chap1.txt

  
  1 Introduction
  
  
  1.1 Introduction to the toric package
  
  This  manual describes the toric 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 GUAVA package
  (for  example,  the  command ToricCode). We refer to the GUAVA manual [CM09]
  and the expository paper [JV02] for more details.
  
  The  toric  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 lib/util.gd file.
  
  toric  is implemented in the GAP language, and runs on any system supporting
  GAP4.3 and above. The toric package is loaded with the command
  
   gap> LoadPackage( "toric" ); 
  
  Please  send  bug  reports,  suggestions  and  other comments about toric to
  mailto:[email protected].
  
  
  1.2 Introduction to constructing toric varieties
  
  Rather  than  sketch  the  theory of toric varieties, we refer to [JV02] and
  [Ful93]  for  details.  However,  we  briefly  describe some terminology and
  notation.
  
  
  1.2-1 Generalities
  
  Let  F  denote  a  field  and  R=F [x_1,...,x_n] be a ring in n variables. A
  binomial equation in R is one of the form
  
  
        x_1^{k_1}...x_n^{k_n}=x_1^{\ell_1}...x_n^{\ell_n}, 
  
  
  where  k_i  ≥ 0, ℓ_j ≥ 0 are integers. A binomial variety is a subvariety of
  affine  n-space  A_F^n 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 a priori 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 R by the
  ``group  algebra''  of this semigroup. By the way, a toric variety is always
  normal (see for example, [Ful93], page 29).
  
  
  1.2-2 Basic combinatorial geometry constructions
  
  Let Q denote the field of rational numbers and Z denote the set of integers.
  Let n>1 denote an integer.
  
  Let V=Q^n having basis f_1=(1,0,...,0), ..., f_n=(0,...,0,1). Let L_0=Z^n⊂ V
  be  the  standard  lattice in V. We identify V and L_0⊗_Z Q. We use ⟨ , ⟩ to
  denote the (standard) inner product on V. Let
  
  
        L_0^*={\rm Hom}(L_0,Z)=\{ v\in V\ |\ \langle v,w \rangle \in Z, \
        \forall w\in L_0\} 
  
  
  denote  the dual lattice, so (fixing the standard basis e_1^*,...,e_n^* dual
  to the f_1,...,f_n) L_0^* may be identified with Z^n.
  
  A cone in V is a set σ of the form
  
  
        \sigma=\{a_1v_1+...+a_mv_m\ |\ a_i\geq 0\}\subset V, 
  
  
  where  v_1,...,v_m  ∈ V is a given collection of vectors, called (semigroup)
  generators  of σ. A rational cone is one where v_1,...,v_m ∈ L_0. A strongly
  convex cone is one which contains no lines through the origin.
  
    By  abuse  of  terminology, from now on a cone of L_0 is a strongly convex
  rational cone. 
  
  A  face of a cone σ is either σ itself or a subset of the form H∩ σ, where H
  is  a  codimension one subspace of V which intersects the cone non-trivially
  and  such  that  the cone is contained in exactly one of the two half-spaces
  determined  by  H.  A  ray  (or  edge)  of a cone is a one-dimensional face.
  Typically,  cones  are  represented in toric by the list of vectors defining
  their  rays. The dimension of a cone is the dimension of the vector space it
  spans.  The toric package can test if a given vector is in a given cone (see
  InsideCone).
  
  If σ is a cone then the dual cone is defined by
  
  
        \sigma^* =\{w \in L_0^*\otimes Q \ |\ \langle v,w \rangle \geq 0,\
        \forall v\in \sigma\}. 
  
  
  The  toric  package can test if a vector is in the dual of a given cone (see
  InDualCone).
  
  Associate to the dual cone σ^* is the semigroup
  
  
        S_\sigma =\sigma^*\cap L_0^* =\{w\in L_0^* \ |\ \langle v,w\rangle
        \geq 0,\ \forall v\in \sigma\}. 
  
  
  Though  L_0^*  has $n$ generators as a lattice, typically S_σ will have more
  than  n  generators  as a semigroup. The toric package can compute a minimal
  list of semigroup generators of S_σ (see DualSemigroupGenerators).
  
  A  fan is a collection of cones which ``fit together'' well. A fan in L_0 is
  a set ∆={σ } of rational strongly convex cones in V= L_0 ⊗ Q such that
  
      if σ ∈ ∆ and τ ⊂ σ is a face of σ then τ ∈ ∆,
  
      if  σ_1, σ_2 ∈ ∆ then the intersection σ_1 ∩ σ_2 is a face of both σ_1
        and σ_2 (and hence belongs to ∆).
  
  In particular, the face of a cone in a fan is a cone is the fan.
  
  If  V  is  the  (set-theoretic) union of the cones in ∆ then we call the fan
  complete.  We  shall assume that all fans are finite. A fan is determined by
  its list of maximal cones.
  
  Notation:  A  fan  ∆  is represented in toric as a set of maximal cones. For
  example,  if  ∆  is the fan with maximal cones σ_1=Q_+⋅ f_1+Q_+⋅ (-f_1+f_2),
  σ_2=Q_+⋅ (-f_1+f_2)+Q_+⋅ (-f_1-f_2), σ_3=Q_+⋅ (-f_1-f_2)+Q_+⋅ f_1, then ∆ is
  represented by [[[1,0],[-1,1]],[[-1,1],[-1,-1]],[[-1,-1],[1,0]]].
  
  The  toric  package can compute all cones in a fan of a given dimension (see
  ConesOfFan).  Moreover,  toric  can compute the set of all normal vectors to
  the faces (i.e., hyperplanes) of a cone (see Faces).
  
  The star of a cone σ in a fan ∆ is the set ∆_σ of cones in ∆ containing σ as
  a face. The toric package can compute stars (see ToricStar).
  
  
  1.2-3 Basic affine toric variety constructions
  
  Let
  
  
        R_\sigma = F [S_\sigma] 
  
  
  denote  the  ``group algebra'' of this semigroup. It is a finitely generated
  commutative  F-algebra.  It is in fact integrally closed ([Ful93], page 29).
  We  may  interprete R_σ as a subring of R=F [x_1,...,x_n] as follows: First,
  identify  each  e_i^*  with  the  variable  x_i.  If  S_σ  is generated as a
  semigroup  by  vectors  of the form ℓ_1 e_1^*+...+ℓ_n e_n^*, where ℓ_i is an
  integer,  then  its  image  in  R  is  generated  by  monomials  of the form
  x_1^ℓ_1dots   x_n^ℓ_n.  The  toric  package  can  compute  these  generating
  monomials  (see  EmbeddingAffineToricVariety).  See Lemma 2.14 in [JV02] for
  more details. This embedding can also be used to resolve singularities - see
  section 5 of [JV02] for more details.
  
  Let
  
  
        U_\sigma={\rm Spec}\ R_\sigma. 
  
  
  This defines an affine toric variety (associated to σ). It is known that the
  coordinate  ring  R_σ  of  the  affine  toric variety U_σ has the form R_σ =
  F[x_1,...,x_n]/J,  where  J  is  an  ideal.  The  toric  package can compute
  generators  of  this  ideal  by  using  the  DualSemigroupGenerators and the
  EmbeddingAffineToricVariety commands.
  
  Roughly speaking, the toric variety X(∆) associated to the fan ∆ is given by
  a collection of affine pieces $U_{\sigma_1},U_{\sigma_2},\dots,U_{\sigma_d}$
  which  ``glue''  together  (where ∆ = {σ_i}). The affine pieces are given by
  the  zero sets of polynomial equations in some affine spaces and the gluings
  are  given  by  maps  ϕ_i,j  :  U_σ_i → U_σ_j which are defined by ratios of
  polynomials  on  open  subsets of the $U_{\sigma_i}$. The toric package does
  not compute these gluings or work directly with these (non-affine) varieties
  X(∆).
  
  A cone σ ⊂ V is said to be nonsingular if it is generated by part of a basis
  for  the  lattice L_0. A fan ∆ of cones is said to be nonsingular if all its
  cones  are nonsingular. It is known that U_σ is nonsingular if and only if σ
  is nonsingular (Proposition 2.1 in [Ful93]).
  
  Example: In three dimensions, consider the cones σ_ϵ_1,ϵ_2,ϵ_3,i,j generated
  by  (ϵ_1⋅  1,ϵ_2⋅  1,ϵ_3⋅  1)  and the standard basis vectors f_i,f_j, where
  ϵ_i=±  1  and 1≤ inot= j≤ 3. There are 8 cones per octant, for a total of 64
  cones.  Let ∆ denote the fan in V=Q^3 determined by these maximal cones. The
  toric variety X(∆) is nonsingular.
  
  
  1.2-4 Riemann-Roch spaces and related constructions
  
  Although  the  toric package does not work directly with the toric varieties
  X(∆), it can compute objects associated with it. For example, it can compute
  the  Euler  characteristic  (see  EulerCharacteristic),  Betti  numbers (see
  BettiNumberToric),   and   the   number   of   GF(q)-rational   points  (see
  CardinalityOfToricVariety) of X(∆),  provided ∆ is nonsingular.
  
  For  an  algebraic  variety X the group of Weil divisors on X is the abelian
  group  Div(X) generated (additively) by the irreducible subvarieties of X of
  codimension  1.  For  a  toric  variety X(∆) with dense open torus T, a Weil
  divisor  D  is T-invariant if D=T⋅ D. The group of T-invariant Weil divisors
  is denoted TDiv(X). This is finitely generated by an explicitly given finite
  set of divisors {D_1,...,D_r} which correspond naturally to certain cones in
  ∆  (see [Ful93] for details). In particular, we may represent such a divisor
  D in TDiv(X) by an k-tuple (d_1,...,d_k) of integers.
  
  Let  ∆  denote a fan in V=Q^n with rays (or edges) τ_i, 1≤ i≤ k, and let v_i
  denote  the  first  lattice point on τ_i. Associated to the T-invariant Weil
  divisor D=d_1D_1+...+d_kD_k, is the polytope
  
  
        P_D = \{ x=(x_1,...,x_n)\ |\ \langle x,v_i \rangle \geq -d_i, \
        \forall 1 \leq i \leq k\}. 
  
  
  The  toric package can compute P_D (see DivisorPolytope), as well as the set
  of    all    lattice    points    contained    in    this    polytope   (see
  DivisorPolytopeLatticePoints).  Also  associated  to  the  T-invariant  Weil
  divisor  D=d_1D_1+...+d_kD_k,  is  the  Riemann-Roch  space, L(D). This is a
  space  of  functions  on  X(∆) whose zeros and poles are ``controlled'' by D
  (for  a more precise definition, see [Ful93]). The toric package can compute
  a  basis  for  L(D)  (see  RiemannRochBasis),    provided  ∆ is complete and
  nonsingular.