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 [email protected].
Rather than sketch the theory of toric varieties, we refer to [JV02] and [Ful93] for details. However, we briefly describe some terminology and notation.
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 \(k_i \geq 0,\ \ell_j \geq 0\) 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).
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 \(L_0=Z^n\subset V\) be the standard lattice in \(V\). We identify \(V\) and \(L_0\otimes_{Z} Q\). We use \(\langle\ ,\ \rangle\) 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 \(\sigma\) of the form
\[ \sigma=\{a_1v_1+...+a_mv_m\ |\ a_i\geq 0\}\subset V, \]
where \(v_1,...,v_m \in V\) is a given collection of vectors, called (semigroup) generators of \(\sigma\). A rational cone is one where \(v_1,...,v_m \in L_0\). 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 \(\sigma\) is either \(\sigma\) itself or a subset of the form \(H\cap \sigma\), 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 \(\sigma\) 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 \(\sigma^*\) 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 \(S_\sigma\) will have more than \(n\) generators as a semigroup. The toric package can compute a minimal list of semigroup generators of \(S_\sigma\) (see DualSemigroupGenerators).
A fan is a collection of cones which ``fit together'' well. A fan in \(L_0\) is a set \(\Delta=\{\sigma \}\) of rational strongly convex cones in \(V= L_0 \otimes Q\) such that
if \(\sigma \in \Delta\) and \(\tau \subset \sigma\) is a face of \(\sigma\) then \(\tau \in \Delta\),
if \(\sigma_1, \sigma_2 \in \Delta\) then the intersection \(\sigma_1 \cap \sigma_2\) is a face of both \(\sigma_1\) and \(\sigma_2\) (and hence belongs to \(\Delta\)).
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 \(\Delta\) 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 \(\Delta\) is represented in toric as a set of maximal cones. For example, if \(\Delta\) is the fan with maximal cones \(\sigma_1=Q_+\cdot f_1+Q_+\cdot (-f_1+f_2)\), \(\sigma_2=Q_+\cdot (-f_1+f_2)+Q_+\cdot (-f_1-f_2)\), \(\sigma_3=Q_+\cdot (-f_1-f_2)+Q_+\cdot f_1\), then \(\Delta\) 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 \(\sigma\) in a fan \(\Delta\) is the set \(\Delta_\sigma\) of cones in \(\Delta\) containing \(\sigma\) as a face. The toric package can compute stars (see ToricStar).
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 \(R_\sigma\) as a subring of \(R=F [x_1,...,x_n]\) as follows: First, identify each \(e_i^*\) with the variable \(x_i\). If \(S_\sigma\) is generated as a semigroup by vectors of the form \(\ell_1 e_1^*+...+\ell_n e_n^*\), where \(\ell_i\) is an integer, then its image in \(R\) is generated by monomials of the form \(x_1^{\ell_1}\dots x_n^{\ell_n}\). 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 \(\sigma\)). It is known that the coordinate ring \(R_\sigma\) of the affine toric variety \(U_\sigma\) has the form \(R_\sigma = F[x_1,...,x_n]/J\), 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 \(X(\Delta)\) associated to the fan \(\Delta\) is given by a collection of affine pieces $U_{\sigma_1},U_{\sigma_2},\dots,U_{\sigma_d}$ which ``glue'' together (where \(\Delta = \{\sigma_i\}\)). The affine pieces are given by the zero sets of polynomial equations in some affine spaces and the gluings are given by maps \(\phi_{i,j} : U_{\sigma_i} \rightarrow U_{\sigma_j}\) 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 \(X(\Delta)\).
A cone \(\sigma \subset V\) is said to be nonsingular if it is generated by part of a basis for the lattice \(L_0\). A fan \(\Delta\) of cones is said to be nonsingular if all its cones are nonsingular. It is known that \(U_\sigma\) is nonsingular if and only if \(\sigma\) is nonsingular (Proposition 2.1 in [Ful93]).
Example: In three dimensions, consider the cones \(\sigma_{\epsilon_1,\epsilon_2,\epsilon_3,i,j}\) generated by \((\epsilon_1\cdot 1,\epsilon_2\cdot 1,\epsilon_3\cdot 1)\) and the standard basis vectors \(f_i,f_j\), where \(\epsilon_i=\pm 1\) and \(1\leq i\not= j\leq 3\). There are 8 cones per octant, for a total of 64 cones. Let \(\Delta\) denote the fan in \(V=Q^3\) determined by these maximal cones. The toric variety \(X(\Delta)\) is nonsingular.
Although the toric package does not work directly with the toric varieties \(X(\Delta)\), 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 \(X(\Delta)\), provided \(\Delta\) 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 \(X(\Delta)\) with dense open torus \(T\), a Weil divisor D is T-invariant if \(D=T\cdot D\). 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 \(\Delta\) (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 \(\Delta\) denote a fan in \(V=Q^n\) with rays (or edges) \(\tau_i\), \(1\leq i\leq k\), and let \(v_i\) denote the first lattice point on \(\tau_i\). 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 \(X(\Delta)\) 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 \(\Delta\) is complete and nonsingular.
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