From Cracked Polytopes to Fano Threefolds

We construct Fano threefolds with very ample anti-canonical bundle and Picard rank greater than one from cracked polytopes - polytopes whose intersection with a complete fan forms a set of unimodular polytopes - using Laurent inversion; a method developed jointly with Coates-Kasprzyk. We also give constructions of rank one Fano threefolds from cracked polytopes, following work of Christophersen-Ilten and Galkin. We explore the problem of classifying polytopes cracked along a given fan in three dimensions, and classify the unimodular polytopes which can occur as 'pieces' of a cracked polytope.


Introduction
We explain how to construct Fano threefolds with very ample anti-canonical bundle from cracked polytopes; special classes of polytopes introduced in [29]. Fixing a complete fan -or shape -we say a polytope is cracked along Σ if its intersection with each maximal cone of Σ is unimodular, see Definition 2.1.
Fixing a polytope P such that the polar polytope P • is cracked along Σ, we use Laurent inversion -developed in joint work with Coates and Kaspzryk [11] -to embed X P into a nonsingular toric variety Y . Such embeddings correspond to special combinatorial decorations of P • called full scaffoldings; see Definition 2.6 and Theorem 2.7. The ideal of X P in the homogeneous co-ordinate ring of Y is determined by the choice of shape: for example, if TV(Σ) is a product of projective spaces, a full scaffolding with this shape realises X P as a toric complete intersection. We show that every Fano threefold with −K X very ample and b 2 ≥ 2 -famously classified by Mori-Mukai [19][20][21][22][23] -can be obtained from a full scaffolding S of a cracked polytope via an explicit deformation of the corresponding toric embedding.
We extend these constructions to the rank one case in §3 by interpreting the toric degenerations constructed by Christophersen-Ilten [5,6] as smoothings of toric varieties associated to cracked polytopes. Moreover, we provide Laurent inversion constructions and toric degenerations of Fano varieties with −K X not very ample in §4.2.
Many of our constructions follow those given in work of Coates-Corti-Galkin-Kasprzyk [9], in which the authors obtain mirror partners for each family of Fano threefolds. These mirror symmetry results rely on explicit constructions which are usually compatible with Laurent inversion. We note that the connection between toric degenerations and mirror symmetry is further explored by Ilten-Lewis-Przyjalkowski [18]. Theorem 1.1. Every smooth Fano threefold with a very ample anti-canonical bundle and b 2 ≥ 2 can be obtained by smoothing a Gorenstein toric Fano variety. In particular these can be constructed as deformations of toric embeddings provided by Laurent inversion, applied to a cracked polytope together with a full scaffolding S. We assume that the shape of the scaffolding S appears in Table 1.
very ample anti-canonical bundle from a cracked polytope and full scaffolding. We consider the Fano threefolds for which −K X is not very ample in §4.2.
We suggest that four-dimensional cracked polytopes form classes of polytopes from which it is natural to algorithmically construct Fano fourfolds. We note, by way of example, that each of the 738 families of Fano fourfolds which appear in [7] can be constructed from a polytope cracked along the fan of a product of projective spaces.
Given a cracked polytope P there is a natural degeneration of X P • to a union of smooth toric varieties. Moreover, when X P • smooths to a Fano threefold X, we expect X to degenerate to this union of smooth toric varieties. This is close to the notion of semi-simple degeneration, although the total space of the degeneration is generally singular. Such degenerations also play a key role in the work of Christophersen-Ilten [5,6], via the work of Altmann-Christophersen [1,2]; in particular they consider the deformation spaces of certain Stanley-Reisner rings. Indeed, cracked polytopes can be regarded as a direct non-simplicial generalisation of the Stanley-Reisner rings considered in [1,2].
The extension of the notion of semi-simple degeneration to singular families plays an important role in mirror symmetry. Indeed, such a notion of toric degeneration plays a key role in the approach pioneered by Gross-Siebert via logarithmic and integral affine geometry [16,17] to the Strominger-Yau-Zaslow conjecture. In [30] we show how to (partially) smooth toric log del Pezzo surfaces using the Gross-Siebert algorithm. In a similar way we can associate a polarised tropical manifold -the input to the Gross-Siebert algorithm -to any cracked polytope. Alternatively one could attempt to smooth the toric variety associated to a cracked polytope by compactifying families of log Calabi-Yau varieties constructed in [14,15] using theta functions. In two dimensions we expect that the toric embeddings defined by scaffolding extend, under appropriate conditions, to compactifications of these families of log Calabi-Yau varieties.
Acknowledgements. We thank Tom Coates and Alexander Kasprzyk for our many conversations about Laurent inversion. The author is supported by a Fellowship by Examination at Magdalen College, Oxford.
Conventions. Throughout this article N will refer to an 3-dimensional lattice, and M := hom(N, Z) will refer to the dual lattice. Given a ring R we write N R := N ⊗ Z R and M R := M ⊗ Z R. For brevity we let [k] denote the set {1, . . . , k} for each k ∈ Z ≥1 . We work over the field C of complex numbers throughout this article. Given a reflexive polytope P ⊂ N R , we assume throughout that X P is the toric variety associated to the fan of cones over faces of P . Cracked polytopes will always be contained in M R ; in particular if Q is a polytope cracked along a fan Σ, Σ is a fan in M R . Given a variety Y , and an identification Pic(Y ) ∼ = Z r , we write O(a 1 , . . . , a r ) for the line bundle of (multi) degree a = (a 1 , . . . , a r ) ∈ Z r .

Cracked polytopes and Laurent Inversion
The method Laurent inversion -introduced in [11] -can be used to construct models of Fano manifolds as follows. First, fix a Fano polytope P , together with a combinatorial decoration of P called a scaffolding S -see Definition 2.3. Loosely, a scaffolding is a collection of polytopes associated to nef divisors on a fixed toric variety Z -the shape -whose convex hull is equal to P . From this information we construct a polytope Q S which projects to P • . Letting Y S denote the toric variety associated to the normal fan of Q S , the toric variety X P embeds into Y S , and the corresponding ideal in the homogeneous co-ordinate ring of Y S is determined by Z. We can then test explicit deformations of the equations cutting out X P in Y S to attempt to construct an embedded smoothing.
For general choices of S, the variety Y S may be highly singular: for example Y S need not be Q-Gorenstein. In [29] we explore the (restrictive) conditions on S which ensure that Y S is non-singular.
Definition 2.1 ([29, Definition 2.1]). Fix a convex polyhedron P ⊂ M R containing the origin in its interior, and a unimodular fan Σ. We say P is cracked along Σ if every tangent cone of P ∩ C is unimodular for every maximal cone C of Σ.
Remark 2.2. Let P be a polytope cracked along a fan Σ. We do not assume that the minimal cone of the fan Σ is not necessarily zero dimensional; these are sometimes called generalised fans. The shape Z is the toric variety associated to the quotientΣ of Σ by its minimal cone. Slightly abusing terminology, we also say that P is cracked along the fanΣ.
We know -[29, Proposition 2.5] -that any cracked polytope is reflexive. In three dimensions the converse holds, in the sense that any reflexive polytope is cracked along some complete unimodular fan. Indeed, consider the fan Σ defined by taking the cone over every face of a maximal triangulation of the boundary of P ; the polytopes obtained by intersecting maximal cones of Σ with P are all standard simplices. Examples of cracked polytopes are shown in Figure 1.
Our principal application for cracked polytopes -constructing toric degenerations of Fano 3-folds -makes heavy use of the notion of scaffolding. We first fix a splitting N =N ⊕ N U , adopting the notation used in [11]. Definition 2.3 ([11, Definition 3.1]). Fix a smooth projective toric variety Z with character latticeN . A scaffolding of a polytope P is a set of pairs (D, χ) -where D is a nef divisor on Z and χ is an element of N U -such that We refer to Z as the shape of the scaffolding, and elements (D, χ) ∈ S as struts. We also assume that there is a unique s = (D, χ) such that v ∈ P D + χ for every vertex v ∈ verts(P ).
Scaffolding a polytope P determines an embedding of X P into an ambient space Y S . This is the main result of [11]; see also the treatment given in [29, §3].
. Given a scaffolding S of P we define a toric variety Y S , associated to the normal fan Σ S of the polytope Q S ⊂ M R := (Div TM Z ⊕ M U ) ⊗ Z R, itself defined by the inequalities where e i denotes the standard basis of Div TM Z ∼ = Z .
We let ρ denote the ray map of the fanΣ determined by Z, and let ρ s := (−D, χ) for each s = (D, χ) ∈ S. We also define a map of lattices θ, setting Theorem 2.5 ([11, Theorem 5.5]). A scaffolding S of a polytope P determines a toric variety Y S and an embedding X P → Y S . This map is induced by the map θ on the corresponding lattices of one-parameter subgroups.
Note that the ideal, in homogeneous co-ordinates on Y S , of X P is determined by the map θ: a hyperplane containing the image of θ defines a function h on the set of ray generators of Σ S . X P then satisfies the equation where products are taken over the ray generators of Σ S , and z v is the homogeneous co-ordinate on Y S corresponding to the ray generated by v.
In [29] we describe the facets of the polar polytope P • to a polytope P cracked along a complete unimodular fan Σ. In particular, each facet F of P • is dual to a vertex F of P , which is contained in a cone of Σ. Assume that σ is minimal among such cones, then σ corresponds to a non-singular toric stratum Z(σ) of the toric variety TV(Σ). The facet F of P • is the Cayley sum P D 1 · · · P D k , where {D i : 1 ≤ i ≤ k} is a set of nef divisors on Z(σ) and k = dim(σ) + 1; see [29,Proposition 2.8]. We call a face A of P • vertical if it is contained in P D i for some facet containing A and some D i as above.
Definition 2.6 ([29, Definition 4.1]). Given a Fano polytope P ⊂ N R cracked along a fan Σ in M R we say a scaffolding S of P with shape Z := TV(Σ) is full if every vertical face of P is contained in a polytope P D + χ for a (unique) element (D, χ) ∈ S.
We show in [29] that full scaffoldings on cracked polytopes give rise to embeddings X P → Y S where Y S is smooth in a neighbourhood of X P . Theorem 2.7 ([29, Theorem 1.1]). Fix a polytope P ⊂ M R , and a rational fan Σ in M R such that the toric variety Z := TV(Σ) is smooth and projective. Given a scaffolding S of P with shape Z, we have that the target of the corresponding embedding is smooth in a neighbourhood of the image of X P if and only if P is cracked along Σ and S is full.
2.1. Torus quotients. Every n-dimensional toric variety X (over C) may be described as the quotient of a Zariski open set of affine space C n+r by a complex torus T := (C ) r . Recalling that, if X is determined by a fan in N , with n + r rays with primitive generators ν 1 , . . . , ν n+r we have an exact sequence , . . . , n + r}, the character lattice of T is L , and right exactness of the above sequence is assumed. This lattice fits into the dual sequence, The map R : (Z n+r ) → L is called the weight data for the toric variety. Assuming that X is a projective variety, the possible fans in N , with rays generated by a subset of {ν 1 , . . . , ν n+r }, such that the associated toric variety is projective are indexed by cones in a fan contained in the effective cone Eff(X) ⊂ Pic(X) R . This fan is called the secondary fan or GKZ decomposition. Fixing a maximal cone (or chamber ) σ in the secondary fan, the corresponding toric variety can be described as the torus quotient where T := (L ⊗ Z C ), the weights of the torus action are specified by R, and the Z(σ) is the irrelevant locus. Choosing a point (or stability condition) ω in the interior of σ, the irrelevant locus is defined by setting where R i = R(e i ) for each standard basis vector e i , i ∈ [n + r]. Some of the constructions described in §4 make use of stability conditions contained in a codimension one cone (or wall ) in the secondary fan. The toric variety corresponding to the wall formed by the intersection of chambers σ 1 and σ 2 is determined by the coarsest common refinement of the fans associated to X σ 1 and X σ 2 .
We can use the GIT presentation of a toric variety to streamline the construction of the variety Y S from a scaffolding S. To do this we first assume that, writing If this condition holds the cone generated jointly by the vectors in B and the standard basis elements {e i : i ∈ [dim Div TM (Z)]} define a smooth torus invariant point in Y S . We assume throughout this section that every scaffolding satisfies this condition. The second Betti number of the toric variety Y S is s − dim N U , where s := |S|. We explain how to form a weight matrix and stability condition which determine the variety Y S directly from the scaffolding S. This construction follows [11, Algorithm 5.1].
Construction 2.8. Given a scaffolding S with shape Z of a polytope P , index the elements of S by [s], and let (D i , χ i ) denote the i th element of S. It follows from our assumptions on S that the ray matrix of Σ S is in echelon form Figure 2. The scaffolding corresponding to the embedding of dP 6 in P 2 × P 2 .
Hence R, the transpose of the kernel matrix, is given by The variety Y S is defined using the a polarising torus invariant divisor given by the sum of all rays corresponding to elements of S. The degree of this divisor is given by the sum of the first s columns of R. That is, the stability condition used to define Y S is given by the sum of (1, . . . , 1) T with the columns of the matrix (χ 1 , . . . , χ r ) T In the case that Z is a product of c projective spaces there is a partition of the columns of R containing the vectors D i ∈ Div TM (Z). In particular, the standard basis in Div TM (Z) partitions into c sets C 1 , . . . , C c , such that C i consists of divisors pulled back from the projection to the i th projective space factor. For each i ∈ [c] the degree of the line bundle L i cutting out X P in Y S is given by the sum of the columns in C i . In particular, there is a distinguished binomial z m 1 − z m 2 in L i , where m 1 is the sum of standard basis vectors in (Z n+r ) corresponding to the columns of C i , and m 2 is the unique lift of L i ∈ L to (Z r ) : the subspace of (Z n+r ) corresponding to the first r columns of R. It is shown in [11] -see also [29, §3] -that X P is the vanishing locus of these binomials. Example 2.9. Fix a 3-dimensional reflexive polytope P , and let Z be a crepant resolution of the toric variety determined by the normal fan of P . In particular,N = N and N U = {0}. Let S := {(D, 0)}, where D ∈ | − K Z | is the toric boundary of Z. Hence P = P D , and the corresponding weight matrix R is equal to 1 1 · · · 1 , and contains n := 1+dim Div T M (Z) columns. The stability condition is equal to 1 ∈ L ∼ = Z, and hence Y S ∼ = P n−1 . This is nothing but the anti-canonical embedding of X P into projective space. Example 2.10. In [11, Example 3.5] we consider two distinct scaffoldings for the polygon P associated with the toric del Pezzo surface of degree six. One of these is illustrated in Figure 2.
The scaffolding illustrated in Figure 2 has shape Z = P 1 × P 1 and -letting D i,a denote the pullback of {a} ⊂ P 1 along the i th projection for each i ∈ {1, 2} and a ∈ {0, ∞} -we define

Fano PALP ID
Equations Shape Fano PALP ID Equations Shape  and stability condition ω = (1, 1). This is a GIT presentation of the toric variety P 2 × P 2 .
The variety X P ∼ = dP 6 is the vanishing locus of the binomials x 1 y 1 = x 0 y 0 and x 2 y 2 = x 0 y 0 , where x i and y i denote homogeneous co-ordinates on the P 2 factors.

Rank one Fano threefolds
Toric degenerations of rank one, index one, Fano toric varieties have been obtained by Ilten and Christophersen [5,6], using the deformation theory of Stanley-Reisner rings developed by Altmann-Christophersen [1,2]. Using these results -and the work of Galkin [13] on small toric degenerations -we obtain cracked polytopes corresponding to each of the 15 rank one Fano threefolds with very ample anti-canonical bundle. In particular, we describe toric degenerations of these 15 Fano threefolds. We remark that, since the toric degenerations in this case occur in the anti-canonical embedding, the use of cracked polytopes in this context is rather trivial; see Example 2.9.
• Z 12 is the blow up of Z 10 := dP 7 × P 1 in a toric invariant line 0 1 ⊂ Z 10 . The sequence of blow up maps described induces the starring operations on these triangulations described in [6]. We define the variety Z 22 to be a crepant resolution of the toric variety determined by the normal fan of the reflexive polytope with ID 1941. Similarly, we define the variety Z 2 to be a crepant resultion of the toric variety determined by the normal fan of the (self-dual) reflexive polytope with ID 427.
The Fano variety P 3 is toric, while Q 3 , B 2 , B 3 , B 4 , V 4 , V 6 , V 8 are well known to be toric complete intersections. These admit toric degenerations to the varieties defined by the equations in Table 2. To describe the scaffolding associated to each of these Fano threefolds, let d be the dimension of the shape variety Z, setN := Z d and N U := Z 3−d . Letting {e 1 , . . . , e 3−d } denote the standard basis of N U , we define where D ∈ | − K Z | is the toric boundary of Z, and χ = (−1, . . . , −1) ∈ N U . This scaffolding is illustrated in the case B 3 in Figure 24 (setting a = 1 and b = 3).
3.1. Pfaffian equations and B 5 . The Fano threefold B 5 is a linear section of the Grassmannian Gr (2,5). We make heavy use of the fact the equations of Gr(2, n) can be written as the 4 × 4 Pffafians of a skew-symmetric n × n matrix; entries of which are the n 2 Plücker co-ordinates of Gr(2, n). Hyperplane sections can then be obtained by replacing entries with linear combinations of a subset of the Plücker co-ordinates. For example, B 5 can be described as the Pfaffians of the matrix for a fixed value of t = 0. Varying t defines a flat family, the central fibre of which is the projective cone over a toric variety with two ordinary double points, obtained from dP 5 by moving the four points at which P 2 is blown up to two pairs of infinitely close points, and contracting the pair of resulting −2 curves. Setting t = 0 recovers five equations generating the ideal of a toric variety in P 5 . This toric variety is isomorphic to X P , where P denote the toric variety with ID 741. The embedding X P → P 5 is the embedding of X P determined by the scaffolding S = {(0, 1), (D, 0)}, where 1 ∈ N U ∼ = Z, and D ∈ −K Z (recalling that Z = dP 7 ) is the toric boundary of Z.

3.2.
Higher genus Fano threefolds. The varieties V 2n−2 for n ∈ {6, 7, 8, 9, 10, 12} are linear sections of the Mukai varieties M n [24]. Toric degenerations of these are related -by work of Ilten-Christophersen [6] -to the convex deltahedra in the cases n < 12, while varieties in the family V 22 admit a toric degeneration to a variety with ordinary double point singularities, see [13]. Given a Fano toric variety Z, let its dual Z be toric variety associated to the normal fan of the convex hull of the ray generators of the fan determined by Z. Proof. If n < 12 we recover the triangulations T n of S 2 used in [6] to construct degenerations of Fano threefolds by removing the origin from N R ∼ = R 3 and radially projecting the fan Σ n determined by Z 2n−2 . The result then follows immediately from [6, Proposition 2.3]. In the case n = 12 we observe that Z 22 contains only ordinary double point singularities, and hence admits a smoothing. It is shown in [13] that the general fibre of this smoothing is a member of the family V 22 .
In the cases n ∈ {6, 7, 8} we can provide an explicit description of the toric degeneration.
(i) V 10 : varieties in this family can be described by the Pfaffians of a 5×5 skew-symmetric matrix, and one quadric equation. We can form a toric degeneration following §3.1. (ii) V 12 : varieties in this family can be described via a system of 9 Pfaffian equations, see 2-21 for a description of a toric degeneration using the same shape variety. (iii) V 14 : varieties in this family can be described as the vanishing of the 4 × 4 Pfaffians of a 6 × 6 skew matrix. An explicit toric degeneration is given by the 4 × 4 Pfaffians of the matrix (1) below.
The vanishing 4 × 4 Pfaffians of the matrix define a toric degeneration of V 14 , a general linear section of Gr (2,6), where x i are homogeneous co-ordinates on P 9 and f 1 , g 1 , and h 1 are general linear forms on P 9 . The scaffolding S in each case is equal to the singleton set {(D, 0)}, where D is the toric boundary of Z.
3.3. The quartic hypersurface in P(1, 1, 1, 1, 2). The toric variety Y S defined by a full scaffolding of a cracked polytope P is non-singular in a neighbourhood of the image of P . This excludes certain constructions of Fano manifolds as hypersurfaces as weighted projective spaces. In particular, consider the scaffolding of the polytope P with ID 3312 with shape Computing the corresponding weight matrix we find Thus X P is the vanishing locus of a section of O(4) in P(1 4 , 2) := P(1, 1, 1, 1, 2). Notice that P • is not cracked along the fan of P 2 . To obtain a construction from a cracked polytope we first embed P(1 4 , 2) into P 10 via The quartic equation x 0 x 1 x 2 x 3 = y 2 defines a projection of this polytope to the reflexive polytope P with ID 427. This polytope is self-dual, and we take the scaffolding of P with shape Z given by a crepant resolution of X P , covering P with a single strut. This scaffolding corresponds to the anti-canonical embedding of X P into P 10 , which is the intersection of the image of the Veronese embedding of P(1 4 , 2) with a (binomial) quadric. Deforming this quadric deforms X P to a general quartic hypersurface in P(1 4 , 2).

Constructions of Fano manifolds
There are 98 Fano threefolds with very ample anti-canonical bundle. In the previous section we described constructions from cracked polytopes of the 15 of these which have Picard rank one. We now explain constructions in the remaining 83 cases. In particular, for each of these 83 Fano threefolds X, we exhibit a fan Σ and polytope P cracked along Σ such that -for some full scaffolding of S with shape Z := TV(Σ) -the toric variety X P admits an embedded smoothing in Y S to X.
Examples from 'Quantum periods for 3-dimensional Fano manifolds'. Explicit constructions of Fano threefolds are provided in [9]. The authors use these constructions to compute (part of) the J-function of each Fano threefold using either the Quantum Lefschetz principle, or the Abelian-non Abelian correspondence. In particular, each Fano threefold X is exhibited either as a complete intersection in a weak Fano toric variety, or as the degeneracy locus of a map of homogeneous vector bundles.
Proposition 4.1. Fix a Fano threefold X, and assume that the model of X in [9] describes X as the zero locus of a section of a split vector bundle Λ = L 1 ⊕ · · · ⊕ L c on a toric variety Y for which −K Y − Λ is ample. There is a reflexive polytope P , shape variety Z, and full scaffolding S of P such that Y S ∼ = Y and the image of the induced inclusion of X P ⊂ Y S admits an embedded smoothing to X.
Proof. Tables 3, 4, and 5 list binomial equations cutting out toric varieties to which Fano varieties in the various families satisfying our hypotheses degenerate. The leading monomial in each case is square-free and defines a subset of the columns C i of the weight matrix listed in [9] for each i ∈ {1, . . . , c}. In every case the sets C i are pairwise disjoint, and disjoint from a subset C of columns which define a basis of Pic(Y ). Reversing Construction 2.8 we obtain a scaffolding associated to each family represented in Tables 3, 4, and 5. The rank one complete intersection cases are listed in Table 2.
It follows from by [29, Theorem 1.1], and smoothness of Y S , that the polytope P • is cracked along the fan determined by Z, and S is full.
Consider a Fano threefold X in the family 2-18. X is a double cover of P 1 × P 2 , branched in a divisor with bidegree (2, 2). The construction in [9] describes this Fano threefold as a hypersurface in the projectivisation of a rank 3 split vector bundle on P 2 .
Consider the scaffolding S with shape Z = P 2 illustrated in the left hand image in Figure 3.
and (x 0 : x 1 : x 2 ) are homogeneous co-ordinates on P 2 . The corresponding hypersurface is given by the vanishing locus of the binomial zy 2 x 3 − y 2 1 x 2 1 , in the toric variety with weight matrix y 1 x 1 x 2 x 3 y 2 z 1 0 0 0 1 1 0 1 1 1 0 1 such that the class (2, 1) is ample. Note that the weight matrix -up to a permutation of the columns -and stability condition ω = (1, 2) are identical to those appearing in [9, p.40]. Thus the general member of the linear system O(2, 2) is a Fano threefold in the family 2-18.
Of the 83 Fano threefolds with very ample anti-canonical divisor and Picard rank > 1, 67 of the constructions given in [9] coincide with constructions from full scaffoldings on cracked polytopes. We summarise these constructions in Tables 3, 4, and 5. The column Equations in each table describes a generating set for the ideal in the homogeneous co-ordinate ring of the ambient variety Y described in [9]. The first monomial of each binomial is always square-free, and may be used to identify columns of the weight matrix defined by Y . If Y is a product of projective spaces the co-ordinates are not named in [9], and we name these x 0 , . . . , x m for the first projective space factor P m , y 0 , . . . , y n for the second, etc.
We now provide constructions from cracked polytopes of the 16 Fano threefolds whose construction in [9] is not directly related to a full scaffolding of a cracked polytope. In five
cases the corresponding construction in [9] does not describe the Fano threefold as a toric complete intersection. In the remaining eleven cases the construction given in [9] expresses the Fano threefold X as the vanishing locus of a section of split vector bundle Λ on a toric variety Y , such that L := −K Y − Λ is nef but not ample. In the latter case the embedding cannot come from a scaffolding S, since the construction of the ambient space uses L to polarise the ambient space.
Remark 4.3. Note that the numbering for the rank 4 Fano threefolds replicates that in [9], which differs from the original list of Mori-Mukai by the insertion of the family 4-2 which was omitted from the original classification (some lists instead append this family as 4-13).
Rank 2, number 8. Varieties in the family 2-8 are either, We make use of the construction given in [9], embeds Fano threefolds in the family 2-8 as hypersurfaces of bi-degree (2,4) in the toric variety Y , defined by the weight matrix and a stability condition in the chamber (0, 1), (1,2) . The coincidence of these two constructions is proved in [9, p.31].
We consider the scaffolding S = {D} of the reflexive polytope P with PALP ID 3262, with shape Z = P 1 × dP 5 ; where dP 5 is the blow up of P 1 × P 1 at three of its torus invariant points, and D ∈ | − K Z | is the toric boundary of Z.
This scaffolding corresponds to the anti-canonical embedding X P → P 9 , see Example 2.9. To prove that X P admits a smoothing in this embedding we consider another scaffolding of P -with shape Z = P 2 × P 1 -shown in Figure 4. Note that P • is not cracked along the fan determined by Z . The scaffolding S defines an embedding X P → Y S where Y S is the toric variety defined by weight matrix   with the embedding ι : X P → Y S , the pull-back of the line bundle O P 10 (1) is the anti-canonical class on X P by adjunction. Moreover X P is the intersection of ϕ O(1,2) (Y S ) with a quadric and a hyperplane in P 1 0. In particular, restricting to this hyperplane, we obtain the anticanonical embedding of X P in P 9 . Restricting to members of a general pencil of hyperplanesand intersecting with a general pencil of quadrics -we see that X P deforms in P 9 to a variety in the family 2-8.   The variety Y S is determined by the weight matrix x 0 x 1 x 2 x 3 x 4 x 5 x 6 y 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 and stability condition ω = (2, 1). The variety Y S is consequently the blow up of P 6 in a codimension 2 linear subspace. The ideal of X P in Y S is obtained by homogenizing the 4 × 4 Pfaffians of the skew-symmetric matrix Consider the contraction Y S → P 6 , and observe that the intersection of the image of X P with the centre V := {x 0 = x 1 = 0} is a cycle of five (−1)-curves. Replacing the two 0 entries with general linear forms this cycle of (−1)-curves becomes a (codimension 3) non-singular curve of genus one, and -blowing up V -we obtain a flat family deforming X P to a Fano threefold in the family 2-14.
Rank 2 number 17. Varieties in the family 2-17 are the blow up of a quadric threefold in an elliptic curve of degree 5. We consider the polytope P with PALP ID 1527, together with the scaffolding S shown in Figure 7 using the shape variety Z = P 1 × dP 7 .
The scaffolding S determines the toric variety Y S ∼ = P 4 ×P 3 . Letting x 0 , . . . , x 4 and y 0 , . . . , y 3 denote homogeneous co-ordinates on the respective projective space factors, X P is the vanishing locus of the binomial x 0 y 0 = x 1 y 1 , and the five 4 × 4 Pfaffians of the skew-symmetric matrix  where t = 0 and f i are general linear equations in x 0 , . . . , x 4 . One of these five Pfaffians describes the threefold x 2 x 3 − x 0 x 4 = 0 in P 4 , while the other 4 equations have bidegree (1,1). It is shown in [9, p.38] that varieties in the family 2-17 may be obtained as the vanishing loci of general sections of the bundle in the variety Gr(2, 4) × P 3 . The Grassmannian Gr(2, 4) ⊂ P 5 is a quadric fourfold, while sections of the line bundle det S define hyperplane sections in P 5 . Moreover, the binomial x 0 y 0 = x 1 y 1 defines a section of the bundle obtained by pulling back (det S O P 3 (1)) to the product of a hyperplane section in P 5 with P 3 . We claim that the remaining four Pfaffian equations define a section of the pull-back of (S O P 3 (1)) to this hyperplane section. Representing a point in Gr (2, 4) (2,5)) in a twisted cubic. Consider the polytope P with PALP ID 1909 together with the scaffolding with shape Z = dP 7 displayed in Figure 8.
The corresponding toric variety Y S is isomorphic to Bl P 3 P 6 . Moreover, the variety X P is the blow up of the vanishing locus of the five 4 × 4 Pfaffians of where x 0 , . . . , x 6 are homogeneous co-ordinates on P 6 , in the locus {x 0 = x 1 = x 6 }. Note that the ideal x 2 x 4 = x 3 x 5 = x 2 x 5 = 0 defines a (degenerate) twisted cubic. Replacing the two zero entries in the above matrix with general homogeneous elements of degree one we obtain a flat deformation of X P → Y S to a blow up of B 5 in a twisted cubic.
on Gr(2, 4) × P 4 . Consider the polytope with PALP ID 702, with the scaffolding shown in Figure 10. This scaffolding has shape Z = Z 12 , the shape used in the construction of Fano threefolds in the family V 12 . The ambient space Y S defined by this scaffolding is isomorphic to P 4 × P 4 with co-ordinates x 0 , . . . , x 4 and y 0 , . . . , x 4 respectively. The equations cutting out X P in Y S can be read off as relations between labelled lattice points in Figure 9. In particular if u 1 + v 1 = u 2 + v 2 , where u i and v i are lattice points labelled with variables z i and w i for each i ∈ {1, 2}, points in X P satisfy the equation z 1 w 1 = z 2 w 2 . There are nine such binomial equations, which can be written as the 4 × 4 Pfaffians of the following pair of matrices (setting t = 0).
Note that these matrices share the Pfaffian x 1 x 3 − x 2 0 + tx 2 x 4 , which defines a toric degeneration of a quadric threefold.
Following the treatment of the variety 2-17, we observe that each set of five Pfaffian equations defines a section of (the pullback to a hyperplane section of) S O P 4 (1). Thus the general member of the family given by the set of 9 Pfaffian equations is isomorphic to a Fano threefold in the family 2-21.
Rank 2 number 22. Varieties in the family 2-22 are the blow up of B 5 in a conic. Consider the polytope P with PALP ID 1856, and scaffolding -with shape Z = dP 7 -displayed in Figure 11. The variety Y S is the blow up of P 6 in a plane; that is, the toric variety determined by the weight matrix 1 0 0 0 0 1 1 1 0 1 1 1 1 1 1 1 and stability condition ω = (1, 2). X P is cut out by the five 4 × 4 Pfaffians of Consider the one parameter family where f 1 and g 1 are general linear forms with no terms in x 4 or x 5 . Varying t, this family contains the line with co-ordinates x 4 and x 5 for all values of t. Blowing up this line we obtain a flat family embedded in Y S × A 1 t with central fibre X P , and general fibre a Fano threefold in the family 2-26.
Rank 3 number 1. Varieties in the family 3-1 are double covers of P 1 × P 1 × P 1 branched along a divisor of tri-degree (2, 2, 2). Our treatment of this family is similar to that of 2-8. Consider the Fano polygon P with PALP ID 3874, illustrated in Figure 13. We give P the 'anti-canonical scaffolding'; covering P with the polyhedron of sections of the toric boundary on the shape variety Z = P 1 × dP 6 . This scaffolding produces the standard anti-canonical map X P → P 8 , see Example 2.9. Similarly to our treatment of family 2-8, we prove this smooths by factoring the embedding through a map to a toric variety obtained from a nonfull scaffolding of P . Figure 13 shows a scaffolding S of P with shape Z := P 1 × P 2 . The scaffolding S consists of three elements, and defines the toric variety Y S with weight matrix and stability condition ω = (1, 1, 1). The hypersurface X P is the vanishing locus of the binomial w 0 w 1 = x 2 0 y 2 0 z 2 0 -a section of the line bundle L 1 with tri-degree (2, 2, 2) -and x 1 y 1 z 1 = x 0 y 0 z 0 -a section of the line bundle L 2 tri-degree (1, 1, 1). Note that the variety Y S is not Q-factorial along the line on which x 0 = y 0 = z 0 = x 1 = y 1 = z 2 = 0. General linear sections though this non-isolated singularity are isomorphic to the affine cone V over P 1 × P 1 × P 1 , polarised by the line bundle of tri-degree (1, 1, 1).
Consider a general section s of E = L 1 ⊕ L 2 , and its vanishing locus X. Projecting away from the point at which all co-ordinates except w 0 vanish, X is an isomorphism onto its image in a toric variety Y . The variety F which appears in the construction in [9, p.57] is obtained from the variety Y by a making one of the three possible small resolutions of the singularity V . Since the variety X does not intersect the singular locus of Y this resolution restricts to an isomorphism of X. The rest of the example follows our treatment of the family 2-8: the complete linear system determined by L 2 defines an embedding Y S → P 9 and varying a quadric section in the anti-canonical embedding of Y S smooths X P .
It is easy to check that this map is homogeneous, and that φ : Pic(Y S ) → Pic(Y ) is given  Proof. There is a morphism π : Y S → P 1 expressing Y S as a P 4 bundle over P 1 , with coordinates (a 0 : a 1 ). Similarly, Y and Y admit projections to the P 1 with co-ordinates (x 0 : x 1 ). Each of these projections commute with the inclusion ι : Y → Y S . Given a point a ∈ P 1 , the intersection π −1 (a) ∩ ι(Y ) is the projective closure of a conifold singularity in P 4 with co-ordinates (b 0 : b 1 : b 2 : c 0 : c 1 ). The (smooth) variety Y is obtained by making either of the two possible small resolutions of this line of conifold singularities. Note however that, for any fibre of π, the divisor b 0 = 0 is disjoint from the singular locus of Y . Since ι b 0 = w, the locus w = 0 is disjoint from the singular locus of Y .
Note that Y is a hypersurface in the class O(1, 2), cut out by det b 1 c 0 b 2 c 1 . Moreover, we have that φ (2, 2) = (2, 2, 2); hence, by Lemma 4.5 any hypersurface cut out by a member of the linear system (2, 2, 2) on Y , is the vanishing locus of a section of E on Y S .  Figure 16. Consider the scaffolding of the polytope P with PALP ID 1836 shown in Figure 15. The corresponding variety Y S is determined by the weight matrix  clear that the weight matrix defining Y is the same as the defining the toric variety Bl B Y S . Moreover, the map defined by setting φ : (x 0 , x 1 , y 0 , y 1 , y 2 , z 0 , z 1 , t) → (x 0 t, x 1 t, y 0 , y 1 , y 2 , z 1 , z 2 ) has pull-back defined by the matrix Hence, considering the ample class ω = (2, 1), φ ω = (2, 1, 2). It remains to analyse the effect of crossing the wall in the secondary fan of Y generated by (1, 0, 0) and (1, 1, 1). We observe that moving the stability condition into this wall contracts the divisor t = 0 (defining the ray generated by (0, 0, 1)) to the locus {y 0 = y 1 = y 2 = 0}.
We claim that general sections of E := O(2, 1) ⊕2 are smooth. If so, the blow-up of the base locus is an isomorphism on general sections, as the restriction of the base locus to a general fibre is a Cartier divisor. Smoothness follows directly from the Jacobian condition. Indeed, sections of E are of the form c 0 f 1 + c 1 g 1 + a 0 f 2 + a 1 g 2 where f j and g j are homogeneous polynomials of degree j ∈ {1, 2} in b 0 , b 1 , b 2 . Taking two such sections the corresponding Jacobian matrix, evaluated at b 0 = b 1 = b 2 and -without loss of generality -a 0 = c 0 = 1, has the form 0 L ; a block matrix consisting of a 2 × 2 zero block and a 2 × 3 matrix L of linear forms in c 1 . Since the locus F in P 5 where a 2 × 3 matrix drops rank has codimension 2, any projective line in this space which misses F determines a matrix L which does not drop rank.
Rank 3 number 14. We consider the reflexive polytope P with PALP ID 142, together with the scaffolding S with shape Z = P 1 shown in Figure 17. This scaffolding expresses X P as a hypersurface of tri-degree (3, 1, 1) in the toric variety Y S with weight matrix Remark 4.6. We could also construct varieties in this family using the scaffolding S obtained by combining the two struts containing the origin in N R into a single line segment of length two. This produces an embedding X P → Y S , where Y S is given by the weight matrix We can recover the construction used in [9, p.71] using a scaffolding of a reflexive polytope. Indeed, consider the polytope P with PALP ID 1091, together with the scaffolding S displayed in Figure 18 with shape Z = P 1 × P 1 . Note that this scaffolding is not full, and P • is not cracked along the fan defined by Z. The toric variety Y S is determined by the weight matrix  1, 1, 1). A stability condition which lies in the cone spanned by (1, 0, 0), (1, 1, 0), (1, 1, 1) determines the toric varietyŶ S used in [9] to construct Fano varieties in 3-16. However ω lies in the wall spanned by a pair of these vectors. Moving ω into the chamber used in [9] resolves the singular locus {x 0 = x 1 = x 2 = y 0 = z 0 = z 1 = 0}. However general sections of O(1, 1, 1) do not vanish along this point, and hence the intersection of two general divisors of tri-degree (1, 1, 1) are isomorphic to varieties in the family 3-16. In order to provide a construction using a cracked polytope, we consider the scaffolding S of P with shape Z = dP 6 , also shown in Figure 18.
The scaffolding S defines the weight matrix x 0 x 1 x 2 x 3 y 0 y 1 y 2 y 3 y We can deform X P in Y S by moving the section of O(1, 1, 1) ⊕2 cutting out X P . In other words, we obtain varieties in the family 3-16 in Y S in codimension 4 by embedding Y S → Y and moving the sections used to cut out Y S .
Rank 4, number 2. Varieties in this family are obtained from P 1 × P 1 × P 1 by blowing up a curve of tri-degree (1, 1, 3). We consider the polytope with PALP ID 1080, together with the scaffolding shown in Figure 19, with shape Z = P 2 . This scaffolding describes X P as a hypersurface of tri-degree  O(1, 1, 2) is not nef, and that its base locus is section of the projection Y S → P 1 × P 1 defined by z 0 = z 1 = 0. Blowing up this base locus we obtain the variety F considered in [9, p.82]. To check smoothness of general hypersurfaces in this linear system, note that general sections of L have the form f = z 2 0 f 1,1 + z 0 z 1 g 1,1 + z 2 1 h 1,1 + z 0 z 2 f 1 + z 1 z 2 g 1 , where f 1,1 and g 1,1 are polynomials of bidegree (1, 1) in x 0 , x 1 , y 0 , y 1 , while f 1 , g 1 are linear polynomials in x 0 , x 1 . Restricting the Jacobian to the locus z 0 = z 1 = 0, we see that the locus {f = 0} is singular precisely when f 1 = g 1 = 0. However this locus is empty for general choices of f 1 and g 1 .
Since the restriction of the base locus of this linear system to a smooth member X is a Cartier divisor in X, its blow up is an isomorphism. Hence such hypersurfaces X are members of the family 4-2, and X P is the central fibre of a toric degeneration in this family.
Rank 4 number 6. Varieties X in the family 4-6 are obtained by blowing up P 2 × P 1 in curves of bidegree (1, 2) and (0, 1) respectively. Consider the polytope P with PALP ID 425, together with the scaffolding S with shape P 1 × P 1 illustrated in Figure 20.
The toric variety Y S is defined by the weight matrix  1, 2). The secondary fan of Y S is illustrated in Figure 21. The variety Y S is isomorphic to P P 1 ×P 2 (O ⊕2 ⊕ O(1, 0)); and the two chambers in the secondary fan correspond to isomorphic varieties -despite the presence of a non-trivial flopping locus. The projection π : Y S → P 2 × P 1 corresponds to projecting out the variables s i for all i ∈ {0, 1, 2}. The toric variety X P is cut out of Y S by the binomial equations These are sections of the line bundles L 1 and L 2 , with weights (1, 1, 1) and (1, 1, 0) respectively. Note that L 1 is nef while L 2 is not. Let X be the vanishing locus of a general section s = l 1 + l 2 of E := L 1 ⊕ L 2 . The section l 1 ∈ Γ(Y S , L 1 ) has the general form s 0 f 1,1 + s 1 g 1,1 + s 2 h 1 , where f 1,1 and g 1,1 have bi-degree (1, 1) in x 0 , x 1 , x 2 and y 0 , y 1 respectively; while h 1 has bi-degree (1, 0). Similarly l 2 has the general form s 0 f 1 + s 1 g 1 , where f 1 and g 1 have bi-degree (1, 1).
Fibres of the restriction of π to X are given by the kernel of the matrix That is, π is a graph away from the locus at which this matrix has rank ≤ 1. This locus in P 2 × P 1 has two connected components, one given by h 1 = f 1,1 g 1 − g 1,1 f 1 = 0, a curve of bidegree (1,2), and the other by f 1 = g 1 = 0, a curve of degree (0, 1). Thus the morphism π exhibits X as a Fano threefold in the family 4-6.
Rank 5 number 1. Varieties in this family are obtained by first blowing up a quadric in a conic -obtaining a variety V in the family 2-29 -and blowing up V in three exceptional lines. Consider the scaffolding S of the polytope with PALP ID 1082 with shape P 2 , illustrated in Figure 22.
That is, we consider general hypersurfaces X of tri-degree (1, 2, 1) in the toric variety Y S defined by the weight matrix s 0 s 1 x 0 x 1 x 2 y 0 y 1 1 1 0 1 1 0 −1 0 0 1 1 1 0 0 0 0 0 0 0 1 1 and stability condition (2, 1, 1). The variety Y S admits a map to P 1 (with co-ordinates (s 0 : s 1 )), giving Y S the structure of a P 2 × P 1 fibre bundle. The variety X also admits a morphism to P 1 , whose fibres are surfaces of bi-degree (2, 1) in P 2 × P 1 . Projecting P 2 × P 1 to P 2 we see that any such smooth fibre is the blow up of P 2 in four (general) points; that is, isomorphic to the del Pezzo surface dP 5 . Hypersurfaces of tri-degree (1, 2, 1) have general form and f i ∈ C[s 0 , s 1 ] are homogeneous polynomials of degree i for each i ∈ {1, 2}. Let X denote the vanishing locus of this polynomial. Note that X contains the surface {x 0 = y 1 = 0}. Fixing a point (s 0 , s 1 ) ∈ P 1 , the dP 5 fibre of the projection X → P 1 is obtained by blowing up the intersection points of the conics C 1 := {x 0 (x 0 f 1 + p 1 ) = 0} and C 2 := {(x 2 0 f 2 + x 0 f 1 q 1 + p 2 ) = 0} in P 2 (with homogeneous co-ordinates (x 0 : x 1 : x 2 )). First consider the case x 0 = p 2 = 0. Choosing a general p 2 , we find two distinct reduced points α 1 , α 2 in C 1 ∩C 2 ; these are independent of the choice of s = (s 0 , s 1 ) ∈ P 1 . The other two solutions depend on s, and lie in the line (x 0 f 1 + p 1 ) = 0. Note that we may choose co-ordinates such that C 1 is defined by {x 0 x 1 = 0}.
Hence we can construct four surfaces, each isomorphic to P 1 × P 1 , contained in X: two surfaces -S 1 and S 2 -swept out by {α i }×P 1 (y 0 :y 1 ) , the surface S 3 swept out by C 1 over P 1 (s 0 :s 1 ) , and the base locus S 4 = {x 0 = y 1 = 0}. Each of these surfaces restrict to exceptional curves in the dP 5 fibres. Note that fibres of X → P 1 are not all smooth -there are two singular fibres -but they are smooth in a neighbourhood of i∈ [4] S i . Hence -applying a relative version of Castelnuovo's criterion -we can have a morphism X → X which contracts the disjoint surfaces S 1 , S 2 , and S 3 to sections of the induced morphism π : X → P 1 (s 0 :s 1 ) . The smooth fibres of π are isomorphic to P 1 × P 1 , while singular fibres have a single nodal singularity; these are isomorphic to P (1, 1, 2). The surface S 4 is the strict transform of a surface S 4 , which intersects every fibre F in a smooth section of − 1 2 K F . Letting ρ(X) denote the Picard rank of X, we have that ρ(X) = ρ(X ) + 3. Since X Pand hence X -has degree 28, we can conclude from the classification of Fano 3-folds that if ρ(X ) ≥ 2, X is in the family 5-1. This is easily seen from the Leray spectral sequence indeed -since H 1 (F, Q) = 0 for all fibres F of π -we have b 2 (X ) = 1 + h 0 (P 1 , R 2 π Q). However h 0 (P 1 , R 2 π Q) ≥ 1 since the surface S 4 defines a non-trivial class in H 2 (F, Q) for every fibre F . Remark 4.7. Comparing our construction with that made by Mori-Mukai [19], they first consider the blow up of a quadric threefold in a conic. Restricting the projection P 4 P 1 this blow-up defines X , a quadric surface bundle over P 1 with two singular fibres (with singularities are disjoint from the exceptional locus). Note that the exceptional locus distinguishes a conic C in each fibre of π. To obtain varieties in 5-1 we then blow-up X in three exceptional lines. These lines are sections of the map X → P 1 defined by a triple of points on the distinguished conic C in each fibre. That is, the surface S 4 is the strict transform of the exceptional locus obtained by the blow-up of the quadric threefold; while S i , i ∈ {1, 2, 3} are obtained by blowing up exceptional lines.

4.1.
Products. The remaining non-toric Fano threefolds X with −K X very ample are products of non-toric del Pezzo surfaces with P 1 . That is, dP k × P 1 for k ∈ {3, 4, 5}. We can easily construct toric degenerations of these from degenerations of dP k for each k. Fix a reflexive polygon Q such that Q • is cracked along the fan of a shape variety Z , together with a scaffolding S of Q with shape Z . We can produce a scaffolding S of conv(Q, (0, 0, 1), (0, 0, −1)) with shape Z := Z × P 1 by setting S = {(π 1 (D), χ) : (D, χ) ∈ S } ∪ {π 2 D} where D is the toric boundary of P 1 , and π i is the i th projection from Z × P 1 . The example of dP 3 × P 1 , together with a scaffolding with shape Z = P 2 × P 1 is illustrated in Figure 23, setting a = 1 and b = 3. We thus produce toric degenerations embedded in the following spaces.
4.2. −K X not very ample. There are 7 families of Fano threefolds X for which −K X is not very ample. These fall into three distinct groups. We first consider the varieties a sextic in P(1, 1, 1, 2, 3) ; and, • V 2 , a sextic in P (1, 1, 1, 1, 3).  Writing x i for homogeneous co-ordinates of degree 1, and y, z for those of degree 2 and 3 respectively, B 1 degenerates to the toric hypersurface x 2 yz = x 6 0 ; while V 2 degenerates to the toric variety x 1 x 2 x 3 z = x 6 0 . These toric varieties correspond to scaffoldings of non-reflexive toric varieties with shape P 2 and P 3 respectively. The scaffolding used to construct B 1 is illustrated in Figure 24 in the case (a, b) = (2, 6). The details of these constructions follow those described in §3.3.
The next three families have Picard rank 2.
• 2-1, the blow up of B 1 is an elliptic curve formed by intersecting two members of − 1 2 K B 1 . • 2-2, a double cover of P 1 × P 2 branched along a divisor of bidegree (2, 4). • 2-3, the blow up of V 2 is an elliptic curve formed by intersecting two members of In each case a toric complete intersection construction is given in [9], and each construction admits a toric degeneration to an embedding described by Laurent inversion. The corresponding scaffoldings have shapes P 2 × P 1 , P 3 , and P 2 × P 1 respectively. Letting (x 0 : x 1 : x 2 : y : z) be homogeneous co-ordinates on P (1, 1, 1, 2, 3), and (s 0 : s 1 ) be co-ordinates on P 1 , varieties in the family 2-1 degenerate to the toric variety given by the binomial equations 1, 1, 2, 3) × P 1 . Varieties in the family 2-3 degenerate to the toric variety given by the binomial equations where (x 0 : x 1 : x 2 : x 3 : y) are homogeneous co-ordinates on P (1, 1, 1, 1, 2). Finally, varieties in the family 2-2 degenerate to the hypersurface x 1 y 1 y 2 w = x 2 0 y 4 0 in the variety F described in [9, p.25].

5.
Classifying cracked 3-topes 5.1. One dimensional shape variety. We refer to polytopes cracked along the fan of P 1 as cracked in half, since their intersection with a pair of half spaces form unimodular polytopes. This class of polytopes is explored in greater detail -and in the four dimensional settingin [10].
Since -by [29, Proposition 2.5] -polytopes cracked in half are reflexive, we can proceed from the classification of reflexive 3-topes. Given a reflexive polytope P ⊂ M R , we define V P to be the vector space spanned by the vertices v ∈ P such that the tangent cone C v to P at v is not unimodular. If P is cracked along P 1 these must lie in a proper linear subspace of M R . Moreover, by [29,Proposition 2.8], no facet of P • contains an interior point. We use Magma to search for reflexive polytopes meeting both these conditions, and obtain a list of 91 reflexive 3-topes. In 73 cases V P is two-dimensional, and hence unique determines the direction of the line segments used to scaffold P • . The remaining polytopes contain a square facet, which admits two possible full scaffoldings.
Testing which of these 91 polytopes are cracked in half, we find there are 82 three dimensional polytopes cracked along the fan of P 1 ; we list these reflexive polytopes in Table 6. These polytopes are specified by the Kreuzer-Skarke list of reflexive 3-topes. Note that, as elsewhere, we index this list from zero. The column Fano indicates the families Fano threefolds X for which there is a mirror Minkowski polynomial -see [8,9] -f such that Newt(f ) is isomorphic to the reflexive polytope with the indicated ID. Note that in each case there is at most one such family of Fano threefolds. Applying Laurent inversion to a full scaffolding PALP ID Fano PALP ID Fano PALP ID Fano  Table 6. Reflexive polytopes cracked in two. on P with shape Z = P 1 , we obtain X P as a Fano hypersurface. We expect to recover X by passing to a general hypersurface, although we have only partial results in this direction. Table 6 with no associated Fano threefold, X P is not smoothable.

Proposition 5.1 ([27]). For each P in
Proof. The list of reflexive 3-topes with no associated Fano in Table 6 is a subset of the list of non-smoothable Fano threefolds which appears in work of Petracci [27, p.10].  Table 6 such that each torus invariant point of X P is either a smooth point, or an ordinary double point, X P smooths to the associated Fano indicated.
Proof. By Namikawa's results [25] all such toric varieties admit a smoothing. The invariants of the smoothed varieties were computed by Galkin in [13].
Note that the set of primitive ray generators is empty in the case Z = P 1 , and need not be a subset of the vertex set of P for any choice of shape Z.
Lemma 5.4. Given a shape variety Z determined by a fan Σ in M R , and a ray ρ ∈ Σ [1], let Z ρ denote the codimension one torus invariant subvariety of Z determined by ρ. There is a canonical inclusion, with bounded image, from the set wrapping polyhedra of reflexive polytopes P cracked along Σ to the set of lattice points in the cone ρ∈Σ [1] Amp(Z ρ ) × (M R /Rρ) .
Proof. Fix a splitting M ∼ = v ⊕ M ρ , and let Σ ρ denote the fan in M ρ determined by Z ρ . The tangent cone at v to a wrapping polyhedron for Σ determines -and is determined by -a piecewise linear function θ : (M ρ ) ⊗ Z R → M R which is linear on each cone of Σ ρ , sends 0 → v, and sends the cones of Σ ρ into their corresponding cones in Σ. The connected component of the complement of the image of θ which contains 0 must be a convex set. Such maps θ are in bijection with points in Amp(Z ρ ) × (M R /Rv) ⊂ Div T Mρ (Z ρ ) ∼ = Z r , for some r ∈ Z ≥0 . Hence the set of possible wrapping polyhedra is contained in the cone required.
To show this region is bounded, first note that each ray τ of Σ ρ corresponds to a cone in Σ of dimension 2; generated by v and some v ∈ M . Since v must be in the same connected component as 0 of M R \ θ((M ρ ) ⊗ Z R), the co-ordinate of θ -regarded as an element of Z r -corresponding to τ is bounded. Each pair (ρ, τ ), where ρ ∈ Σ[1] and τ ∈ Σ ρ [1] defines a linear inequality satisfied by any tuple of piecewise linear maps θ which define a wrapping polyhedron. The intersection of these half spaces with Amp(Z ρ )×(M R /Rρ) defines a polytope, R Σ , which contains the image of each wrapping polyhedron.
Given a fan Σ, we call a polygon contained in a two dimensional cone of Σ a panel. Given an element ϕ ∈ R Σ , we let S(ϕ) denote the set of tuples of panels whose tangent cone at the generator of ray ρ of Σ is given by the projection of ϕ to the cone Amp(Z ρ ) × (M R /Rρ).
Definition 5.5. Let Q be a unimodular hollow polytope in M R . We call Q a (reflexive) piece if 0 ∈ Q and -for any facet F of Q with inner normal vector w -w(F ) = 0 if 0 ∈ F , and w(F ) = −1 otherwise.
The set of reflexive pieces has an obvious iterative structure: faces of reflexive pieces which contain the origin are themselves reflexive pieces. Thus the classification of reflexive pieces of dimension n makes use of the classification in dimensions < n. If Q is a 3-tope there are four cases, depending on the minimal dimension d of the face of Q containing 0. In particular either (i) Q is a reflexive polytope; (ii) the origin is the unique interior point of a facet of Q; (iii) the origin is the unique relative interior lattice point of an edge of Q, or; (iv) the origin is a vertex of Q, and every edge of Q containing v has lattice length 1.
Note that this generalises both the notion of reflexive polytope (the first case) and the notion of top [4] (the second case).
Assuming that the minimal face of Q containing 0 has dimension d, we say that a piece Q has type 3 − d. Given a smooth cone -with minimal face of dimension d -and choices panels {p 1 , . . . , p d } in each of its facets we can attempt to classify all possible pieces of type 3 − d whose facets are given by the specified panels. We let P(p 1 , . . . , p d ) denote the set of possible pieces with facets given by the polygons p 1 , . . . , p d .
Algorithm 5.6. Fix a complete fan Σ in N such that the dimension of the minimal cone of Σ is at most one.
(i) Compute the integral points in the polytope R Σ . (ii) Exploit symmetries of Σ to obtain a minimal subset R of R Σ , containing a representative of every isomorphism class of cracked polytope in N R . (iii) Compute the set S(ϕ) for each point ϕ ∈ R, and iterate over this set of tuples of panels. (iv) For each pair ϕ ∈ R, p ∈ S(ϕ), and maximal cone σ ∈ Σ let {p 1 , . . . , p d } be the multiset of panels contained in facets of σ (note that d ∈ {2, 3}). There is a finite subset A(ϕ, p, σ) of P(p 1 , . . . , p d ) such that for each polytope Q in this subset, w(v) ≥ −1 for all inner normal vectors w to facets of Q and vertices v of polygons in p. (v) Iterate over all functions from the set of maximal cones σ in Σ to A(ϕ, p, σ). Test whether the union of these polytopes is itself a convex, reflexive, and cracked polytope.

Classifying Pieces.
In order to implement Algorithm 5.6 in dimension n we require a database of pieces in dimension ≤ n. We now treat the classification of pieces in dimension ≤ 3. Note that the classification in dimension n divides into cases depending on the dimension k of the minimal face containing Q. The cases k = n and k = n − 1 form known classes: indeed, if k = n, the corresponding pieces are polar dual to smooth polytopes, which have a well-known classification up to dimension 8 by Øbro [26]. If k = n − 1 the definition of reflexive piece coincides precisely with the notion of a top [4,12] which is also a unimodular polytope; we call such polytopes unimodular tops.
In dimension one there are two possible cases, depending on the dimension k of the minimal face of P containing 0: • If k = 1, P = conv(−1, 1) is a line segment of length two.
It is well-known that hollow polytopes in dimension two are either Cayley polytopes or equal to T := conv((0, 0), (2, 0), (0, 2)) up to integral affine linear transformations. We have three cases for pieces P in R 2 , depending on the dimension k of the minimal face of P containing 0: • If k = 2, P is a reflexive polytope, of which five are unimodular.
In dimension three we have four possible cases depending on k. In the case k = 3, P is a unimodular reflexive polytope, of which there are 18. If k = 2, P is a unimodular top. We do not describe the classification of unimodular tops in dimension 3, as the algorithm used in case Z = P 1 -see §5.1 -does not rely on this classification. Moreover, this classification is contained in that of all three dimensional tops made by Bouchard-Skarke [3].
Lemma 5.7. Let P i , i ∈ [k] be a collection of d-dimensional lattice polytopes in R d . If P := P 1 · · · P k ⊂ R d+k is a unimodular polytope, there is a non-singular projective toric variety Z such that P i is the polyhedron of sections of an ample divisor D i on Z for all i ∈ [k].
Proof. Since, for any i 1 , i 2 ∈ [k], P i 1 P i 2 is a face of P , we assume without loss of generality that k = 2. Since P 1 is unimodular, its normal fan defines a non-singular projective toric variety Z. We claim that P 2 = P D for some ample divisor on Z.
Note that verts(P ) = verts(P 1 ) verts(P 2 ). Moreover, each vertex of P 1 is contained in d edges of P 1 and (d + 1) edges of P . Hence, fixing a facet F of P different from P 1 and P 2 , F ∩ P 1 is equal to a facet G of P 1 . G contains (d − 1) edges of P 1 incident to v.
The normal fan of P consequently contains a ray for each facet of P 1 (or P 2 ), as well as rays ρ 1 , ρ 2 dual to P 1 and P 2 respectively. Moreover, each vertex of P 1 is dual to a maximal cone, generated by ρ 1 and rays corresponding to facets of P 1 containing v. Since the same applies to vertices of P 2 , the toric variety associated to the normal fan of P has the structure of a fibre bundle over P 1 , in particular the fibres over 0 and ∞ are isomorphic.
We claim such pieces P are determined by the facet F = P ∩ {u : u, e 3 = −1}. Indeed, fixing this polygon F it is easy to verify that P = A ∩ (F × R).
We summarise the above calculations in the following Proposition.
Proposition 5.10. If P is a 3-dimensional piece such that the origin is contained in an edge of P , then P belongs to one of the infinite families P (α, l, j), one of the three exceptional cases shown in Figure 25, or one of the polytopes listed in Lemma 5.9.  Finally, assume that k = 0. For each l ∈ Z ≥0 and j ∈ {1, 2}, we define the Cayley polytopes Q(α, l, j) to be the intersection of P (α, l, j) with the half-space {u ∈ R 3 : e 3 , u ≥ 0}.
Proposition 5.11. If P is a 3-dimensional piece such that the origin is contained in an edge of P , then P belongs to the infinite family Q(α, l, j). The polytope Q(α, l, j) is a reflexive piece if and only if one of the following hold.
Note that the only case which occurs in the fourth case is the standard simplex.
Proof. In a suitable co-ordinate system, the vertex set of a piece P contains 0 and each of the three standard basis vectors. The polygon F i := {e i = 0} ∩ P is a two dimensional reflexive piece, which are classified above.
In particular we may assume that each polygon F i is either a standard triangle or a Cayley sum of line segments. These polygons may be oriented relative to each other in two distinct ways, illustrated in Figure 27. We show that the first case does not include any piece which is not a special case of the second. Polytopes in the first case contain vertices (1, 0, k 1 ), (k 2 , 1, 0), and (0, k 3 , 1). Note that we can assume that k i ≥ 2. If k i > 2 for any i ∈ [3], the lattice point (1, 1, 1) is in the interior of the convex hull of the vertices of P , and hence k 1 = k 2 = k 3 = 2. However, as P is contained in the half space {u ∈ R 3 : (1, 1, 1) · u ≤ 3}, P is a sub-polytope of the convex hull P of the vertices shown in Figure 28. Note that every vertex of this polytope is contained in a panel, and hence P = P . Since P is not unimodular it does not contribute to the list of pieces.
In the second case illustrated in Figure 29, we observe that P is a Cayley polytope. Indeed, assuming that P contains the vertices (1, 0, k 1 ), (k 2 , 1, 0), and (0, 1, k 3 ), P is the Cayley sum of the facets contained in H 0 and H 1 , where H k := {u : e 2 , u = k}. These are both 2dimensional if α 1 ≥ 0 and α 2 ≥ 0; and in this case it follows from Lemma 5.7 that P is of the form Q(α, l, j) for some l ∈ Z ≥0 and j ∈ [2]. The classification of the remaining possible pieces follows from a case-by-case analysis. The case α = (−1, −1) is trivial. If α = (0, −1), P is contained in the product of a standard simplex and a ray, and equal to some Q(α, 1, l). If α 1 > 0 and α 2 = −1 we note that the polytopes Q(α, 1, l) are not unimodular, while Q(α, 2, l) is a Cayley polytope P 1 P 2 , such that P 1 is a standard simplex. By Lemma 5.7 P 2 is a dilate of a standard simplex, and hence l = α 1 + 1.