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Path: gap4r8 / pkg / Toric-1.9.4 / doc / toric.xml
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<?xml version="1.0" encoding="UTF-8"?>
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<!DOCTYPE Book SYSTEM "gapdoc.dtd">
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<Book Name="toric">
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<#Include SYSTEM "title.xml">
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<TableOfContents/>
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<Body>
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<Chapter><Heading>Introduction</Heading>
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<Section>
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  <Heading>Introduction to the <Package>toric</Package> package</Heading>
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<P/>
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This manual describes the <Package>toric</Package> package for working with
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toric varieties in &GAP;. Toric varieties can be dealt
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with more easily than general varieties since often
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times questions about a toric variety can be reformulated in terms
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of combinatorial geometry. 
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Some coding theory commands related to toric varieties
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are contained in the error-correcting codes 
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<Package>GUAVA</Package> package
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(for example, the command <C>ToricCode</C>). 
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We refer to the <Package>GUAVA</Package> manual
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<Cite Key="Gua05"/> and the expository 
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paper <Cite Key="JV02"/> for more details.
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33
<P/>
34
The <Package>toric</Package> package also contains
35
several commands unrelated to toric varieties
36
(mostly for list manipulations). 
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These will not be described in this documention but 
38
they are briefly documented in the <C>lib/util.gd</C>
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file.
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<P/>
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<Package>toric</Package> is implemented in the &GAP; language, 
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and runs on any system supporting &GAP;4.3 and above. 
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The <Package>toric</Package> package is loaded with the command
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<Verb> gap> LoadPackage( "toric" ); </Verb>
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<P/>
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Please send bug reports, suggestions and other comments about
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<Package>toric</Package> to 
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<Email>[email protected]</Email>.
51

52
</Section>
53
  
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<Section>
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  <Heading>Introduction to constructing toric varieties</Heading>
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57
Rather than sketch the theory of 
58
toric varieties, we refer to
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<Cite Key="JV02"/> and <Cite Key="F93"/> for details.
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However, we briefly describe some terminology and notation.
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<Subsection>
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<Heading>Generalities</Heading>
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65
Let <M>F</M> denote a field and
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<M>R=F [x_1,...,x_n]</M> be a ring in <M>n</M> variables.
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A <B>binomial equation</B>
68
in <M>R</M> is one of the form
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70
<Display>
71
x_1^{k_1}...x_n^{k_n}=x_1^{\ell_1}...x_n^{\ell_n},
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</Display>
73
where <M>k_i \geq 0,\ \ell_j \geq 0</M> are integers.
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A binomial variety
75
is a subvariety of affine <M>n</M>-space
76
<M>A_F^n</M> defined by a finite set of binomial 
77
equations (such a variety need not be normal).
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A typical ``toric variety'' is binomial, though they 
79
will be introduced via an <E>a priori</E>
80
independent construction.
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The basic idea of the construction is to
82
replace each such binomial equation as above by a relation in a 
83
semigroup contained in a lattice
84
and replace <M>R</M> by the ``group algebra'' of this 
85
semigroup. By the way, a toric variety is 
86
always normal (see for example, <Cite Key="F93"/>, 
87
page 29).
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89
</Subsection>
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<Subsection>
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<Heading>Basic combinatorial geometry constructions</Heading>
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Let <M>Q</M> denote the field of rational numbers and <M>Z</M> denote the
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set of integers. Let <M>n>1</M> denote an integer.
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97
<P/>
98
Let <Math>V=Q^n</Math> having basis
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<M>f_1=(1,0,...,0)</M>, ..., <M>f_n=(0,...,0,1)</M>.
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Let <M>L_0=Z^n\subset V</M> be the standard lattice in <M>V</M>.
101
We identify <M>V</M> and <M>L_0\otimes_{Z} Q</M>.
102
We use <M>\langle\ ,\ \rangle</M> to denote
103
the (standard) inner product on <M>V</M>. Let
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105
<Display>
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L_0^*={\rm Hom}(L_0,Z)=\{ v\in V\ |\ \langle v,w \rangle \in Z,
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\ \forall w\in L_0\}
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</Display>
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denote the <B>dual lattice</B>, so (fixing the standard basis
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<M>e_1^*</M>,...,<M>e_n^*</M> dual to the <M>f_1</M>,...,<M>f_n</M>)
111
<M>L_0^*</M> may be identified with <M>Z^n</M>.
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113
<P/>
114
A <B>cone</B> in <M>V</M> is a set <M>\sigma</M> of the form
115

116
<Display>
117
\sigma=\{a_1v_1+...+a_mv_m\ |\ a_i\geq 0\}\subset V,
118
</Display>
119
where <M>v_1,...,v_m \in V</M> is a given collection of 
120
vectors, called (semigroup) <B>generators</B> of <M>\sigma</M>. 
121
A <B>rational cone</B> is one where <M>v_1,...,v_m \in L_0</M>.
122
A <B>strongly convex</B> cone is one which contains
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no lines through the origin.
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<P/>
126
<E>
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By abuse of terminology, from now on
128
a <B>cone</B> of <M>L_0</M> is a strongly convex rational cone.
129
</E>
130
<Index>cone</Index>
131
<P/>
132
A <B>face</B> of a cone <M>\sigma</M> is either <M>\sigma</M> 
133
itself or a subset of the form <M>H\cap \sigma</M>, 
134
where <M>H</M> is a codimension one subspace of <M>V</M>
135
which intersects the cone non-trivially and such that the cone
136
is contained in exactly one of the two half-spaces determined by 
137
<M>H</M>. A <B>ray</B> (or edge) of a cone is a one-dimensional face.
138
<Index>ray</Index>
139
Typically, cones are represented in <Package>toric</Package>
140
by the list of vectors defining their rays.
141
The <B>dimension</B> of a cone is the dimension of the
142
vector space it spans.
143
The <Package>toric</Package> package can test if a 
144
given vector is in a given cone (see <C>InsideCone</C>).
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<P/>
147
If <M>\sigma</M> is a cone then the <B>dual cone</B> is defined by
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<Display>
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\sigma^* =\{w \in L_0^*\otimes Q
151
\ |\ \langle v,w \rangle \geq 0,\ \forall v\in \sigma\}.
152
</Display>
153
The <Package>toric</Package> package can test if a 
154
vector is in the dual of a given cone (see <C>InDualCone</C>).
155
<Index>dual cone</Index>
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<P/>
158
Associate to the dual cone <M>\sigma^*</M> is the semigroup 
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160
<Display>
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S_\sigma =\sigma^*\cap L_0^*
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=\{w\in L_0^*
163
\ |\ \langle v,w\rangle \geq 0,\ 
164
\forall v\in \sigma\}.
165
</Display>
166
Though <M>L_0^*</M> has $n$ generators <E>as a lattice</E>,
167
typically <M>S_\sigma</M> will have more than <M>n</M>
168
generators <E>as a semigroup</E>. 
169
The <Package>toric</Package> package can compute a 
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minimal list of semigroup generators of <M>S_\sigma</M>
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(see <C>DualSemigroupGenerators</C>).
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<Index>semigroup associated to cone</Index>
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174
<P/>
175
A fan is a collection of cones which ``fit together'' well.
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177
A <B>fan</B> in <M>L_0</M> is a set 
178
<M>\Delta=\{\sigma \}</M> of rational strongly convex
179
cones in <M>V= L_0 \otimes Q</M>
180
such that
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<List>
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184
<Item>
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if <M>\sigma \in \Delta</M> and <M>\tau \subset \sigma</M> is a face
186
of <M>\sigma</M> then <M>\tau \in \Delta</M>,
187
</Item>
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189
<Item>
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if <M>\sigma_1, \sigma_2 \in \Delta</M> then 
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the intersection <M>\sigma_1 \cap \sigma_2</M> is a face of 
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both <M>\sigma_1</M> and <M>\sigma_2</M> (and hence
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belongs to <M>\Delta</M>).
194
</Item>
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</List>
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In particular, the face of a cone in a fan is a cone is the fan.
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<P/>
198
If <M>V</M> is the (set-theoretic) union of the
199
cones in <M>\Delta</M> then we call the fan <B>complete</B>.
200
We shall assume that all fans are finite.
201
A fan is determined by its list of maximal cones.
202
<Index>fan</Index>
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204
<P/>
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<E>Notation</E>:   A fan <M>\Delta</M> is represented in
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<Package>toric</Package> as a set of maximal cones.
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For example, if <M>\Delta</M> is the fan with maximal cones
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<M>\sigma_1=Q_+\cdot f_1+Q_+\cdot (-f_1+f_2)</M>,
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<M>\sigma_2=Q_+\cdot (-f_1+f_2)+Q_+\cdot (-f_1-f_2)</M>,
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<M>\sigma_3=Q_+\cdot (-f_1-f_2)+Q_+\cdot f_1</M>,
211
then <M>\Delta</M> is represented by
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<M>[[[1,0],[-1,1]],[[-1,1],[-1,-1]],[[-1,-1],[1,0]]]</M>.
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214
<P/>
215
The <Package>toric</Package> package can compute all cones
216
in a fan of a given dimension (see <C>ConesOfFan</C>).
217
Moreover, <Package>toric</Package> can compute
218
the set of all normal vectors to the faces (i.e., hyperplanes)
219
of a cone (see <C>Faces</C>).
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<P/>
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The <B>star</B> of a cone <M>\sigma</M> in a 
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fan <M>\Delta</M> is the set <M>\Delta_\sigma</M>
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of cones in <M>\Delta</M> containing <M>\sigma</M> as a face.
225
The <Package>toric</Package> package can compute stars 
226
(see <C>ToricStar</C>).
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<Index>star</Index>
228

229
<P/>
230

231
</Subsection>
232

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<Subsection>
234
<Heading>Basic affine toric variety constructions</Heading>
235

236
Let
237
<Display>
238
R_\sigma = F [S_\sigma]
239
</Display>
240
denote the ``group algebra'' of this semigroup. It is a
241
finitely generated commutative F-algebra.
242
It is in fact integrally closed (<Cite Key="F93"/>, page 29).
243
We may interprete <M>R_\sigma</M> as a subring of 
244
<M>R=F [x_1,...,x_n]</M> as follows: First, 
245
identify each <M>e_i^*</M> with the variable <M>x_i</M>.
246
If <M>S_\sigma</M> is generated as a semigroup by 
247
vectors of the form <M>\ell_1 e_1^*+...+\ell_n e_n^*</M>,
248
where <M>\ell_i</M> is an integer, then its image in 
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<M>R</M> is generated by monomials of the 
250
form <M>x_1^{\ell_1}\dots x_n^{\ell_n}</M>.
251
The <Package>toric</Package> package can compute these
252
generating monomials (see <C>EmbeddingAffineToricVariety</C>).
253
See Lemma 2.14 in <Cite Key="JV02"/> for more details.
254
This embedding can also be used to resolve singularities -
255
see section 5 of <Cite Key="JV02"/> for more details.
256

257
<P/>
258
Let
259
<Display>
260
U_\sigma={\rm Spec}\ R_\sigma.
261
</Display>
262
This defines an <B>affine toric variety</B> (associated 
263
to <M>\sigma</M>). It is known that the coordinate ring
264
<M>R_\sigma</M> of the affine toric variety <M>U_\sigma</M>
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has the form <M>R_\sigma = F[x_1,...,x_n]/J</M>, where <M>J</M>
266
is an ideal. The <Package>toric</Package> package can compute 
267
generators of this ideal by using the <C>DualSemigroupGenerators</C>
268
and the <C>EmbeddingAffineToricVariety</C>
269
commands.
270
<Index>affine toric variety</Index>
271

272
<P/>
273
Roughly speaking, the toric variety <M>X(\Delta)</M> associated to
274
the fan <M>\Delta</M> is given by a collection of affine
275
pieces $U_{\sigma_1},U_{\sigma_2},\dots,U_{\sigma_d}$
276
which ``glue'' together (where <M>\Delta = \{\sigma_i\}</M>).  
277
The affine pieces are given by the zero
278
sets of polynomial equations in some affine spaces
279
and the gluings are given by maps 
280
<M>\phi_{i,j} : U_{\sigma_i} \rightarrow U_{\sigma_j}</M>
281
which are defined by ratios of polynomials on open subsets of the 
282
$U_{\sigma_i}$.
283
The <Package>toric</Package> package does <E>not</E> 
284
compute these gluings or work directly with these 
285
(non-affine) varieties <M>X(\Delta)</M>.
286

287
<P/>
288
A cone <M>\sigma \subset V</M> is said to be <B>nonsingular</B> if
289
it is generated by part of a basis for the lattice <M>L_0</M>.
290
A fan <M>\Delta</M> of cones is said to be <B>nonsingular</B> if
291
all its cones are nonsingular. It is known that <M>U_\sigma</M>
292
is nonsingular if and only if <M>\sigma</M> is nonsingular
293
(Proposition 2.1 in <Cite Key="F93"/>).
294
<Index>cone, nonsingular</Index>
295

296
<P/>
297
<B>Example:</B>
298
In three dimensions, consider the cones
299
<M>\sigma_{\epsilon_1,\epsilon_2,\epsilon_3,i,j}</M>
300
generated by <M>(\epsilon_1\cdot 1,\epsilon_2\cdot 1,\epsilon_3\cdot 1)</M>
301
and the standard basis vectors <M>f_i,f_j</M>,
302
where <M>\epsilon_i=\pm 1</M> and
303
<M>1\leq i\not= j\leq 3</M>. There are 8 cones per octant,
304
for a total of 64 cones. Let <M>\Delta</M> denote the
305
fan in <M>V=Q^3</M> determined by these maximal cones.
306
The toric variety <M>X(\Delta)</M> is nonsingular.
307
<P/>
308

309
</Subsection>
310

311
<Subsection>
312
<Heading>Riemann-Roch spaces and related constructions</Heading>
313

314
Although the <Package>toric</Package> package does not 
315
work directly with the toric varieties <M>X(\Delta)</M>,
316
it can compute objects associated with it.
317
For example, it can compute the 
318
Euler characteristic (see <C>EulerCharacteristic</C>), Betti numbers 
319
(see <C>BettiNumberToric</C>), and the
320
number of GF(q)-rational points 
321
(see <C>CardinalityOfToricVariety</C>) of <M>X(\Delta)</M>,
322
<E> provided <M>\Delta</M> is nonsingular.</E>
323

324
<P/>
325
For an algebraic variety <M>X</M> the group of <B>Weil divisors</B> 
326
on <M>X</M> is
327
the abelian group <M>Div(X)</M> generated (additively) by the
328
irreducible subvarieties of X of codimension <M>1</M>.
329
For a toric variety <M>X(\Delta)</M> with dense open torus <M>T</M>, a Weil divisor
330
D is <B>T-invariant</B> if <M>D=T\cdot D</M>.
331
The group of <M>T</M>-invariant Weil divisors is denoted 
332
<M>TDiv(X)</M>. This is finitely generated by an explicitly
333
given finite set of divisors 
334
<M>\{D_1,...,D_r\}</M> which correspond naturally to certain
335
cones in <M>\Delta</M> (see <Cite Key="F93"/> for details).
336
In particular, we may represent such a divisor <M>D</M> in <M>TDiv(X)</M>
337
by an <M>k</M>-tuple <M>(d_1,...,d_k)</M> of integers.
338
<Index>Weil divisors</Index>
339

340
<P/>
341
Let <M>\Delta</M> denote a fan in <M>V=Q^n</M> with rays (or edges)
342
<M>\tau_i</M>, <M>1\leq i\leq k</M>, and let 
343
<M>v_i</M> denote the first lattice point on <M>\tau_i</M>.
344
Associated to the T-invariant Weil divisor <M>D=d_1D_1+...+d_kD_k</M>,
345
is the <B>polytope</B>
346
<Display>
347
P_D = \{ x=(x_1,...,x_n)\ |\ \langle x,v_i \rangle \geq -d_i, 
348
\ \forall 1 \leq i \leq k\}.
349
</Display>
350
The <Package>toric</Package> package can compute 
351
<M>P_D</M> (see <C>DivisorPolytope</C>), as well as the set of all 
352
lattice points contained in this polytope
353
(see <C>DivisorPolytopeLatticePoints</C>).
354
Also associated to the <M>T</M>-invariant Weil divisor 
355
<M>D=d_1D_1+...+d_kD_k</M>, is the Riemann-Roch space,
356
<M>L(D)</M>. This is a space of functions on 
357
<M>X(\Delta)</M> whose zeros and poles are 
358
``controlled'' by <M>D</M> (for a more precise definition, see
359
<Cite Key="F93"/>). The <Package>toric</Package> package can compute 
360
a basis for <M>L(D)</M> (see <C>RiemannRochBasis</C>),
361
<E> provided <M>\Delta</M> is complete and nonsingular.</E>
362
<Index>polytope associated to divisor</Index>
363

364
</Subsection>
365
</Section>
366

367
</Chapter>
368

369
<Chapter>
370
<Heading>Cones and semigroups</Heading>
371

372

373
  <Section>
374
  <Heading>Cones</Heading>
375
  
376
   This section introduces the <Package>toric</Package>
377
commands which deal with cones and related
378
combinatorial-geometric objects.
379

380
Recall, a <B>cone</B> is a strongly convex polyhedral
381
cone (<Cite Key="F93"/>, page 4).
382

383
<ManSection Label="InsideCone">
384
   <Func Name="InsideCone" Arg=" v, L "/>
385
		   
386
        <Description>
387
This command returns `true` if the vector <A>v</A> belongs to the interior 
388
of the (strongly convex polyhedral)
389
cone generated by the vectors in <A>L</A>.
390
        </Description>
391
      </ManSection>			  
392

393
This procedure does not check if <A>L</A> generates a 
394
strongly convex polyhedral cone.
395

396
 <Example>
397
gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
398
gap> InsideCone(v,L);
399
false
400
gap> L:=[[1,0],[3,4]];;
401
gap> v:=[1,-7]; InsideCone(v,L);
402
[ 1, -7 ]
403
false
404
gap> v:=[4,-3]; InsideCone(v,L);
405
[ 4, -3 ]
406
false
407
gap> v:=[4,-4]; InsideCone(v,L);
408
[ 4, -4 ]
409
false
410
gap> v:=[4,1]; InsideCone(v,L);
411
[ 4, 1 ]
412
true
413
</Example>
414
<!--
415
L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; 
416
v:=[0,0,1];;
417
InsideCone(v,L);
418
false
419
L:=[[1,0],[3,4]];;
420
v:=[1,-7]; InsideCone(v,L);
421
v:=[4,-3]; InsideCone(v,L);
422
v:=[4,-4]; InsideCone(v,L);
423
v:=[4,1];  InsideCone(v,L);
424
-->
425

426
<ManSection Label="InDualCone">
427
   <Func Name="InDualCone" Arg=" v, L "/>
428
		   
429
        <Description>
430
This command
431
returns `true` if <A>v</A> belongs to the dual of the cone generated by
432
the vectors in <A>L</A>.
433
        </Description>
434
      </ManSection>			  
435

436
 <Example>
437
gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
438
gap> InDualCone(v,L);
439
true
440
gap> L:=[[1,0],[3,4]];
441
[ [ 1, 0 ], [ 3, 4 ] ]
442
gap> v:=[1,-7]; InDualCone(v,L);
443
[ 1, -7 ]
444
false
445
gap> v:=[4,-3]; InDualCone(v,L);
446
[ 4, -3 ]
447
true
448
gap> v:=[4,-4]; InDualCone(v,L);
449
[ 4, -4 ]
450
false
451
gap> v:=[4,1]; InDualCone(v,L);
452
[ 4, 1 ]
453
true
454
</Example>
455
<!--
456
L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];; v:=[0,0,1];;
457
InDualCone(v,L);
458
L:=[[1,0],[3,4]];
459
v:=[1,-7]; InDualCone(v,L);
460
v:=[4,-3]; InDualCone(v,L);
461
v:=[4,-4]; InDualCone(v,L);
462
v:=[4,1]; InDualCone(v,L);
463
--> 
464

465
<ManSection>
466
   <Func Name="PolytopeLatticePoints" Arg=" A, Perps "/>
467
		   
468
        <Description>
469
<E>Input</E>: <A>Perps</A><M>=[v_1,...,v_k]</M> is the list of ``inward normal" 
470
vectors perpendicular to the walls of a polytope <M>P</M> in the
471
vector space <M>L_0^*\otimes Q</M>,
472
<Br></Br>
473
<A>A</A><M>=[a_1,...,a_k]</M> is a k-tuple of integers, where
474
<M>a_i</M> denotes the amount the i-th ``wall" (defined by the normal <M>v_i</M>)
475
is shifted from the origin (each <M>a_i</M> is assumed non-negative).
476
<Br></Br>
477
For example, the polytope <M>P</M> with faces
478
<C>[x=0, x=a, y=0, y=b]</C>
479
has <A>Perps</A><M>=[[1,0],[-1,0],[0,1],[0,-1]]</M> and <A>A</A><M>=[0,a,0,b]</M>.
480
<Br></Br>
481
<E>Output</E>: the list of points in <M>P \cap L_0^*</M>.
482
        </Description>
483
</ManSection>			  
484

485
<Example>
486
gap> Perps:=[[1,0],[-1,0],[0,1],[0,-1]];
487
[ [ 1, 0 ], [ -1, 0 ], [ 0, 1 ], [ 0, -1 ] ]
488
gap> A:=[0,4,0,3];
489
[ 0, 4, 0, 3 ]
490
gap> PolytopeLatticePoints(A,Perps);
491
[ [ 0, 0 ], [ 0, 1 ], [ 0, 2 ], [ 0, 3 ], [ 1, 0 ], [ 1, 1 ], [ 1, 2 ],
492
  [ 1, 3 ], [ 2, 0 ], [ 2, 1 ], [ 2, 2 ], [ 2, 3 ], [ 3, 0 ], [ 3, 1 ],
493
  [ 3, 2 ], [ 3, 3 ], [ 4, 0 ], [ 4, 1 ], [ 4, 2 ], [ 4, 3 ] ]
494
gap> Length(last);
495
20
496
</Example>
497
<!--
498
Perps:=[[1,0],[-1,0],[0,1],[0,-1]];
499
A:=[0,4,0,3];
500
PolytopeLatticePoints(A,Perps);
501
Length(last);
502
-->
503

504
<ManSection>
505
   <Func Name="Faces" Arg=" Rays "/>
506
		   
507
        <Description>
508
<E>Input</E>: <A>Rays</A> is a list of rays for the fan <M>\Delta</M>
509
<Br></Br>
510
<E>Output</E>: All the normals to the faces (hyperplanes of the cone).
511
        </Description>
512

513
</ManSection>			  
514

515
<Example>
516
gap> Cones1:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
517
gap> Faces(Cones1[1]);
518
[ [ 1/2, 1 ], [ 2, 1 ] ]
519
gap> Faces(Cones1[2]);
520
[ [ -2, -1 ], [ -1, 1 ] ]
521
gap> Cones2:=[[[ 2,0,0],[0,2,0],[0,0,2]], [[2,0,0], [0,2,0], [2,-2,1],[1,2,-2]]];;
522
gap> Faces(Cones2[1]);
523
[ [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, 0, 0 ] ]
524
gap> Faces(Cones2[2]);
525
[ [ 1/3, 5/6, 1 ], [ 1/2, 0, -1 ], [ 2, 0, 1 ] ]
526
</Example>
527
<!--
528
Cones1:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
529
Faces(Cones1[1]);
530
Faces(Cones1[2]);
531
Cones2:=[[[ 2,0,0],[0,2,0],[0,0,2]], [[2,0,0], [0,2,0], [2,-2,1],[1,2,-2]]];;
532
Faces(Cones2[1]);
533
Faces(Cones2[2]);
534
-->
535

536

537

538
<ManSection>
539
   <Func Name="ConesOfFan" Arg=" Delta k "/>
540
		   
541
        <Description>
542
<E>Input</E>: <A>Delta</A> is the fan of cones,
543
<Br></Br>
544
<A>k</A> is the dimension of the cones desired.
545
<Br></Br>
546
<E>Output</E>: The <A>k</A>-dimensional cones in the fan.
547
        </Description>
548

549
</ManSection>			  
550

551

552

553
<ManSection>
554
<Func Name="NumberOfConesOfFan" Arg=" Delta k "/>
555
		   
556
        <Description>
557
<E>Input</E>: <A>Delta</A> is the fan of cones in <M>V=Q^n</M>,
558
<Br></Br>
559
 <A>k</A> is the dimension of the cones counted.
560
<Br></Br>
561
<E>Output</E>: The number of <A>k</A>-dimensional cones in the fan.
562
        </Description>
563
</ManSection>			  
564

565
Idea:   The fan <A>Delta</A> is represented as a set of maximal cones.
566
For each maximal cone, look at the <A>k</A>-dimensional faces
567
obtained by taking <M>n</M> choose <A>k</A> subsets of the
568
rays describing the cone. Certain of these <A>k</A>-subsets 
569
yield the desired cones.
570

571
<Example>
572
gap> Delta0:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
573
gap> NumberOfConesOfFan(Delta0,2);
574
6
575
gap> ConesOfFan(Delta0,2);
576
[ [ [ 0, 0, 2 ], [ 0, 2, 0 ] ], [ [ 0, 0, 2 ], [ 2, 0, 0 ] ], 
577
  [ [ 0, 2, 0 ], [ 1, 2, -2 ] ], [ [ 0, 2, 0 ], [ 2, -2, 1 ] ],
578
  [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 1, 2, -2 ], [ 2, -2, 1 ] ] ]
579
gap> ConesOfFan(Delta0,1);
580
[ [ [ 0, 0, 2 ] ], [ [ 0, 2, 0 ] ], [ [ 1, 2, -2 ] ], 
581
  [ [ 2, -2, 1 ] ], [ [ 2, 0, 0 ] ] ]
582
gap> NumberOfConesOfFan(Delta0,1);
583
5
584
</Example>
585
<!--
586
Delta0:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
587
NumberOfConesOfFan(Delta0,2);
588
ConesOfFan(Delta0,2);
589
ConesOfFan(Delta0,1);
590
NumberOfConesOfFan(Delta0,1);
591

592
-->
593

594

595
<ManSection>
596
   <Func Name="ToricStar" Arg=" sigma, Delta "/>
597
		   
598
        <Description>
599
<E>Input</E>:  <A>sigma</A> is a cone in the fan,
600
represented by its set of maximal (i.e., highest dimensional) cones.
601
<Br></Br>
602
<A>Delta</A> is the fan of cones in <M>V=Q^n</M>.
603
<Br></Br>
604
<E>Output</E>: The star of the cone <A>sigma</A> in <A>Delta</A>,
605
i.e., the cones <M>\tau</M> which have <A>sigma</A> as a face. 
606

607
        </Description>
608

609
</ManSection>			  
610

611
<Example>
612
gap> MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ], 
613
>                 [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
614
gap> #this is the set of maximal cones in the fan Delta
615
gap> ToricStar([[1,0]],MaxCones);
616
[  ]
617
gap> ToricStar([[2,0,0],[0,2,0]],MaxCones);
618
[ [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ],
619
  [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 2, -2, 1 ], [ 1, 2, -2 ] ] ]
620
gap> MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[1,1,-2] ] ];;
621
gap> ToricStar([[2,0,0],[0,2,0]],MaxCones);
622
[ [ [ 0, 2, 0 ], [ 2, 0, 0 ] ], [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 0, 0, 2 ] ],
623
  [ [ 2, 0, 0 ], [ 0, 2, 0 ], [ 1, 1, -2 ] ] ]
624
gap> ToricStar([[1,0]],MaxCones);
625
[  ]
626
</Example>
627
<!--
628
MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[2,-2,1],[1,2,-2] ] ];;
629
ToricStar([[1,0]],MaxCones);
630
ToricStar([[2,0,0],[0,2,0]],MaxCones);
631
MaxCones:=[ [ [2,0,0],[0,2,0],[0,0,2] ], [ [2,0,0],[0,2,0],[1,1,-2] ] ];;
632
ToricStar([[2,0,0],[0,2,0]],MaxCones);
633
ToricStar([[1,0]],MaxCones);
634
-->
635

636
</Section>
637

638
<Section>
639
  <Heading>Semigroups</Heading>
640

641
<ManSection>
642
   <Func Name="DualSemigroupGenerators" Arg=" L "/>
643
		   
644
        <Description>
645
<E>Input</E>: <A>L</A> is a list of integral <M>n</M>-vectors generating a 
646
cone <M>\sigma</M>. 
647
<Br></Br>
648
<E>Output</E>: the generators of <M>S_\sigma</M>,  
649
<P/>
650
Idea:  let <M>M</M> be the maximum of the absolute values of the 
651
coordinates of the <A>L</A>[i]'s, for each vector <M>v</M> in 
652
<M>[1..M]^n</M>, test if <M>v</M> is in the dual cone 
653
<M>\sigma^*</M>. If so, 
654
add <M>v</M> to list of possible generators. 
655
Once this for loop is finished, one can check 
656
this list for redundant generators. The  
657
trick is to simply omit those elements 
658
which are of the form <M>d_1+d_2</M>, where <M>d_1</M> and <M>d_2</M> 
659
are ``small" elements in the integral dual cone. 
660
<P/>
661
This program is not very efficient and should not be used 
662
in ``large examples'' involving semigroups with ``many'' generators. 
663
For example, if you take
664
<A>L:=[[1,2,3,4],[0,1,0,7],[3,1,0,2],[0,0,1,0]];</A> then
665
<C>DualSemigroupGenerators(L);</C> can exhaust GAP's memory
666
allocation.
667
</Description>
668
</ManSection>			  
669

670
<Example>
671
gap> L:=[[1,0],[3,4]];; DualSemigroupGenerators([[1,0],[3,4]]);
672
[ [ 0, 0 ], [ 0, 1 ], [ 1, 0 ], [ 2, -1 ], [ 3, -2 ], [ 4, -3 ] ]
673
gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
674
gap> DualSemigroupGenerators(L);
675
[ [ 0, 0, 0 ], [ 0, 0, 1 ], [ 0, 1, 0 ], [ 1, -1, 0 ], [ 1, 0, -1 ] ]
676
</Example>
677
<!--
678
L:=[[1,0],[3,4]];; DualSemigroupGenerators([[1,0],[3,4]]);
679
L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
680
DualSemigroupGenerators(L);
681
-->
682

683
</Section>
684

685
</Chapter>
686

687
<Chapter>
688
<Heading>Affine toric varieties</Heading>
689

690
This chapter concerns <Package>toric</Package>
691
commands which deal with the coordinate rings of
692
affine toric varieties <M>U_\sigma</M>.
693

694
<Section>
695
  <Heading>Ideals defining affine toric varieties</Heading>
696
  
697
<ManSection>
698
           <Func Name="EmbeddingAffineToricVariety" Arg=" L "/>
699
		   
700
        <Description>
701
<E>Input</E>: <A>L</A> is a list generating a cone 
702
(as in <C>DualSemigroupGenerators</C>).
703
<Br></Br>
704
<E>Output</E>: the toroidal embedding of <M>X=Spec(I)</M>, where I is the
705
ideal of the affine toric variety (given as a list of multinomials).
706
        </Description>
707
</ManSection>			  
708

709
<Example>
710
gap> phi:=EmbeddingAffineToricVariety([[1,0],[3,4]]);
711
[ x_2, x_1, x_1^2/x_4, x_1^3/x_4^2, x_1^4/x_4^3 ]
712
gap> L:=[[1,0,0],[1,1,0],[1,1,1],[1,0,1]];;
713
gap> phi:=EmbeddingAffineToricVariety(L);
714
[ x_3, x_2, x_1/x_5, x_1/x_6 ]
715
</Example>
716

717

718
  </Section>
719
</Chapter>
720

721
 <Chapter>
722
<Heading>Toric varieties <M>X(\Delta)</M> </Heading>
723

724
This chapter concerns <Package>toric</Package>
725
commands which deal with certain objects associated to the
726
(non-affine) toric varieties <M>X(\Delta)</M>.
727

728
<Section>
729
  <Heading>Riemann-Roch spaces</Heading>
730

731
Let <M>\Delta</M> denote a complete nonsingular fan.
732

733
<ManSection>
734
           <Func Name="DivisorPolytope" Arg=" D, Rays "/>
735
		   
736
        <Description>
737
<E>Input</E>: <A>Rays</A> is the list of smallest integer
738
vectors in the rays for the fan <M>\Delta</M>
739
which determine the Weil divisors of <M>X(\Delta)</M>.
740
<Br></Br>
741
<A>D</A> is the list of coefficients for the a Weil divisor.
742
<Br></Br>
743
<E>Output</E>: the linear expressions in the affine coordinates of the
744
space of the cone which must be positive for a point to be in the
745
desired polytope.
746
        </Description>
747
</ManSection>			  
748

749
<Example>
750
gap> DivisorPolytope([6,6,0],[[2,-1],[-1,2],[-1,-1]]);
751
[ 2*x_1-x_2+6, -x_1+2*x_2+6, -x_1-x_2 ]
752
</Example>
753

754
See also Example 6.13 in <Cite Key="JV02"/>.
755

756
<ManSection>
757
           <Func Name="DivisorPolytopeLatticePoints" Arg=" D, Delta, Rays "/>
758
		   
759
        <Description>
760
<E>Input</E>: <A>Delta</A> is the fan <Br></Br>
761
       <A>Rays</A> is the <E>ordered</E> list of rays for <A>Delta</A> <Br></Br>
762
   <A>D</A> is the list of coefficients for a Weil divisor.
763
<Br></Br>
764
<E>Output</E>: the list of points in <M>P_D \cap L_0^*</M>
765
which parameterize the elements in the Riemann-Roch space
766
<M>L(D)</M>, where <M>P_D</M>
767
is the polytope associated to the divisor <M>D</M>
768
(see <C>DivisorPolytope</C>).
769
        </Description>
770
</ManSection>			  
771

772
<Example>
773
gap> Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
774
gap> Delta0:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
775
gap> P_Div:=DivisorPolytopeLatticePoints(Div,Delta0,Rays);
776
[ [ -6, -6 ], [ -5, -5 ], [ -5, -4 ], [ -4, -5 ], [ -4, -4 ], [ -4, -3 ],
777
  [ -4, -2 ], [ -3, -4 ], [ -3, -3 ], [ -3, -2 ], [ -3, -1 ], [ -3, 0 ],
778
  [ -2, -4 ], [ -2, -3 ], [ -2, -2 ], [ -2, -1 ], [ -2, 0 ], [ -2, 1 ],
779
  [ -2, 2 ], [ -1, -3 ], [ -1, -2 ], [ -1, -1 ], [ -1, 0 ], [ -1, 1 ],
780
  [ 0, -3 ], [ 0, -2 ], [ 0, -1 ], [ 0, 0 ], [ 1, -2 ], [ 1, -1 ], [ 2, -2 ] ]
781
</Example>
782
<!--
783
Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
784
Delta0:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
785
P_Div:=DivisorPolytopeLatticePoints(Div,Delta0,Rays);
786
-->
787

788
<ManSection>
789
           <Func Name="RiemannRochBasis" Arg=" D, Delta, Rays "/>
790
		   
791
        <Description>
792
<E>Input</E>: <A>Delta</A> is a complete and nonsingular fan <Br></Br>
793
<A>D</A> is the list of coefficients for the Weil divisor<Br></Br>
794
<A>Rays</A> is a list of rays for the fan used to describe 
795
the Weil divisors.
796
<Br></Br>
797
<E>Output</E>: A basis (a list of monomials)
798
for the Riemann-Roch space of the divisor represented by <A>D</A>.
799
        </Description>
800
</ManSection>			  
801

802
For details on how the Weil divisors can be expressed in terms of
803
the rays of the fan, please see section 3.3 in
804
<Cite Key="F93"/>.
805

806
This procedure does not check if the fan is complete and nonsingular.
807

808
<Example>
809
gap> Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
810
gap> Delta0:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
811
gap> RiemannRochBasis(Div,Delta0,Rays);
812
[ 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),
813
  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),
814
  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),
815
  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,
816
  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,
817
  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 ]
818
</Example>
819
<!--
820
Div:=[6,6,0];; Rays:=[[2,-1],[-1,2],[-1,-1]];;
821
Delta:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
822
RiemannRochBasis(Div,Delta,Rays);
823
-->
824

825
 </Section>
826

827
<Section>
828
  <Heading>Topological invariants</Heading>
829

830
Throughout this section, <M>X(\Delta)</M> <E>must be non-singular</E>.
831

832
<ManSection>
833
           <Func Name="EulerCharacteristic" Arg=" Delta "/>
834
		   
835
        <Description>
836
<E>Input</E>: <A>Delta</A> is a nonsingular fan of cones,
837
represented by its list of maximal cones.
838
<Br></Br>
839
<E>Output</E>: the Euler characteristic of the toric variety 
840
<M>X(\Delta)</M>, where <M>\Delta</M> is a fan determined by 
841
<A>Delta</A>.
842
        </Description>
843
</ManSection>			  
844

845
<Example>
846
gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
847
gap> EulerCharacteristic(Cones);
848
3
849
</Example>
850

851
Note: <M>X(\Delta)</M> <E>must be non-singular</E>
852
here.
853

854
<ManSection>
855
           <Func Name="BettiNumberToric" Arg=" Delta, k "/>
856
		   
857
        <Description>
858
<E>Input</E>: <A>Delta</A> represents a nonsingular 
859
fan <M>\Delta</M> (represented by maximal cones),
860
<Br></Br>
861
     <A>k</A> is an integer.
862
<Br></Br>
863
<E>Output</E>: the <A>k</A>-th Betti number of the toric variety 
864
<M>X(\Delta)</M>.
865

866
</Description>
867
</ManSection>			  
868

869
The <C>BettiNumberToric</C> procedure does not check if 
870
<A>Delta</A> is nonsingular. It is possible that this procedure outputs
871
nonsense when <A>Delta</A> is not represented by maximal cones or 
872
is nonsingular.
873

874
<Example>
875
gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
876
gap> BettiNumberToric(Cones,1);
877
0
878
gap> BettiNumberToric(Cones,2);
879
1
880
gap> Cones:=[[[2,-1],[-1,1]],[[-1,1],[-1,0]],[[-1,0],[2,-1]]];;
881
gap> BettiNumberToric(Cones,1);
882
0
883
gap> BettiNumberToric(Cones,2);
884
1
885
</Example>
886
<!--
887
The first example is singular, second is nonsingular.
888

889
-->
890

891
Not to be confused with the Betti number of a polycyclically 
892
presented torsion free group, already available in &GAP;.
893

894
 </Section>
895

896
<Section>
897
  <Heading>Points over a finite field</Heading>
898

899

900
<ManSection>
901
           <Func Name="CardinalityOfToricVariety" Arg=" Cones q "/>
902
		   
903
        <Description>
904
<E>Input</E>: <A>Cones</A> is the list of maximal cones of a 
905
fan <M>\Delta</M>, <A>q</A> is a prime power.
906
<Br></Br>
907
<E>Output</E>: The size of the set of <M>GF(q)</M>-rational 
908
points of the toric variety <M>X(\Delta)</M>.
909
        </Description>
910
</ManSection>			  
911

912
Note: <M>X(\Delta)</M> <E>must be non-singular</E>
913
here.
914

915
<Example>
916
gap> Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
917
gap> CardinalityOfToricVariety(Cones,3);
918
13
919
gap> CardinalityOfToricVariety(Cones,4);
920
21
921
gap> CardinalityOfToricVariety(Cones,5);
922
31
923
gap> CardinalityOfToricVariety(Cones,7);
924
57
925
</Example>
926
<!--
927
Cones:=[[[2,-1],[-1,2]],[[-1,2],[-1,-1]],[[-1,-1],[2,-1]]];;
928
CardinalityOfToricVariety(Cones,3);
929
CardinalityOfToricVariety(Cones,4);
930
CardinalityOfToricVariety(Cones,5);
931
CardinalityOfToricVariety(Cones,7);
932
-->
933

934
 </Section>
935

936
</Chapter>
937

938
</Body>
939
<Bibliography Databases="toric"/>
940
<TheIndex/>                                               
941
</Book>
942

943
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