -->
<Copyright>
<!-- OPTIONAL -->
©right; 2004-2005 David Joyner.
</Copyright>
<Acknowledgements>
<!-- OPTIONAL -->
<P/>
The code for the <Package>toric</Package> package was written during the
summer of 2002.
It was put into &GAP; package format in the summer of 2004.
<Package>toric</Package> is released under the
GNU General Public License (GPL).
This file is part of <Package>toric</Package>, though as documentation
it is released under the
GNU Free Documentation License
(see <URL>http://www.gnu.org/licenses/licenses.html#FDL</URL>).
<P/>
<Package>toric</Package> is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
<P/>
<Package>toric</Package> is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
<P/>
You should have received a copy of the GNU General Public License
along with <Package>toric</Package>; if not, write to the Free Software
Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
<P/>
For more details, see
<URL>http://www.fsf.org/licenses/gpl.html</URL>.
<P/>
<P/>This documentation was prepared with the
<Package>GAPDoc</Package> package of Frank L\"ubeck and Max Neunh\"offer.
</Acknowledgements>
</TitlePage>
<TableOfContents/> <!-- OPTIONAL -->
<!-- The document -->
<Body> <!-- REQUIRED -->
<Chapter><Heading>Introduction</Heading>
<Section>
<Heading>Introduction to the <Package>toric</Package> package</Heading>
<P/>
This manual describes the <Package>toric</Package> package for working with
toric varieties in &GAP;. Toric varieties can be dealt
with more easily than general varieties since often
times questions about a toric variety can be reformulated in terms
of combinatorial geometry.
Some coding theory commands related to toric varieties
are contained in the error-correcting codes
<Package>GUAVA</Package> package
(for example, the command <C>ToricCode</C>).
We refer to the <Package>GUAVA</Package> manual
<Cite Key="Gua05"/> and the expository
paper <Cite Key="JV02"/> for more details.
<P/>
The <Package>toric</Package> package also contains
several commands unrelated to toric varieties
(mostly for list manipulations).
These will not be described in this documention but
they are briefly documented in the <C>lib/util.gd</C>
file.
<P/>
<Package>toric</Package> is implemented in the &GAP; language,
and runs on any system supporting &GAP;4.3 and above.
The <Package>toric</Package> package is loaded with the command
<Verb> gap> LoadPackage( "toric" ); </Verb>
<P/>
Please send bug reports, suggestions and other comments about
<Package>toric</Package> to
<Email>support@gap-system.org</Email>.
</Section>
<Section>
<Heading>Introduction to constructing toric varieties</Heading>
Rather than sketch the theory of
toric varieties, we refer to
<Cite Key="JV02"/> and <Cite Key="F93"/> for details.
However, we briefly describe some terminology and notation.
<Subsection>
<Heading>Generalities</Heading>
Let <M>F</M> denote a field and
<M>R=F [x_1,...,x_n]</M> be a ring in <M>n</M> variables.
A <B>binomial equation</B>
in <M>R</M> is one of the form
<Display>
x_1^{k_1}...x_n^{k_n}=x_1^{\ell_1}...x_n^{\ell_n},
</Display>
where <M>k_i \geq 0,\ \ell_j \geq 0</M> are integers.
A binomial variety
is a subvariety of affine <M>n</M>-space
<M>A_F^n</M> defined by a finite set of binomial
equations (such a variety need not be normal).
A typical ``toric variety'' is binomial, though they
will be introduced via an <E>a priori</E>
independent construction.
The basic idea of the construction is to
replace each such binomial equation as above by a relation in a
semigroup contained in a lattice
and replace <M>R</M> by the ``group algebra'' of this
semigroup. By the way, a toric variety is
always normal (see for example, <Cite Key="F93"/>,
page 29).
</Subsection>
<Subsection>
<Heading>Basic combinatorial geometry constructions</Heading>
Let <M>Q</M> denote the field of rational numbers and <M>Z</M> denote the
set of integers. Let <M>n>1</M> denote an integer.
<P/>
Let <Math>V=Q^n</Math> having basis
<M>f_1=(1,0,...,0)</M>, ..., <M>f_n=(0,...,0,1)</M>.
Let <M>L_0=Z^n\subset V</M> be the standard lattice in <M>V</M>.
We identify <M>V</M> and <M>L_0\otimes_{Z} Q</M>.
We use <M>\langle\ ,\ \rangle</M> to denote
the (standard) inner product on <M>V</M>. Let
<Display>
L_0^*={\rm Hom}(L_0,Z)=\{ v\in V\ |\ \langle v,w \rangle \in Z,
\ \forall w\in L_0\}
</Display>
denote the <B>dual lattice</B>, so (fixing the standard basis
<M>e_1^*</M>,...,<M>e_n^*</M> dual to the <M>f_1</M>,...,<M>f_n</M>)
<M>L_0^*</M> may be identified with <M>Z^n</M>.
<P/>
A <B>cone</B> in <M>V</M> is a set <M>\sigma</M> of the form
<Display>
\sigma=\{a_1v_1+...+a_mv_m\ |\ a_i\geq 0\}\subset V,
</Display>
where <M>v_1,...,v_m \in V</M> is a given collection of
vectors, called (semigroup) <B>generators</B> of <M>\sigma</M>.
A <B>rational cone</B> is one where <M>v_1,...,v_m \in L_0</M>.
A <B>strongly convex</B> cone is one which contains
no lines through the origin.
<P/>
<E>
By abuse of terminology, from now on
a <B>cone</B> of <M>L_0</M> is a strongly convex rational cone.
</E>
<Index>cone</Index>
<P/>
A <B>face</B> of a cone <M>\sigma</M> is either <M>\sigma</M>
itself or a subset of the form <M>H\cap \sigma</M>,
where <M>H</M> is a codimension one subspace of <M>V</M>
which intersects the cone non-trivially and such that the cone
is contained in exactly one of the two half-spaces determined by
<M>H</M>. A <B>ray</B> (or edge) of a cone is a one-dimensional face.
<Index>ray</Index>
Typically, cones are represented in <Package>toric</Package>
by the list of vectors defining their rays.
The <B>dimension</B> of a cone is the dimension of the
vector space it spans.
The <Package>toric</Package> package can test if a
given vector is in a given cone (see <C>InsideCone</C>).
<P/>
If <M>\sigma</M> is a cone then the <B>dual cone</B> is defined by
<Display>
\sigma^* =\{w \in L_0^*\otimes Q
\ |\ \langle v,w \rangle \geq 0,\ \forall v\in \sigma\}.
</Display>
The <Package>toric</Package> package can test if a
vector is in the dual of a given cone (see <C>InDualCone</C>).
<Index>dual cone</Index>
<P/>
Associate to the dual cone <M>\sigma^*</M> is the semigroup
<Display>
S_\sigma =\sigma^*\cap L_0^*
=\{w\in L_0^*
\ |\ \langle v,w\rangle \geq 0,\
\forall v\in \sigma\}.
</Display>
Though <M>L_0^*</M> has $n$ generators <E>as a lattice</E>,
typically <M>S_\sigma</M> will have more than <M>n</M>
generators <E>as a semigroup</E>.
The <Package>toric</Package> package can compute a
minimal list of semigroup generators of <M>S_\sigma</M>
(see <C>DualSemigroupGenerators</C>).
<Index>semigroup associated to cone</Index>
<P/>
A fan is a collection of cones which ``fit together'' well.
A <B>fan</B> in <M>L_0</M> is a set
<M>\Delta=\{\sigma \}</M> of rational strongly convex
cones in <M>V= L_0 \otimes Q</M>
such that
<List>
<Item>
if <M>\sigma \in \Delta</M> and <M>\tau \subset \sigma</M> is a face
of <M>\sigma</M> then <M>\tau \in \Delta</M>,
</Item>
<Item>
if <M>\sigma_1, \sigma_2 \in \Delta</M> then
the intersection <M>\sigma_1 \cap \sigma_2</M> is a face of
both <M>\sigma_1</M> and <M>\sigma_2</M> (and hence
belongs to <M>\Delta</M>).
</Item>
</List>
In particular, the face of a cone in a fan is a cone is the fan.
<P/>
If <M>V</M> is the (set-theoretic) union of the
cones in <M>\Delta</M> then we call the fan <B>complete</B>.
We shall assume that all fans are finite.
A fan is determined by its list of maximal cones.
<Index>fan</Index>
<P/>
<E>Notation</E>: A fan <M>\Delta</M> is represented in
<Package>toric</Package> as a set of maximal cones.
For example, if <M>\Delta</M> is the fan with maximal cones
<M>\sigma_1=Q_+\cdot f_1+Q_+\cdot (-f_1+f_2)</M>,
<M>\sigma_2=Q_+\cdot (-f_1+f_2)+Q_+\cdot (-f_1-f_2)</M>,
<M>\sigma_3=Q_+\cdot (-f_1-f_2)+Q_+\cdot f_1</M>,
then <M>\Delta</M> is represented by
<M>[[[1,0],[-1,1]],[[-1,1],[-1,-1]],[[-1,-1],[1,0]]]</M>.
<P/>
The <Package>toric</Package> package can compute all cones
in a fan of a given dimension (see <C>ConesOfFan</C>).
Moreover, <Package>toric</Package> can compute
the set of all normal vectors to the faces (i.e., hyperplanes)
of a cone (see <C>Faces</C>).
<P/>
The <B>star</B> of a cone <M>\sigma</M> in a
fan <M>\Delta</M> is the set <M>\Delta_\sigma</M>
of cones in <M>\Delta</M> containing <M>\sigma</M> as a face.
The <Package>toric</Package> package can compute stars
(see <C>ToricStar</C>).
<Index>star</Index>
<P/>
</Subsection>
<Subsection>
<Heading>Basic affine toric variety constructions</Heading>
Let
<Display>
R_\sigma = F [S_\sigma]
</Display>
denote the ``group algebra'' of this semigroup. It is a
finitely generated commutative F-algebra.
It is in fact integrally closed (<Cite Key="F93"/>, page 29).
We may interprete <M>R_\sigma</M> as a subring of
<M>R=F [x_1,...,x_n]</M> as follows: First,
identify each <M>e_i^*</M> with the variable <M>x_i</M>.
If <M>S_\sigma</M> is generated as a semigroup by
vectors of the form <M>\ell_1 e_1^*+...+\ell_n e_n^*</M>,
where <M>\ell_i</M> is an integer, then its image in
<M>R</M> is generated by monomials of the
form <M>x_1^{\ell_1}\dots x_n^{\ell_n}</M>.
The <Package>toric</Package> package can compute these
generating monomials (see <C>EmbeddingAffineToricVariety</C>).
See Lemma 2.14 in <Cite Key="JV02"/> for more details.
This embedding can also be used to resolve singularities -
see section 5 of <Cite Key="JV02"/> for more details.
<P/>
Let
<Display>
U_\sigma={\rm Spec}\ R_\sigma.
</Display>
This defines an <B>affine toric variety</B> (associated
to <M>\sigma</M>). It is known that the coordinate ring
<M>R_\sigma</M> of the affine toric variety <M>U_\sigma</M>
has the form <M>R_\sigma = F[x_1,...,x_n]/J</M>, where <M>J</M>
is an ideal. The <Package>toric</Package> package can compute
generators of this ideal (see <C>IdealAffineToricVariety</C>).
<Index>affine toric variety</Index>
<P/>
Roughly speaking, the toric variety <M>X(\Delta)</M> associated to
the fan <M>\Delta</M> is given by a collection of affine
pieces $U_{\sigma_1},U_{\sigma_2},\dots,U_{\sigma_d}$
which ``glue'' together (where <M>\Delta = \{\sigma_i\}</M>).
The affine pieces are given by the zero
sets of polynomial equations in some affine spaces
and the gluings are given by maps
<M>\phi_{i,j} : U_{\sigma_i} \rightarrow U_{\sigma_j}</M>
which are defined by ratios of polynomials on open subsets of the
$U_{\sigma_i}$.
The <Package>toric</Package> package does <E>not</E>
compute these gluings or work directly with these
(non-affine) varieties <M>X(\Delta)</M>.
<P/>
A cone <M>\sigma \subset V</M> is said to be <B>nonsingular</B> if
it is generated by part of a basis for the lattice <M>L_0</M>.
A fan <M>\Delta</M> of cones is said to be <B>nonsingular</B> if
all its cones are nonsingular. It is known that <M>U_\sigma</M>
is nonsingular if and only if <M>\sigma</M> is nonsingular
(Proposition 2.1 in <Cite Key="F93"/>).
<Index>cone, nonsingular</Index>
<P/>
<B>Example:</B>
In three dimensions, consider the cones
<M>\sigma_{\epsilon_1,\epsilon_2,\epsilon_3,i,j}</M>
generated by <M>(\epsilon_1\cdot 1,\epsilon_2\cdot 1,\epsilon_3\cdot 1)</M>
and the standard basis vectors <M>f_i,f_j</M>,
where <M>\epsilon_i=\pm 1</M> and
<M>1\leq i\not= j\leq 3</M>. There are 8 cones per octant,
for a total of 64 cones. Let <M>\Delta</M> denote the
fan in <M>V=Q^3</M> determined by these maximal cones.
The toric variety <M>X(\Delta)</M> is nonsingular.
<P/>
</Subsection>
<Subsection>
<Heading>Riemann-Roch spaces and related constructions</Heading>
Although the <Package>toric</Package> package does not
work directly with the toric varieties <M>X(\Delta)</M>,
it can compute objects associated with it.
For example, it can compute the
Euler characteristic (see <C>EulerCharacteristic</C>), Betti numbers
(see <C>BettiNumberToric</C>), and the
number of GF(q)-rational points
(see <C>CardinalityOfToricVariety</C>) of <M>X(\Delta)</M>,
<E> provided <M>\Delta</M> is nonsingular.</E>
<P/>
For an algebraic variety <M>X</M> the group of <B>Weil divisors</B>
on <M>X</M> is
the abelian group <M>Div(X)</M> generated (additively) by the
irreducible subvarieties of X of codimension <M>1</M>.
For a toric variety <M>X(\Delta)</M> with dense open torus <M>T</M>, a Weil divisor
D is <B>T-invariant</B> if <M>D=T\cdot D</M>.
The group of <M>T</M>-invariant Weil divisors is denoted
<M>TDiv(X)</M>. This is finitely generated by an explicitly
given finite set of divisors
<M>\{D_1,...,D_r\}</M> which correspond naturally to certain
cones in <M>\Delta</M> (see <Cite Key="F93"/> for details).
In particular, we may represent such a divisor <M>D</M> in <M>TDiv(X)</M>
by an <M>k</M>-tuple <M>(d_1,...,d_k)</M> of integers.
<Index>Weil divisors</Index>
<P/>
Let <M>\Delta</M> denote a fan in <M>V=Q^n</M> with rays (or edges)
<M>\tau_i</M>, <M>1\leq i\leq k</M>, and let
<M>v_i</M> denote the first lattice point on <M>\tau_i</M>.
Associated to the T-invariant Weil divisor <M>D=d_1D_1+...+d_kD_k</M>,
is the <B>polytope</B>
<Display>
P_D = \{ x=(x_1,...,x_n)\ |\ \langle x,v_i \rangle \geq -d_i,
\ \forall 1 \leq i \leq k\}.
</Display>
The <Package>toric</Package> package can compute
<M>P_D</M> (see <C>DivisorPolytope</C>), as well as the set of all
lattice points contained in this polytope
(see <C>DivisorPolytopeLatticePoints</C>).
Also associated to the <M>T</M>-invariant Weil divisor
<M>D=d_1D_1+...+d_kD_k</M>, is the Riemann-Roch space,
<M>L(D)</M>. This is a space of functions on
<M>X(\Delta)</M> whose zeros and poles are
``controlled'' by <M>D</M> (for a more precise definition, see
<Cite Key="F93"/>). The <Package>toric</Package> package can compute
a basis for <M>L(D)</M> (see <C>RiemannRochBasis</C>),
<E> provided <M>\Delta</M> is complete and nonsingular.</E>
<Index>polytope associated to divisor</Index>
</Subsection>
</Section>
</Chapter>
<Chapter>
<Heading>Cones and semigroups</Heading>
<Section>
<Heading>Cones</Heading>
This section introduces the <Package>toric</Package>
commands which deal with cones and related
combinatorial-geometric objects.
Recall, a <B>cone</B> is a strongly convex polyhedral
cone (<Cite Key="F93"/>, page 4).
<ManSection Label="InsideCone">
<Func Name="InsideCone" Arg=" v, L "/>
<Description>
This command returns `true` if the vector <A>v</A> belongs to the interior
of the (strongly convex polyhedral)
cone generated by the vectors in <A>L</A>.
</Description>
</ManSection>
This procedure does not check if <A>L</A> generates a
strongly convex polyhedral cone.
<Example>
gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
gap> InsideCone(v,L);
false
gap> L:=[[1,0],[3,4]];;
gap> v:=[1,-7]; InsideCone(v,L);
[ 1, -7 ]
false
gap> v:=[4,-3]; InsideCone(v,L);
[ 4, -3 ]
false
gap> v:=[4,-4]; InsideCone(v,L);
[ 4, -4 ]
false
gap> v:=[4,1]; InsideCone(v,L);
[ 4, 1 ]
true
</Example>
<!--
L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
v:=[0,0,1];;
InsideCone(v,L);
false
L:=[[1,0],[3,4]];;
v:=[1,-7]; InsideCone(v,L);
v:=[4,-3]; InsideCone(v,L);
v:=[4,-4]; InsideCone(v,L);
v:=[4,1]; InsideCone(v,L);
-->
<ManSection Label="InDualCone">
<Func Name="InDualCone" Arg=" v, L "/>
<Description>
This command
returns `true` if <A>v</A> belongs to the dual of the cone generated by
the vectors in <A>L</A>.
</Description>
</ManSection>
<Example>
gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
gap> InDualCone(v,L);
true
gap> L:=[[1,0],[3,4]];
[ [ 1, 0 ], [ 3, 4 ] ]
gap> v:=[1,-7]; InDualCone(v,L);
[ 1, -7 ]
false
gap> v:=[4,-3]; InDualCone(v,L);
[ 4, -3 ]
true
gap> v:=[4,-4]; InDualCone(v,L);
[ 4, -4 ]
false
gap> v:=[4,1]; InDualCone(v,L);
[ 4, 1 ]
true
</Example>
<!--
L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
InDualCone(v,L);
L:=[[1,0],[3,4]];
v:=[1,-7]; InDualCone(v,L);
v:=[4,-3]; InDualCone(v,L);
v:=[4,-4]; InDualCone(v,L);
v:=[4,1]; InDualCone(v,L);
-->
<ManSection>
<Func Name="PolytopeLatticePoints" Arg=" A, Perps "/>
<Description>
<E>Input</E>: <A>Perps</A><M>=[v_1,...,v_k]</M> is the list of ``inward normal"
vectors perpendicular to the walls of a polytope <M>P</M> in the
vector space <M>L_0^*\otimes Q</M>,
<Br></Br>
<A>A</A><M>=[a_1,...,a_k]</M> is a k-tuple of integers, where
<M>a_i</M> denotes the amount the i-th ``wall" (defined by the normal <M>v_i</M>)
is shifted from the origin (each <M>a_i</M> is assumed non-negative).
<Br></Br>
For example, the polytope <M>P</M> with faces
<C>[x=0, x=a, y=0, y=b]</C>
has <A>Perps</A><M>=[[1,0],[-1,0],[0,1],[0,-1]]</M> and <A>A</A><M>=[0,a,0,b]</M>.
<Br></Br>
<E>Output</E>: the list of points in <M>P \cap L_0^*</M>.
</Description>
</ManSection>
<Example>
gap> Perps:=[[1,0],[-1,0],[0,1],[0,-1]];
[ [ 1, 0 ], [ -1, 0 ], [ 0, 1 ], [ 0, -1 ] ]
gap> A:=[0,4,0,3];
[ 0, 4, 0, 3 ]
gap> PolytopeLatticePoints(A,Perps);
[ [ 0, 0 ], [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 1, 0 ], [ 1, 1 ], [ 1, 2 ],
[ 1, 3 ], [ 2, 0 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 0 ], [ 3, 1 ],
[ 3, 2 ], [ 3, 3 ], [ 4, 0 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ] ]
gap> Length(last);
20
</Example>
<!--
Perps:=[[1,0],[-1,0],[0,1],[0,-1]];
A:=[0,4,0,3];
PolytopeLatticePoints(A,Perps);
Length(last);
-->
<ManSection>
<Func Name="Faces" Arg=" Rays "/>
<Description>
<E>Input</E>: <A>Rays</A> is a list of rays for the fan <M>\Delta</M>
<Br></Br>
<E>Output</E>: All the normals to the faces (hyperplanes of the cone).
</Description>
</ManSection>
<Example>
gap> Cones1:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> Faces(Cones1[1]);
[ [ 1/2, 1 ], [ 2, 1 ] ]
gap> Faces(Cones1[2]);
[ [ -2, -1 ], [ -1, 1 ] ]
gap> Cones2:=[[[ 2,0,0],[0,2,0],[0,0,2]], [[2,0,0], [0,2,0], [2,-2,1],[1,2,-2]]];;
gap> Faces(Cones2[1]);
[ [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, 0, 0 ] ]
gap> Faces(Cones2[2]);
[ [ 1/3, 5/6, 1 ], [ 1/2, 0, -1 ], [ 2, 0, 1 ] ]
gap>
</Example>
<!--
Cones1:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
Faces(Cones1[1]);
Faces(Cones1[2]);
Cones2:=[[[ 2,0,0],[0,2,0],[0,0,2]], [[2,0,0], [0,2,0], [2,-2,1],[1,2,-2]]];;
Faces(Cones2[1]);
Faces(Cones2[2]);
-->
<ManSection>
<Func Name="ConesOfFan" Arg=" Delta k "/>
<Description>
<E>Input</E>: <A>Delta</A> is the fan of cones,
<Br></Br>
<A>k</A> is the dimension of the cones desired.
<Br></Br>
<E>Output</E>: The <A>k</A>-dimensional cones in the fan.
</Description>
</ManSection>
<ManSection>
<Func Name="NumberOfConesOfFan" Arg=" Delta k "/>
<Description>
<E>Input</E>: <A>Delta</A> is the fan of cones in <M>V=Q^n</M>,
<Br></Br>
<A>k</A> is the dimension of the cones counted.
<Br></Br>
<E>Output</E>: The number of <A>k</A>-dimensional cones in the fan.
</Description>
</ManSection>
Idea: The fan <A>Delta</A> is represented as a set of maximal cones.
For each maximal cone, look at the <A>k</A>-dimensional faces
obtained by taking <M>n</M> choose <A>k</A> subsets of the
rays describing the cone. Certain of these <A>k</A>-subsets
yield the desired cones.
<Example>
gap> Delta0:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
gap>
gap> NumberOfConesOfFan(Delta0,2);
6
gap> ConesOfFan(Delta0,2);
[ [ [ 0, 0, 2 ], [ 0, 2, 0 ] ], [ [ 0, 0, 2 ], [ 2, 0, 0 ] ],
[ [ 0, 2, 0 ], [ 1, 2, -2 ] ], [ [ 0, 2, 0 ], [ 2, -2, 1 ] ],
[ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 1, 2, -2 ], [ 2, -2, 1 ] ] ]
gap> ConesOfFan(Delta0,1);
[ [ [ 0, 0, 2 ] ], [ [ 0, 2, 0 ] ], [ [ 1, 2, -2 ] ],
[ [ 2, -2, 1 ] ], [ [ 2, 0, 0 ] ] ]
gap> NumberOfConesOfFan(Delta0,1);
5
</Example>
<!--
Delta0:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
NumberOfConesOfFan(Delta0,2);
ConesOfFan(Delta0,2);
ConesOfFan(Delta0,1);
NumberOfConesOfFan(Delta0,1);
-->
<ManSection>
<Func Name="ToricStar" Arg=" sigma, Delta "/>
<Description>
<E>Input</E>: <A>sigma</A> is a cone in the fan,
represented by its set of maximal (i.e., highest dimensional) cones.
<Br></Br>
<A>Delta</A> is the fan of cones in <M>V=Q^n</M>.
<Br></Br>
<E>Output</E>: The star of the cone <A>sigma</A> in <A>Delta</A>,
i.e., the cones <M>\tau</M> which have <A>sigma</A> as a face.
</Description>
</ManSection>
<Example>
gap> MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ],
[ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
gap> #this is the set of maximal cones in the fan Delta
gap> ToricStar([[1,0]],MaxCones);
[ ]
gap> ToricStar([[2,0,0],[0,2,0]],MaxCones);
[ [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ],
[ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 2, -2, 1 ], [ 1, 2, -2 ] ] ]
gap>
gap> MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[1,1,-2] ] ];;
gap> ToricStar([[2,0,0],[0,2,0]],MaxCones);
[ [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ],
[ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 1, 1, -2 ] ] ]
gap> ToricStar([[1,0]],MaxCones);
[ ]
</Example>
<!--
MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
ToricStar([[1,0]],MaxCones);
ToricStar([[2,0,0],[0,2,0]],MaxCones);
MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[1,1,-2] ] ];;
ToricStar([[2,0,0],[0,2,0]],MaxCones);
ToricStar([[1,0]],MaxCones);
-->
</Section>
<Section>
<Heading>Semigroups</Heading>
<ManSection>
<Func Name="DualSemigroupGenerators" Arg=" L "/>
<Description>
<E>Input</E>: <A>L</A> is a list of integral <M>n</M>-vectors generating a
cone <M>\sigma</M>.
<Br></Br>
<E>Output</E>: the generators of <M>S_\sigma</M>,
<P/>
Idea: let <M>M</M> be the maximum of the absolute values of the
coordinates of the <A>L</A>[i]'s, for each vector <M>v</M> in
<M>[1..M]^n</M>, test if <M>v</M> is in the dual cone
<M>\sigma^*</M>. If so,
add <M>v</M> to list of possible generators.
Once this for loop is finished, one can check
this list for redundant generators. The
trick is to simply omit those elements
which are of the form <M>d_1+d_2</M>, where <M>d_1</M> and <M>d_2</M>
are ``small" elements in the integral dual cone.
<P/>
This program is not very efficient and should not be used
in ``large examples'' involving semigroups with ``many'' generators.
For example, if you take
<A>L:=[[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]];</A> then
<C>DualSemigroupGenerators(L);</C> can exhaust GAP's memory
allocation.
</Description>
</ManSection>
<Example>
gap> L:=[[1,0],[3,4]];; DualSemigroupGenerators([[1,0],[3,4]]);
[ [ 0, 0 ], [ 0, 1 ], [ 1, 0 ], [ 2, -1 ], [ 3, -2 ], [ 4, -3 ] ]
gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
gap> DualSemigroupGenerators(L);
[ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, -1, 0 ], [ 1, 0, -1 ] ]
gap>
</Example>
<!--
L:=[[1,0],[3,4]];; DualSemigroupGenerators([[1,0],[3,4]]);
L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
DualSemigroupGenerators(L);
-->
</Section>
</Chapter>
<Chapter>
<Heading>Affine toric varieties</Heading>
This chapter concerns <Package>toric</Package>
commands which deal with the coordinate rings of
affine toric varieties <M>U_\sigma</M>.
<Section>
<Heading>Ideals defining affine toric varieties</Heading>
<ManSection>
<Func Name="IdealAffineToricVariety" Arg=" L "/>
<Description>
<E>Input</E>: <A>L</A> is a list generating a cone (as in <C>DualSemigroupGenerators</C>).
<Br></Br>
<E>Output</E>: the &GAP; ideal defining the toric variety associated to the cone
generated by the vectors in <A>L</A>.
<P/>
This computation is not very efficient and should not be used for
ideals with many generators. For example, if you take
<A>L:=[[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]];</A> then
<C>IdealAffineToricVariety(L);</C> can exhaust GAP's memory
allocation.
</Description>
</ManSection>
<Example>
gap> J:=IdealAffineToricVariety([[1,0],[3,4]]);
[ two-sided ideal in PolynomialRing(..., [ x_1, x_2 ]), (3 generators) ]
gap> GeneratorsOfIdeal(J);
[ -x_2^2+x_1, -x_2^3+x_1^2, -x_2^4+x_1^3 ]
</Example>
<ManSection>
<Func Name="EmbeddingAffineToricVariety" Arg=" L "/>
<Description>
<E>Input</E>: <A>L</A> is a list generating a cone
(as in <C>DualSemigroupGenerators</C>).
<Br></Br>
<E>Output</E>: the toroidal embedding of <M>X=Spec</M>(<C>IdealAffineToricVariety(L)</C>)
(given as a list of multinomials).
</Description>
</ManSection>
<Example>
gap> phi:=EmbeddingAffineToricVariety([[1,0],[3,4]]);
[ x_2, x_1, x_1^2/x_4, x_1^3/x_4^2, x_1^4/x_4^3 ]
gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
gap> phi:=EmbeddingAffineToricVariety(L);
[ x_3, x_2, x_1/x_5, x_1/x_6 ]
</Example>
</Section>
</Chapter>
<Chapter>
<Heading>Toric varieties <M>X(\Delta)</M> </Heading>
This chapter concerns <Package>toric</Package>
commands which deal with certain objects associated to the
(non-affine) toric varieties <M>X(\Delta)</M>.
<Section>
<Heading>Riemann-Roch spaces</Heading>
Let <M>\Delta</M> denote a complete nonsingular fan.
<ManSection>
<Func Name="DivisorPolytope" Arg=" D, Rays "/>
<Description>
<E>Input</E>: <A>Rays</A> is the list of smallest integer
vectors in the rays for the fan <M>\Delta</M>
which determine the Weil divisors of <M>X(\Delta)</M>.
<Br></Br>
<A>D</A> is the list of coefficients for the a Weil divisor.
<Br></Br>
<E>Output</E>: the linear expressions in the affine coordinates of the
space of the cone which must be positive for a point to be in the
desired polytope.
</Description>
</ManSection>
<Example>
gap> DivisorPolytope([6,6,0],[[2,-1],[-1,2],[-1,-1]]);
[ 2*x_1-x_2+6, -x_1+2*x_2+6, -x_1-x_2 ]
</Example>
See also Example 6.13 in <Cite Key="JV02"/>.
<ManSection>
<Func Name="DivisorPolytopeLatticePoints" Arg=" D, Delta, Rays "/>
<Description>
<E>Input</E>: <A>Delta</A> is the fan <Br></Br>
<A>Rays</A> is the <E>ordered</E> list of rays for <A>Delta</A> <Br></Br>
<A>D</A> is the list of coefficients for a Weil divisor.
<Br></Br>
<E>Output</E>: the list of points in <M>P_D \cap L_0^*</M>
which parameterize the elements in the Riemann-Roch space
<M>L(D)</M>, where <M>P_D</M>
is the polytope associated to the divisor <M>D</M>
(see <C>DivisorPolytope</C>).
</Description>
</ManSection>
<Example>
gap> Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
gap> Delta0:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> P_Div:=DivisorPolytopeLatticePoints(Div,Delta0,Rays);
[ [ -6, -6 ], [ -5, -5 ], [ -5, -4 ], [ -4, -5 ], [ -4, -4 ], [ -4, -3 ],
[ -4, -2 ], [ -3, -4 ], [ -3, -3 ], [ -3, -2 ], [ -3, -1 ], [ -3, 0 ],
[ -2, -4 ], [ -2, -3 ], [ -2, -2 ], [ -2, -1 ], [ -2, 0 ], [ -2, 1 ],
[ -2, 2 ], [ -1, -3 ], [ -1, -2 ], [ -1, -1 ], [ -1, 0 ], [ -1, 1 ],
[ 0, -3 ], [ 0, -2 ], [ 0, -1 ], [ 0, 0 ], [ 1, -2 ], [ 1, -1 ], [ 2, -2 ] ]
gap>
</Example>
<!--
Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
Delta0:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
P_Div:=DivisorPolytopeLatticePoints(Div,Delta0,Rays);
-->
<ManSection>
<Func Name="RiemannRochBasis" Arg=" D, Delta, Rays "/>
<Description>
<E>Input</E>: <A>Delta</A> is a complete and nonsingular fan <Br></Br>
<A>D</A> is the list of coefficients for the Weil divisor<Br></Br>
<A>Rays</A> is a list of rays for the fan used to describe
the Weil divisors.
<Br></Br>
<E>Output</E>: A basis (a list of monomials)
for the Riemann-Roch space of the divisor represented by <A>D</A>.
</Description>
</ManSection>
For details on how the Weil divisors can be expressed in terms of
the rays of the fan, please see section 3.3 in
<Cite Key="F93"/>.
This procedure does not check if the fan is complete and nonsingular.
<Example>
gap> Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
gap> Delta:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> RiemannRochBasis(Div,Delta,Rays);
[ 1/(x_1^6*x_2^6), 1/(x_1^5*x_2^5), 1/(x_1^5*x_2^4), 1/(x_1^4*x_2^5),
1/(x_1^4*x_2^4), 1/(x_1^4*x_2^3), 1/(x_1^4*x_2^2), 1/(x_1^3*x_2^4),
1/(x_1^3*x_2^3), 1/(x_1^3*x_2^2), 1/(x_1^3*x_2), 1/x_1^3, 1/(x_1^2*x_2^4),
1/(x_1^2*x_2^3), 1/(x_1^2*x_2^2), 1/(x_1^2*x_2), 1/x_1^2, x_2/x_1^2,
x_2^2/x_1^2, 1/(x_1*x_2^3), 1/(x_1*x_2^2), 1/(x_1*x_2), 1/x_1, x_2/x_1,
1/x_2^3, 1/x_2^2, 1/x_2, 1, x_1/x_2^2, x_1/x_2, x_1^2/x_2^2 ]
</Example>
<!--
Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
Delta:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
RiemannRochBasis(Div,Delta,Rays);
-->
</Section>
<Section>
<Heading>Topological invariants</Heading>
Throughout this section, <M>X(\Delta)</M> <E>must be non-singular</E>.
<ManSection>
<Func Name="EulerCharacteristic" Arg=" Delta "/>
<Description>
<E>Input</E>: <A>Delta</A> is a nonsingular fan of cones,
represented by its list of maximal cones.
<Br></Br>
<E>Output</E>: the Euler characteristic of the toric variety
<M>X(\Delta)</M>, where <M>\Delta</M> is a fan determined by
<A>Delta</A>.
</Description>
</ManSection>
<Example>
gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> EulerCharacteristic(Cones);
3
</Example>
Note: <M>X(\Delta)</M> <E>must be non-singular</E>
here.
<ManSection>
<Func Name="BettiNumberToric" Arg=" Delta, k "/>
<Description>
<E>Input</E>: <A>Delta</A> represents a nonsingular
fan <M>\Delta</M> (represented by maximal cones),
<Br></Br>
<A>k</A> is an integer.
<Br></Br>
<E>Output</E>: the <A>k</A>-th Betti number of the toric variety
<M>X(\Delta)</M>.
</Description>
</ManSection>
The <C>BettiNumberToric</C> procedure does not check if
<A>Delta</A> is nonsingular. It is possible that this procedure outputs
nonsense when <A>Delta</A> is not represented by maximal cones or
is nonsingular.
<Example>
gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> BettiNumberToric(Cones,1);
0
gap> BettiNumberToric(Cones,2);
1
gap> Cones:=[[[2,-1],[-1,1]],[[-1,1],[-1,0]],[[-1,0],[2,-1]]];;
gap> BettiNumberToric(Cones,1);
0
gap> BettiNumberToric(Cones,2);
1
</Example>
<!--
The first example is singular, second is nonsingular.
-->
Not to be confused with the Betti number of a polycyclically
presented torsion free group, already available in &GAP;.
</Section>
<Section>
<Heading>Points over a finite field</Heading>
<ManSection>
<Func Name="CardinalityOfToricVariety" Arg=" Cones q "/>
<Description>
<E>Input</E>: <A>Cones</A> is the list of maximal cones of a
fan <M>\Delta</M>, <A>q</A> is a prime power.
<Br></Br>
<E>Output</E>: The size of the set of <M>GF(q)</M>-rational
points of the toric variety <M>X(\Delta)</M>.
</Description>
</ManSection>
Note: <M>X(\Delta)</M> <E>must be non-singular</E>
here.
<Example>
gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
gap> CardinalityOfToricVariety(Cones,3);
13
gap> CardinalityOfToricVariety(Cones,4);
21
gap> CardinalityOfToricVariety(Cones,5);
31
gap> CardinalityOfToricVariety(Cones,7);
57
</Example>
<!--
Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
CardinalityOfToricVariety(Cones,3);
CardinalityOfToricVariety(Cones,4);
CardinalityOfToricVariety(Cones,5);
CardinalityOfToricVariety(Cones,7);
-->
</Section>
</Chapter>
</Body>
<Bibliography Databases="toric"/>
<TheIndex/>
</Book>
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