Canonical toric Fano threefolds

An inductive approach to classifying toric Fano varieties is given. As an application of this technique, we present a classification of the toric Fano threefolds with at worst canonical singularities. Up to isomorphism, there are 674,688 such varieties.

A. M. Kasprzyk that there are (up to isomorphism) 634 varieties, of which 233 are Q-factorial and 100 are Gorenstein.
All the above classifications are subsets of a more general case: toric Fano varieties with at worst canonical singularities. Here the surface case reduces to the Gorenstein case. This paper describes an inductive approach to achieving a classification in higher dimensions. As an application, the classification for threefolds is calculated. There are 674,688 isomorphism classes. As well as encapsulating the three-dimensional classifications mentioned above, it is worth observing that 12,190 of the resulting varieties are Q-factorial (of which the Picard number is bounded by ρ ≤ 7). The classification is available online via the Graded Ring Database ( [Bro07]) at http://grdb.lboro.ac.uk/.
The various classifications are summarised in Table 1.

Fano Polytopes
A toric variety is a normal variety X that contains an algebraic torus as a dense open subset, together with an action of the torus on X that extends the natural action of the torus on itself. For further details see [Oda78,Dan78,Ful93]. We shall briefly review the properties we need, and in so doing fix our notation. Let M ∼ = Z n be the lattice of characters of the torus, with dual lattice N := Hom(M, Z). Every toric variety X of dimension n has an associated fan ∆ in N R := N ⊗ Z R. The converse also holds; to any fan ∆ there is an associated toric variety X(∆). Let {ρ i } i∈I be the set of rays of ∆. For each i ∈ I there exists a unique primitive lattice element of ρ i , which by a traditional abuse of notation we continue to denote ρ i . X is Fano if and only if {ρ i } i∈I correspond to the vertices of a convex polytope in N R (see, for example, [Dan78]).
A normal variety X is Q-factorial if every prime divisor Γ ⊂ X has a positive integer multiple cΓ that is a Cartier divisor. Once again, for the toric case there exists a well-known description in terms of the fan. The toric variety X is Q-factorial if and only if the fan ∆ is simplicial.
We say that a fan ∆ is terminal if each cone σ ∈ ∆ satisfies the following: (i) The rays ρ 1 , . . . , ρ k of σ are contained in an affine hyperplane H : (u(v) = 1) for some u ∈ M Q .
A toric variety X is terminal (i.e., has at worst terminal singularities) if and only if the fan ∆ is terminal. Relaxing condition (ii) slightly to allow lattice points on H, one obtains the definition of a canonical fan. X has (at worst) canonical singularities if and only if the fan ∆ is canonical ( [Rei83]).
Definition 2.1 Let P ⊂ N R be a convex lattice polytope containing only the origin as a strictly interior lattice point (i.e., P • ∩ N = {0}). We call such a polytope Fano. If in addition the only boundary lattice points of P are the vertices (i.e., ∂P ∩ N = vert P), then we call P a terminal Fano polytope. Otherwise we call P a canonical Fano polytope.
Clearly there is an equivalence between terminal (resp. canonical) Fano polytopes and toric Fano varieties with at worst terminal (resp. canonical) singularities. Two toric Fano n-folds are isomorphic if and only if the corresponding Fano polytopes are unimodular equivalent, i.e., equivalent up to a linear unimodular transformation from GL(n, Z).
In [Kas06a] a classification of toric Fano threefolds with at worst terminal singularities was given. The method employed relied on an approach first outlined in [BB]. It depends on the polytopal description of a toric Fano variety and can be summarised in two steps: (i) classify all the "minimal" polytopes; (ii) inductively "grow" these minimal polytopes.
Let us explain this algorithm in more detail. First we shall define what we mean by minimal.

Definition 2.2
Let P be a canonical (resp. terminal) Fano n-tope. We say that P is minimal if, for all ρ ∈ vert P, the polytope conv(P ∩ N \ {ρ}) obtained by subtracting ρ from P is not a canonical (resp. terminal) Fano n-tope.
Notice that in the canonical case we are only required to check that the origin is not contained in the interior of any of the smaller polytopes obtained by subtracting a vertex. Our use of Fano and minimal will often be relative to some obvious subspace. Such occurrences should not cause any confusion. This is a common theme when considering lattice polytopes: for example, when talking about the volume of a face, one usually means the lattice volume of the face in the appropriate sublattice.

Example 2.3
Let P := conv{±e 1 , ±e 2 }, where e 1 and e 2 form a basis for N. P is the terminal Fano polygon associated with P 1 × P 1 . Let P ′ := conv{±e 1 } ⊂ P. P ′ is the one-dimensional terminal Fano polytope associated with P 1 . Both P and P ′ are examples of minimal Fano polytopes (in dimensions two and one, respectively).
Given a Fano polytope P, one can enlarge (or "grow") it to P ′ = conv(P ∪ {v}) by the addition of a lattice point v ∈ N and evaluate whether P ′ is also a Fano polytope. Clearly, if one starts with the minimal Fano polytopes, one will achieve a complete classification using this technique.
The number of possible lattice points that can be added to P to create a Fano polytope is finite. Assume that P ′ is Fano, and consider the ray passing through the origin and −v. It will intersect ∂P in a point x on some face F not containing v. Let S ⊂ vert P ∩ F be of smallest size such that x ∈ conv S; say |S| = d, where d ≤ n is as small as possible. Then conv(S ∪ {v}) is a d-simplex containing the origin strictly in its (relative) interior. In other words, conv(S ∪ {v}) is a Fano d-simplex; there are finitely many of these by, for example, [BB92,Bor00].
Thus we have an algorithm for finding all possible Fano polytopes P ′ that can be obtained from P. What we require is a classification of the Fano d-simplices, for d ≤ n (actually it is sufficient to know the possible weights). Such a classification can be obtained from the techniques in [BB92] (see also [Con02,Kas08]). What remains to be described is a method for constructing the minimal Fano polytopes. We shall prove an inductive description of these minimal Fano polytopes in Proposition 3.2. It shall be seen that an understanding of these minimal Fano polytopes reduces to an understanding of the Fano d-simplices for all d ≤ n.
Finally, in Section 4, we shall find all minimal canonical Fano 3-topes. A computer can then be used to establish a complete classification of toric Fano threefolds with canonical singularities. The resulting classification is summarised in Section 5.

Decomposition of Minimal Fano Polytopes
The results in this section should be compared with [KS97]. It should be stressed that the results ignore the lattice point structure of the Fano polytope; only the property that the Fano polytope contains the origin in its interior is relevant.
x i indicates that the vertex x i is omitted. The following lemma is immediate.
We are now in a position to prove the main result of this section.
Proposition 3.2 Any minimal canonical (resp. terminal) Fano n-tope P is either a simplex, or can be written as P = conv(S ∪ P ′ ) for some S a minimal canonical (resp. terminal) Fano k-simplex and P ′ a minimal canonical (resp. terminal) Fano (n − k + r)-tope, where 0 ≤ r < k < n. Moreover, dim( S ∩ P ′ ) ≤ r, and r equals the number of common vertices of S and P ′ .
Proof We assume that P is not a simplex. Let x 0 , . . . , x l be the vertices of P, where l > n. Without loss of generality we may assume that x 0 , . . . , x n do not lie in a hyperplane and that 0 ∈ conv{x 0 , . . . , x n }.
Minimality of P ensures that 0 / ∈ conv{x 0 , . . . , x n } • . Hence the origin must lie on some facet, and we may assume (with a possible reordering) that 0 ∈ conv{x 0 , . . . , x k } • for some k < n. We obtain the k-simplex S := conv{x 0 , . . . , x k }. S is minimal and Fano, since P is. If P is terminal, then S must be terminal.
By minimality of P and Lemma 3.1, we have that The first case gives us that P ′′ is a minimal Fano (n − k)-tope (which is necessarily terminal if P is terminal), so by setting P ′ = P ′′ we are done. For the second possibility we may assume that σ = cone{x k−r+1 , . . . , x k } and construct the polytope P ′ := conv{x k−r+1 , . . . , x l }. By construction dim P ′ = n − k + r and by Lemma 3.1, we have that 0 ∈ P ′ • . Hence P ′ is our desired minimal Fano (n − k + r)-tope.
From Proposition 3.2 we may conclude the following two corollaries, which are well-known results of Steinitz.

Corollary 3.3 Any minimal Fano polytope P has at most 2 dim P vertices.
Corollary 3.4 Let P be a minimal Fano polytope such that |vert P| = 2 dim P. Then P is centrally symmetric.
For k > 1, no k-simplex is centrally symmetric. Hence Corollary 3.4 is actually an "if and only if ".
A characterisation of centrally symmetric simplicial reflexive Fano polytopes is given in [Nil06]. These polytopes can always be embedded in the n-cube conv{±e 1 ± · · · ± e n }.

Minimal Canonical Fano Threefolds
For the convenience of the reader we begin by summarising the main results of this section in the following theorem (see also Tables 2 and 4). We shall begin by describing which of the Fano tetrahedra are minimal. We do this by restricting the possible weights that may occur. where the x i are the vertices of S, labelled in some order.
Before we continue, we need to be familiar with the Fano triangles. The Fano polytopes are well documented in the literature, more often than not appearing alongside an original method of proof. Consult, for example, [KS97, Sat00, PRV00, Nil05]. The triangles are illustrated in Figure 1.
Suppose that P = conv{x 0 , x 1 , x 2 , x 3 } is not terminal. Minimality dictates that no edge of P can contain more than one interior lattice point. Let x ∈ ∂P ∩ N \ vert P. Since the fan ∆ of P is complete, so x ∈ −σ for some cone σ ∈ ∆ of smallest possible dimension. In particular, dim σ ≤ 2, otherwise P is not minimal, hence σ cone{x 0 , x 1 } without loss of generality. Because of minimality we may suppose that any non-vertex lattice point in conv{x 1 , (i) If x is the only non-vertex lattice point in the face, then we may regard conv{x 1 , x 2 , x 3 } as the Fano triangle (a) in Figure 1, with x playing the role of the origin. Hence, and we obtain weights (1, 1, 1, 3).
x on the edge joining x 1 and x ′ . We may choose x ′ to be as far from x 1 as possible.
(a) x ′ lies on the edge joining x 2 and x 3 : In this case, There are only two possible Fano triangles: (b) and (c) in Figure 1. The former gives: 2x 0 + x 1 + 1 2 (x 2 + x 3 ) = 0, and hence P has weights (1, 1, 2, 4). The latter gives: yielding weights (1, 1, 4, 6). (b) x ′ does not lie on the edge joining x 2 and x 3 : There are no lattice points on the line segment between x 0 and x ′ , hence the Fano triangle conv{x ′ , x 0 , x 1 } can only be (b) (observe that (c) is impossible, since there are no lattice points between x 1 and x = −x 0 ), and so This gives weights (1, 1, 3, 5).
dim σ = 2: We have that σ = conv{x 0 , x 1 } and may assume that −x 0 and −x 1 are not lattice points in the polytope, otherwise we can reduce to the previous case. Let us choose x to be as far from x 1 as is possible. Furthermore, minimality gives that any non-vertex lattice point in conv{x 1 , x 2 , x 3 } must be contained in −cone{x 0 , x 1 }.
(i) Suppose that x lies on that edge joining x 2 and x 3 . Then x = (1/2)(x 2 + x 3 ). In this case, since the edge joining x 0 and x 1 contains at most one interior lattice point, the Fano triangle conv{x, x 0 , x 1 } must be equivalent to (a), (b), or (c) from Figure 1 (note that (d) is impossible, since −x 0 or −x 1 would be a lattice point in the polytope). Triangle (a) gives equation x + x 0 + x 1 = 0, yielding weights (1, 1, 2, 2). For (b) we obtain 2x +x 0 +x 1 = 0, giving weights (1, 1, 1, 1). Finally we consider (c). Notice that −x 0 is not in the face by assumption, hence either x + 2x 0 + 3x 1 = 0 or 2x + x 0 + 3x 1 = 0. The second possibility gives us the lattice points −x 1 and x 0 + x 1 + x on the face conv{x 0 , x 2 , x 3 }, where the second point is closer to x 0 than the first. This contradicts minimality. Hence the only possibility is (1, 1, 4, 6). (ii) If x does not lie on the edge joining x 2 and x 3 , then x is, say, in the interior of conv{x 1 , x 2 , x 3 }, and the only possible Fano triangles for conv{x, x 0 , x 1 } are (a), (b), and (c) (since the edge joining x 0 and x must be lattice point free). Triangle (a) tells us that x is the only non-vertex lattice point in the face conv{x 1 , x 2 , x 3 }, so we obtain This gives weights (1, 1, 3, 4).

A. M. Kasprzyk
Since −x 0 is not in the face, (b) gives us that the face has only one non-vertex lattice point. Hence: yielding weights (2, 2, 3, 5). Possibility (c) contradicts the assumption that −x 0 and −x 1 are not in the polytope.
Proof The terminal Fano tetrahedra are listed in [Kas06a, Table 4]. We need only consider the canonical cases.
From the proof of Proposition 4.3 we can see when the vertices of a minimal tetrahedron generate the lattice N. When this is the case, Proposition 4.4 tells us that the tetrahedron corresponds to weighted projective space. This is the only possibility for all weights except (1, 1, 1, 1) (in the notation of the proof, we are considering dim σ = 2, case (i)(b)). This gives a tetrahedron whose vertices generate an index two sublattice. This corresponds to a fake weighted projective space of index two; [Con02] describes how to compute the vertices of the tetrahedron.
It should be emphasised that not every Fano tetrahedron is minimal. As mentioned in [BB92,p. 278], there are a total of 225 Fano tetrahedra; see the appendix of [BB] for the complete list. This has been verified by the author using the bounds described in [Kas08]. There are 104 distinct weights, which are listed in Table 3.
Proof Let P X be the polytope associated with X. There exists a minimal polytope Q such that Q ⊂ P X , hence P ∨ X ⊂ Q ∨ . Inspection gives vol Q ∨ ≤ 12, hence (−K X ) 3 ≤ 3! · 12.

Theorem 4.7 ([Pro05]) Let X be a Gorenstein Fano threefold with at worst canonical singularities. Then
For the following two results minimality ensures that any such Fano polytope must be at worst terminal; these were classified in [Kas06a, Lemmas 3.4 and 3.5].  Figure 1(b), along with a pair of points ±x not lying in the same subspace as the triangle, is equivalent to

Lemma 4.10 Any minimal Fano polytope containing the minimal Fano triangle shown in
1 My thanks to Professor Victor Batyrev for this observation.
Proof Arrange matters such that P := conv{e 1 , e 2 , −2e 1 − e 2 , x, −x}; x := (a, b, c) is such that 0 ≤ a, b < c. Clearly a = 0, b = 0, c = 1 is a solution. Let us assume that c > 1. Since x = e 3 , we cannot have e 3 ∈ P, since removing x would then yield a smaller canonical Fano polytope with vertex e 3 , contradicting minimality.
Hence e 3 / ∈ P and consider the line connecting e 3 to the origin. If a ≥ 2b, this line intersects conv{−e 1 , −2e 1 − e 2 , x} at the point ke 3 , where k = c/(a − b + 1). This tells us that k < 1, thus a − b ≥ c, which contradicts our assumptions.

Lemma 4.12
Any minimal Fano polytope containing one copy of each of the two minimal Fano triangles (Figure 1(a) and (b)) is equivalent to Proof Arrange matters so that P := conv{e 1 , e 2 , −2e 1 − e 2 , x, y}. There are two cases to consider: Observe that in case (i), the line joining e 1 and −2e 1 − e 2 intersects span {e 2 } at the point −(1/3)e 2 , whereas the line joining x and y intersects span {e 2 } at −(1/2)e 2 . Hence P \ {−2e 1 − e 2 } is still Fano, which contradicts the minimality of P. Indeed, this case reduces to those polytopes discussed in Lemma 4.9.
We now address case (ii). We have that x = (a, b, c) , y = (−a − 1, −b, −c) , and can insist that 0 ≤ a, b < c. Clearly a = 0, b = 0, c = 1 is a solution, so suppose that c > 1. By minimality e 3 / ∈ P. Note that the point −e 1 lies on the line joining e 2 and −2e 1 − e 2 , whilst the line joining x to y intersects the plane span {e 1 , e 2 } at −(1/2)e 1 . Hence this line (without the end points) is contained strictly in the interior of P.
Finally, consider the point e 2 + e 3 . This point must lie outside P. If e 2 + e 3 were contained in P, then conv{e 1 , e 2 + e 3 , −2e 1 − e 2 , y} would be a Fano tetrahedron. The line joining the point with the origin intersects conv{e 1 , −2e 1 − e 2 , x} or conv{−e 1 , e 2 , x}. In the first case the point of intersection is given by k(e 2 +e 3 ), where k = c/(c − a − b + 1). Hence a + b ≤ 0, which is an impossibility (since c = 1).
The alternative is that the line intersects conv{−e 1 , e 2 , x}. This occurs at the point k(e 2 + e 3 ), where k = c/(a − b + c + 1), and we see that a ≥ b. By considering equation (4.7) we obtain our final contradiction.

Lemma 4.13
Any minimal Fano polytope containing two copies of the minimal Fano triangle of type P(1, 1, 2) is equivalent to Proof Fix the lattice such that P := conv{e 1 , e 2 , −2e 1 − e 2 , x, y}. Again there are two cases to consider. If x + y + 2e 2 = 0, then −e 2 is contained on the boundary of P.
We already know that −e 1 lies on the boundary of P, and hence minimality reduced us to the case considered in Lemma 4.10. Thus x + y + 2e 1 = 0 and x = (a, b, c) , y = (−a − 2, −b, −c), where 0 ≤ a, b < 0. Clearly a = 0, b = 0, c = 1 is a solution. Let us assume that c > 1.
In particular a = 0, since the alternative would force c = 1. Finally we consider the point −e 1 − e 3 . The line connecting this point with the origin intersects conv{−e 1 , e 2 , y} if a + 2 ≤ c, or conv{e 1 , e 2 , y} if a + 2 > c. The first possibility gives the point of intersection as k(−e 1 − e 3 ), where k = c/(b + c − a − 1). If −e 1 − e 3 lies on the boundary of P, we see that b = a + 1. This contradicts equation (4.10). Hence it must be that −e 1 − e 3 lies outside P. In this case, b ≥ a + 2, and once again this contradicts equation (4.10). It must be that a + 2 > c, which implies that a = c − 1. Equation (4.9) forces b ≤ 1, and by applying equation (4.8) we see that the only possibility is a = 1, b = 1, c = 2.

Canonical Toric Fano Threefolds
Using the results of Section 4, a computer classification of all canonical Fano polytopes of dimension three is possible. This was a significant undertaking; a month of computation on a parallel computing system was required. The code, written in C, is available from the author upon request. It should be emphasised that several known results exist as sub-classifications, and that the resulting list can be independently checked using packages such as PALP [KS04]. We summarise the algorithm below.  Tables 2 and 4, perform the following recursive algorithm (i) Identify unimodular equivalence: We have been given a canonical Fano polytope P, and inductively are constructing a set P that will ultimately contain all possible canonical Fano polytopes, up to unimodular equivalent. Thus for each Q ∈ P, check whether there exists a transformation in GL(3, Z) sending the vertices of P bijectively onto the vertices of Q. If P is new, then add it to P and proceed to step (ii). Obviously invariants of the two polytopes such as their volume, degree, whether they are both simplicial, etc., can be used to greatly reduce the number of comparisons required. (ii) Successively choose new vertices: We have been given a canonical Fano polytope P and wish to extend P via the addition of a new vertex.
(a) For each vertex v of P such that P ′ := conv (P ∪ {−v}) is a canonical Fano polytope with −v ∈ vert P ′ , recurse on step (i) with P ′ . (b) For each pair of distinct vertices v 1 and v 2 , check which of the following six sums give a lattice point v ∈ N (cf. Figure 1): In each case, if P ′ := conv (P ∪ {v}) is a canonical Fano polytope with v ∈ vert P ′ , then recurse on step (i) with P ′ . (c) For each choice of pair-wise distinct vertices v 1 , v 2 , and v 3 , and for each weight (λ 0 , λ 1 , λ 2 , λ 3 ) in Table 3, check whether any of the four sums: give a lattice point v ∈ N. In each case, if P ′ := conv (P ∪{v}) is a canonical Fano polytope with v ∈ vert P ′ , then recurse on step (i) with P ′ . The final classification is available online, in a searchable format, via the Graded Ring Database at http://grdb.lboro.ac.uk/. The key results are summarised below; for further details consult the online database.
Theorem 5.2 Up to isomorphism, there exist exactly 674,688 toric Fano threefolds. Of these, 18 are smooth, 634 have at worst terminal singularities, 4,319 are Gorenstein, and 12,190 are Q-factorial. Amongst the Q-factorial varieties, the rank of the Picard group is bounded by ρ ≤ 7; this bound is attained in exactly two cases: once when the variety is terminal, once when the variety is canonical.
A significant portion of this work was funded by an Engineering and Physical Sciences Research Council (EPSRC) studentship, and forms part of the author's Ph. D. thesis ( [Kas06b]). The computational resources required for the classification were funded by an ACEnet Postdoctoral Research Fellowship.