Box Graphs and Resolutions I

Box graphs succinctly and comprehensively characterize singular fibers of elliptic fibrations in codimension two and three, as well as flop transitions connecting these, in terms of representation theoretic data. We develop a framework that provides a systematic map between a box graph and a crepant algebraic resolution of the singular elliptic fibration, thus allowing an explicit construction of the fibers from a singular Weierstrass or Tate model. The key tool is what we call a fiber face diagram, which shows the relevant information of a (partial) toric triangulation and allows the inclusion of more general algebraic blowups. We shown that each such diagram defines a sequence of weighted algebraic blowups, thus providing a realization of the fiber defined by the box graph in terms of an explicit resolution. We show this correspondence explicitly for the case of SU(5) by providing a map between box graphs and fiber faces, and thereby a sequence of algebraic resolutions of the Tate model, which realizes each of the box graphs.

The concise and representation-theoretic description of singular fibers in terms of box graphs is highly suggestive of the existence of a more unified, elegant approach to resolutions of singular elliptic fibrations. The goal of this paper and of the followup [16] is to develop resolution methods for singular elliptic fibrations which provide an explicit map between a given box graph and an associated resolution of the singular fibration.
The framework that we propose is a hybrid between toric resolutions 1 and algebraic blowups: we use partial toric triangulations, represented in terms of fiber face diagrams, which in turn determine a resolution sequence of weighted projective blowups. The various subcases that fall into this framework are: • Standard toric triangulations, which have a description in terms of weighted blowups as is known from e.g. [17] 2 • Standard algebraic resolutions, which correspond to the specialization to unit weights • Algebraic resolutions leading to a realization of the fiber as a complete intersection, which appeared already in the resolution studied in [14,13,4].
Our proposal is to use the top [18,19] corresponding to a degenerate fiber as an organizing tool for weighted blowups, which realize the different box graphs, or equivalently Coulomb phases.
This realization by direct blowups guarantees in particular projectivity of the resolved space. Each phase or box graph can be mapped to a resolution by computing the splitting of fiber components over codimension two loci in the base. Even though it is not possible to obtain all box graphs by a triangulation of the top, we can use partial triangulations to map out the entire network of the corresponding resolutions. Such partial triangulations correspond to only partial resolutions, after which singular loci are still present. We may then continue the resolution process in ways which can not be obtained through straightforward triangulation of the top, e.g. turning the Tate form into a complete intersection. Keeping this in mind, we hence display the partial triangulation of the top which is relevant to obtain each phase.
The main advantage to this way of or organizing the resolutions is that it is systematic and is amenable to generalization [16].
In all but one 3 of the box graphs/phases for su (5), the associated resolutions of the Tate form are given as a hypersurface 4 or complete intersection of codimension two. In the remaining case, we need to blow up along a divisor realized as a determinantal variety. This turns the Tate model into a non-complete intersection.
In summary, we propose the following correspondence between box graphs and algebraic resolutions of singular elliptic fibers, via fiber face diagrams: Box Graphs The box graphs determine the codimension two fibers, or equivalently Coulomb branch phases. From the splitting of the fibers in codimension two we determine an associated fiber face diagram, which is based on the top of the fiber in codimension one. This in turn determines a sequence of algebraic resolutions of the Tate form. In the present paper we develop this direct correspondence for su (5), with codimension two fibers associated to the representations 5 and 10, construct the fiber face diagrams, and associated associated weighted blowups.
Through direct comparison of the fibers in codimension two with the box graph we establish the correspondence. Finally, it is possible to also map all the flops into flops of the resolved geometries, and both networks are in agreement.
The plan of this paper is as follows. Section 2 is a lightning review of box graphs, with a focus on the su(5) case. In section 3 we discuss crepant weighted blowups and how to systematically determine these for a given singularity. In section 4 we discuss the precise correspondence between triangulations, fiber faces and weighted blowups for SU (5). Finally in section 5 we discuss the determinantal blowups. The main result is table 1. Here, the correspondence is succinctly summarized for all cases, as well as the networks of flop transitions in box graph and fiber face presentation as given in figures 4 and 11.
Note added: As we were completing this paper, another work [20] appeared which claims to also construct all the su(5) resolutions, based on the earlier work on su(n), n = 2, 3, 4 [21]. In v2 of [20] it is erroneously claimed that the resolutions in the present paper are restricted to "the special case of singular Calabi-Yau hypersurfaces in compact toric varieties". The crepant resolutions we construct can be applied to any singular elliptic fibration for which the fiber is embedded in P 123 .

Box graph primer
The main result of [3] is the chacracterization of singular fibers in higher codimension of an elliptic fibration in terms of representation theoretic objects, the box graphs. The goal of this paper is to develop a precise map between explicit resolutions of singular fibrations and the data describing singular fibers in higher codimension that is encoded in the box graphs. We will start with a brief primer on how to use box graphs to determine the codimension two and three fibers. Consider simple Lie algebras g ⊂ g, and let R be a representation of g, with weights λ i , i = 1, · · · , d = dim(R), such that the adjoint of g decomposes as 5 g → g ⊕ u (1) adj( g) → adj(g) ⊕ adj(u(1)) ⊕ R + ⊕ R − , (2.1) For the present paper, the case of interest is g = su(n), and R = n or Λ 2 n, in which case g = su (6) and so (10), respectively. The representation graphs, including the action of the simple roots, are shown in figure 1. In this case the weights will be denoted by L i , i = 1, · · · , n, and L i + L j , i < j, respectively, with the tracelessness condition A box graph for the pair (su(n), n) is a sign (color) decorated representation graph of n, i.e.
which satisfies the following two conditions: • Flow rules: If i = +, then j = + for all j < i. Likewise, i = −, then j = − for all j > i: • Diagonal condition: The signs i cannot all be the same. This follows from the fact that L i = 0 for su(n), and thus the trace should not have a definite sign.
A box graph for (su(n), Λ 2 n) is again a sign-decoration or coloring of the representation graph of Λ 2 n, with weights again satisfying the constraints: • Flow rules: If i,j = +, then k,l = + for all k < i and l < j, i.e. " + signs flow up and to the left".
Likewise if i,j = −, then k,l = − for all k > i and l > j, i.e. "-signs flow down and to the right".
• Diagonal condition: The signs along the diagonals (defined below) cannot all be the same. And example is shown in figure 1. This is again related to the trace, and differentiates between su(n) and u(n) box graphs: For n = 2k: ( 1,2k , 2,2k−1 , · · · , k,k+1 ) = (+, · · · , +), (−, · · · , −) For n = 2k + 1: The box graphs can equivalently be described in terms of the convex path, that separates the + and − sign boxes. For su(n), this path has to cross the diagonals (2.5), and therefore is called an anti-Dyck path.
Each box graph corresponds to a small resolution of an elliptic fibration with codimension one singular fiber specified by the Lie algebra g via Kodaira's classification. Here, we will summarize the rules for how to determine the splitting of the codimension one fiber into the codimension two fiber, as well as the intersections of the fiber components. Let us denote the curves associated to the simple roots α i and weights λ by The initial fiber is given by I n , where the intersection matrix between F i and the curve associated to the zero section F 0 = − F i , is given by the affine su(n) Cartan matrix. Given a box graph, we can read off which curves F i split along the codimension two loci, and secondly, what the intersections of the irreducible fiber component are: • Fiber splitting rules: If adding the simple root α i crosses from a + to − box (i.e. it crosses the anti-Dyck path) then the associated curve F i splits. If not, then F i remains irreducible.
• Extremal generators: The extremal generators of the cone of effective curves above the codimension two locus that the box graph describes are the irreducible F i , as well as the extremal curves, which are defined as follows: a curve C λ is extremal, if changing the sign of the box associated to λ maps the graph to another decorated representation graph, that satisfies the flow rules. These extremal curves, which always lie along the anti-Dyck path, will  (5), with the action of the simple roots L i − L i+1 on the diagram as shown along the edges. In the representation graph for 10, the red boxes correspond to the 'diagonal' (2.5), i.e. the signs of these three boxes cannot be the same in an su(5) box graph. .
be marked by an X in the box graph. An extremal curve cannot necessarily be flopped (sign changed), as this might violate the diagonal condition. If it can be flopped, it will be marked by a black X, otherwise by a red X.
• Intersections: The extremal curves C ± λ intersect the irreducible F i by ±1 if adding the corresponding root to λ retains/changes the sign. We define the intersection with the (representationtheoretically prefered) sign convention, where D i is the divisor dual to the curve F i (2.7)

Box graphs and singular fibers for su(5)
For the fundamental representation 5, the box graphs are based on the representation graph shown in figure 1. Here, L i are the weights, and L i − L i+1 the simple roots, which act between the weights. Similarly, for 10 the representation graph can be written in terms of the weights L i,j = L i + L j , with i < j, i, j = 1, · · · , 5, and the simple roots act as indicated in figure 1.
The box graphs for su(5) with fundamental 5 and/or anti-symmetric 10 representation, which characterize the fibers in codimension two and three of the elliptic fibration with I 5 Kodaira fiber in codimension one, were determined in [3]. The box graphs for each of these situations are shown in figures 2, 3 and 4, respectively. The main result in [3,5] is that the box graphs determine the complete set of small resolutions, which is characterized by the fibers in codimension two and three. The extremal generators, which in the geometry correspond to the curves that can be flopped, are marked with a black X, whereas red X's indicate cone generators which cannot be flopped as they would yield u(5) phases. The green line marks the anti-Dyck path. Figure 3: Box graphs for su(5) with 10 representation. Two box graphs that are connected by a black line can be flopped into each other. The extremal generators, which in the geometry correspond to the curves that can be flopped are marked with a black X, whereas red X's indicate cone generators which cannot be flopped as they would yield u(5) phases.

Singular fibers for 5 Representation
The possible box graphs are shown in figure 2. We denote the curves with a ± sign associated to the weight L i by C ± i . It is clear that these are all possibilities that satisfy the flow rule and diagonal condition. In each diagram there is exactly one simple root that splits by the rules specified in section 2.1. For instance, in the first box graph, the blue (+) and yellow (-) separation is between L 4 and L 5 , i.e. adding α 4 = L 4 − L 5 changes the sign, and thus F 4 splits into C + 4 + C − 5 . The resulting splittings, extremal generators of the cone of effective curves, and the new intersections are as follows, and give rise to I 6 fibers in all cases, as shown in (2.8).

# Box Graph
Splitting Generators Intersections (2.8) Note that for each resolution, there is one simple roots that split into two weights C + i and C − i+1 , which are marked with X in the box graphs. These intersect each other transversally, and with the remaining irreducible roots to form an I 6 Kodaira fiber in codimension two.

Singular fibers for 10 Representation
The fibers of the 10 representation are obtained similarly from the box graphs in figure 3.
There is a Z 2 symmetry that corresponds to reversal of the ordering of simple roots of su(5), so that we only need to discuss half of the box graphs. The resulting fibers are all I * 1 , consistent 11 with the local enhancement to so (10), and the splittings produce the correct multiplicities: # Box Graph Splitting Generators Intersections

Combined box graphs and flops
The possible combined box graphs are obtained by consistently combining the ones from 5 and 10, which turns out to be equivalent to consistent su(6) box graphs with the 15 representation [3]. This structure encodes also codimension three information, as was shown there, and allows to compute all possibly non-Kodaira fibers along the e 6 enhacement loci. The combined flop graph is shown in figure 4. Each box graph is combined from one 10 and one 5 box grahs, carrying labels (arabic, roman), and the combined resolved geometry has to exhibit both types of splittings, as determined in (2.8) and (2.9).
The flops are either with respect to curves corresponding to 5, or to 10 weights. This again is easily read off from the flop network figure 4: if two box graphs are connected, they differ by either their arabic or roman numeral. Correspondingly, a 10 or 5 curve is flopped.
We labeled all connecting lines with the curves that are being flopped. In figure 4, each connecting line is labeled by the curve, C i or C ij , that is being flopped.   Figure 4: Box graphs for su(5) with both 5 and 10 representation. The extremal generators, which in the geometry correspond to the curves that can be flopped, are marked with a black X, whereas red X's indicate cone generators, which cannot be flopped as they would yield u(5) phases. The lines connecting the box graphs are labeled by the curve that is being flopped, i.e. C i (C ij ) corresponds to flopping a 5 (10)curve.

13
In this section we explain how to determine weighted blowups that give rise to crepant resolutions. One of the organizational tools is to use the connection between toric triangulations, which we define to be toric resolutions, based on fine triangulations of polytopes 6 , and weighted blowups. Such toric triangulations form a strict subclass of possible crepant resolutions. However, the way we will characterize these will be generalized and extended to resolutions, do not necessarily arise from a (fine) triangulation. These generalizations will be discussed for SU (5) in the next sections and in general in [16]. Basic definitions and facts from toric geometry (as well as an explanation of our notation) are contained in appendix A.

Cones and Toric Resolutions
Consider a toric variety described in terms of a fan. For any given fan Σ, we may consider a refinement Σ in which we consistently subdivide cones. By construction, there is a projection π : Σ → Σ such that any cone of Σ is mapped to a single cone of Σ. Hence there is an associated toric morphism which gives rise to a proper birational map T Σ → T Σ , i.e. we may think of a refinement of a fan as a (generalized) blowup and a fusing of appropriate cones as a blowdown. A simple refinement of a 3-dimensional cone is shown in figure 5.
Let us now consider an algebraic subvariety X of a toric variety T Σ . X has singularities if singularities of T Σ meet X or the defining equations of X are not transversal. We can try to (partially) resolve such singularities by refining the cones of the fan Σ, which is what we will discuss in the following. Consider a singularity of X coming from the non-transversality 7 of one of its defining equations P (z 1 , · · · z n ) = 0 , (3.1) along a locus z 1 = · · · z k = 0. This singularity has the codimension k − 1. We can now easily describe blowups along this locus by a toric morphism of the ambient space. As z 1 = · · · z k = 0 are allowed to vanish simultaneously by assumption, they must share a common cone σ = v 1 , · · · , v k . Hence we want to refine the cone σ by introducing a new one-dimensional cone with generator v E and appropriate higher-dimensional cones. In the simplest case, where v E 6 These are resolutions that are commonly referred to as toric resolutions, for instance in the context of triangulations of tops and polytopes. However, we will consider more general toric resolutions, that do not directly correspond to such triangulations, but to more general refinements of cones. As the resolved are projective, which follows form the direct blowup procedures, one can of course construct an extended polytope whose triangulation yields the resolution. However, for a systematic analysis of all possible crepant resolutions, our approach is more efficient. 7 To check this, we have to go to a patch where we can use a set of affine coordinates.  Figure 5: The subdivision of a three-dimensional cone σ by introducing a new one-dimensional cone in its interior. On the left, a (simplicial) three-dimensional cone generated by the three lattice vectors v 1 , v 2 and v 3 is displayed. We have also included the lattice point v E we wish to use for the subdivision, which is drawn in red. Note that this point does not need to lie on the hyperplane supporting v 1 , v 2 and v 3 . The refinement of σ including v E is shown on the right. This refinement introduces three three-dimensional cones, three two-dimensional cones and the one-dimensional cone generated by σ. This figure can also be used as an example of a toric blowdown: if we have three three-dimensional cones sitting in a fan as shown on the right, we can blow down the coordinate corresponding to the lattice vector in the interior. This will eliminate three two-dimensional cones and glue three three-dimensional cones into a single one. Note that the combinatorics will be more complicated if σ is a three-dimensional cone in a fan of four dimensions or more.
is in the interiour of an n-dimensional cone, This means that the Stanley-Reisner ideal now contains z 1 · · · z n , or written in terms of projective relations more commonly used for algebraic resolutions, [z 1 , · · · , z n ]. We have shown two elementary examples of such subdivisions in figures 5 and 6. We will frequently be interested in displaying such cones and their subdivisions, for which we will introduce cone diagrams.

Cone diagrams
We now define cone diagrams, which are one of the tools that we will use to systematially describe resolutions of singular fibrations. Instead of depicting the entire cone of a fan, as in figure 5, we will consider diagrams, such as the one shown in figure 6, which are more convenient visualizations using a projection. This is done such that the relevant combinatorics is kept intact 8 and the relative locations of the various cones are faithfully represented.
We Figure 6: The subdivision of two three-dimensional cones by introducing a new onedimensional cone interior to their intersection. Contrary to figure 5, we are using a projection in which p-dimensional cones are mapped to (p − 1) -dimensional simplices, keeping their combinatoric intact. We refer to such illustrations as cone diagrams. In the specific example shown, we wish to introduce a new one-dimensional cone generated by v E , which sits in the interior of a two-dimensional cone generated by v 2 and v 4 . If these cones are part of a threedimensional fan, this kind of blowup will subdivide each of the two adjacent cones in two. Starting from the fan on the right, it is possible to blow down v E . Note that this will necessarily take us back to the figure on the left as the resulting cones after the blowdown must be strongly convex. Note that the four vertices v 1 · · · v 4 are not necessarily on a hyperplane.
will call such pictorial representations of one or several cones cone diagrams 9 . We will use cone diagrams to describe partial triangulations (or resolutions), and most importantly, to characterize which triangulations (or crepant resolutions) can still be applied to further resolve the geometry. It is important to keep in mind that such a representation is not possible for any collection of cones in fan. In the situations we encounter, however, this presentation allows us to restrict ourselves to the salient information.

Toric Resolutions as Weighted Blowups
Let us return to subdivisions such as the one displayed in figure 8. We will now realize this toric resolution in terms of a weighted blowup in the coordinates z i . For a cone σ and a point v E in the inside of σ, we can write where ρ E is the one-dimensional cone associated to v E . In the toric variety corresponding to the refined fan Σ , we have a new homogeneous coordinate z E . Due to the above relation, there is also a new C * action with the weights Depending on the details at hand, this will reproduce customary algebraic blowups, but also naturally includes cases with non-trivial weights, see [17] for a classic exposition. For such a weighted blowup we will use the notation In these cases, the ambient space can potentially become singular. The power of describing these data in terms of a fan is that it is easy to trace the fate of the singularity as we are blowing up and determine the singular strata of the ambient space Σ .
Whereas a refinement of cones in a fan Σ can also be conveniently captured in terms of projective relations, the situation is more subtle for blowdowns. Here, the language of fans allows us to determine when such a blowdown can be carried out at the level of the ambient space: we need to be able to consistently eliminate cones from the fan and/or glue

Crepant weighted blowups
In this paper, we are interested in crepant resolutions, so that we only want to consider (partial) resolutions keeping the canonical class invariant. The anticanonical bundle of a toric variety is where the sum goes over all one-dimensional cones in Σ, i.e. all toric divisors. If we perform a blowup associated with a refinement Σ → Σ which introduces a single one-dimensional cone with generator v E , the anticanonical class of T Σ hence receives the contribution This tells us that the above only is a crepant (partial) resolution of X if its class after proper In other words, the proper transform must allow us to 'divide out' the right power of the exceptional coordinate z E to make P (z i ) aquire the weight (−a E + i a i ) under the C * action (3.4).
, (3.9) i.e. we simply need to use ( Hence only blowups related to the introduction of new generators v E satisfying the above relation can be crepant. For a given singularity, this will single out a finite number of crepant weighted blowups. After performing such a weighted blowup (cone refinement), the set m j of monomials is not changed, i.e. at every step of a sequence of blowups we find the same condition (3.10) for the next step. We hence learn that we can only use weighted blowups originating from the set of v E satisfying (3.10) in any step of a sequence of blowups.
Note that even though we have used toric language, the result stands on its own. We may completely discard all of the toric language at this point and merely proceed to carry out the weighted blowups we have found. We will however, continue to use the diagrams associated with the fan spanned by the v, as these conveniently encode the projective relations (i.e. the SR ideal) of the ambient space coordinates.
In the discussion above, we have assumed that the locus we want to blow up can be described by the vanishing of a set of homogeneous coordinates of the ambient space. The above discussion is still applicable, however, if we appropriately enlarge the dimension of the ambient space we are working with.
Let us give a schematic example and describe the blowup of a hypersurface X given by P = 0 in a toric variety along the locus for some homogeneous polynomial φ(z i ). The trick is to introduce another coordinate z φ which lifts φ(z i ) to a coordinate of the ambient space. We hence ask this new coordinate to fulfill the equation z φ = φ(z i ), which by homogeneity also uniquely fixes the weights of z φ .
After fixing v φ we lift the generators v i of one-dimensional cones in Σ to v i in n+1 dimensions such that the scaling relations involving z φ are reproduced. The lift of the fan Σ, Σ , is then obtained as follows. For every p-dimensional cone of Σ we add z φ as an extra vertex, turning it into a p + 1-dimensional cone of the lifted fan Σ . We have now increased the dimension of the ambient space by one and gained a further equation. In particular, we have managed to place the locus we intend to blow up along the intersection z 1 , · · · , z k = z φ = 0 of toric divisors. We can now perform a blowup by introducing a new generator v E and subdividing the cone v 1 , v 2 , · · · , v k , v φ appropriately. The resolved complete intersection is then given by two equations of the form The description as a complete intersection was redundant before this blowup (we could simply solve the equation of z φ and discard this coordinate), however becomes non-trivial after the blowup. Resolutions of this type will form another subclass of algebraic resolutions that are necessary in order to construct all possible small resolutions.
If we restrict ourselves to the case of Calabi-Yau varieties, the above discussion boils down to the reflexive polytopes of [22][23][24].

Flops
We now turn to a discussion of flops in the toric context. A flop is realized by blowing down a subvariety of codimension two of X and resolving to a different manifoldX. In toric geometry, such objects correspond to two-dimensional cones 11 . To see if we can do a flop, we hence have to ask if we can consistently remove a two-dimensional cone from Σ and replace it with a different one. The prototypical example is shown in figure 7. We will encounter more complicated examples in the rest of this paper.

Fiber Faces and weighted blowups for SU (5)
In this section we will apply the general insights obtained in the last section, and construct an explicit algebraic resolution sequence for each box graph of su(5) with both 5 and 10 representation.

Top Cone and Fiber Faces
A singular Weierstrass model with a fiber of type I 5 over S = {ζ 0 = 0} is best consumed in Tate form [11,25] The above equation embeds the elliptic fiber into the weighted projective space P 123 with homogeneous coordinates (w, x, y) for every point in the base. We can make contact with the techniques reviewed in the last section by the following construction, which borrows from the idea of tops first introduced in [18,19] and has been widely adapted in the literature on F-theory 12 12 In particular, see [26] for a recent paper, which discusses this in a similar spirit to ours.

20
We first introduce the vectors and construct a fan from the cones p x , p y , p x , p w and p w , p y . The corresponding toric variety is the weighted projective space P 123 and we can think of both P 123 and the elliptic curve (4.1) as being fibered over the base. In order to be able to resolve the I 5 fiber in (4.1) using toric methods, we have to introduce a toric coordinate corresponding to ζ 0 . We use the and construct a fan Σ I 5 from the cones v The power of this construction is that (4.1) captures the behaviour of the elliptic fiber and allows us to find resolutions without having to explicitely specify the base. This works as follows. We describe  The singularity in (4.1) is located at x = y = ζ 0 = 0. We hence want to refine the cone x, y, ζ 0 in such a way as to resolve the singularity crepantly. As this cone plays a key role we will be refered to it as the top cone.
As we have discussed in section 3, to refine this top cone, we have to demand that the generators v i of one-dimensional cones introduced in the refinement process satisfy v i , m j ≥ −1 Together with v x , v y , v ζ 0 and v w , these points span the well-known SU (5) top for P 123 fiber embeddings.
We have hence shown that any refinement of the cone x, y, ζ 0 , which introduces a onedimensional cone generated by any one of the four lattice vectors above, will induce a crepant blowup of (4.1). A projection, which we call cone diagram (introduced in section 3. In the following we will provide a map between box graphs and triangulations of the fiber face. As with the triangulations of the top cones, black (red) points correspond to points that have (not) been used in a triangulation. Black lines connecting points correspond to actual triangulations, whereas black lines connecting to red points correspond to triangulations in- volving v x or v y . An example corresponding to a single triangulation for the SU (5) top is shown in figure 10.

Starting resolutions
In order to get a feeling for these methods, let us demonstrate which options we have for the first blowup. Introducing one of the four coordinates ζ 1 , ζ 2 ,ζ 1 ,ζ 2 corresponds to the four weighted  These blowups will subdivide the top cone v x , v y , v ζ 0 in the way shown in figure 9 in terms of cone diagrams. The alternative presentation in terms of fiber face diagrams is shown in figure 10.

Fiber Faces and weighted blowups for Box Graphs
We are now in the position to determine an explicit weighted blowup for each box graph in the network of small resolutions (or Coulomb phase analysis) [4,3], detailed in section 2 and shown in figure 4. The only exception to this is the graph corresponding to (11, IV) (and by reversing the order of the simple roots (6,I)), which we will discuss in the next section. This analysis provides a global construction of each box graph for su(5) with 5 and 10 matter, confirming the flops performed in patches in [4]. The main advantage of the present approach is that it will have a natural generalization [16].
As explained in Section 4.1, weighted crepant blowups of (4.1) can be found by successive refinements of the top cone v x , v y , v ζ 0 , using the four vectors (4.6). The sequence of blowups is determined from the fiber face diagram, which captures the essential information for the singularity resolution of the triangulation of the top cone v x , v y , v ζ 0 . The complete set of fiber faces and the network of flops among them is shown in figure   11. There are two situations that can arise: • Standard toric resolutions correspond to finely triangulated fiber faces, where all points are black and connected by black lines, i.e. are used in the triangulation.
• Partially triangulated fiber faces which contain red nodes correspond to partial toric resolutions, where the vertices corresponding to the red points have not been used in the triangulation. These are further resolved by algebraic blowups involving sections, that are not points in the triangulation, as we shall discuss momentarily. As discussed Figure 11: Fiber face diagrams for the SU (5) model with 5 and 10 representation, i.e. resolutions of the singular I 5 model with codimension 2 loci corresponding to I 6 and I * 1 fibers. Note that the codimension three fibers for these models are not necessarily of Kodaira type but monodromy reduced [3]. Each fiber face appears twice, and they only differ by reordering of the simple roots, i.e. on the right hand half of the network, the α i are associted to the ζ j ,ζ k with orientation that is clockwise, in the other half anti-clockwise. The fiber face diagram encodes the explicit weighted blowup sequence for a resolution. The lines connecting different fiber faces correspond to flops, so that this image corresponds to the box graph diagram figure 4.

25
in Section 3, such phases may be realized as complete intersections of codimension two in toric varieties.
We now discuss these two situation in turn, highlighting the simplicity and generalizability of our approach: If we successively introduce all four of the vectors v i in (4.6), we will obtain a resolution of (4.1). As the weight system of the corresponding toric variety is determined by the lattice vectors generating the one-dimensional cones alone, we can write it down without specifying the order of resolutions we are performing: x y ζ 0 ζ 1 ζ 2ζ1ζ2 This already determines the structure of the resolved Tate form Different sequences of weighted blowups using all four of v ζ 1 , v ζ 2 , vζ 1 , vζ 2 correspond to fine triangulations of the point configuration shown in figure 8, i.e. a triangulation, which uses all points. Even though there are 4! = 24 different sequences of weighted blowups, there are only 3 inequivalent triangulations corresponding to three different phases. In order to find all phases, we clearly have to use a more general strategy.
The remaining cases correspond to partially triangulated fiber face diagrams, containing red nodes, i.e. points that are not used in the triangulation. Let us start with the observation that partial resolutions of (4.1) (or, equivalently, blowdowns of (4.9)) can be described by simply deleting the absent coordinates from (4.9) and (4.8), and we also need to remove the C * -actions corresponding to these coordinates from (4.8). As we will see, in each of these cases, there is a small resolution which either turns the Tate model into a complete intersection, such as explained in section 3.4, or into a determinantal variety, which will be discussed in section 5.
The fiber face diagrams of section 4.1 are particularly well suited for the description of such resolutions, which go beyond the finely triangulated diagrams corresponding to standard toric resolutions.
To show that the partially triangulated fiber face resolutions admit a description in terms of complete intersections, we write the resolved Tate model in two particularly interesting factored ways: Here, we have introduced the notation (4.12) It is clear now that any resolution sequence starting with Res ζ 1 , Res ζ 2 or Resζ 1 has a partially resolved form given by either (4.10) (with either ζ 2 or ζ 1 set to zero) or (4.11). In all these cases, the equation takes the form of a conifold, and thus there is an alternative resolution sequence, which involves eitherŷ and P , or W and S, which is not a toric resolution. The resolutions of this type correspond to fiber face diagrams which contain red nodes (i.e. resolutions where some elementary vertices are not used in the triangulation process).
We will detail this process and the correspondence to the box graphs in the following for SU (5), and in [16] in general. The summary of the results for SU (5) can be found in table 1, which shows triplets of box graphs, fiber faces and algebraic resolutions. Note that by reversal of the ordering of the assignment of the exceptional sections to the simple roots each resolution corresponds to two box graphs, as detailed in the table. In the following we simply list only one half, associated to one ordering of the simple roots.

Box Graph (4, III)
The corresponding fiber face diagram is . Note that this is a fine triangulation, and thus corresponds to a toric resolution discussed in [12]. As we argued in general, these toric triangulations have an algebraic realization in terms of weighted blowups. We shall now present these here.  Table 1: Correspondence of box graphs, fiber faces, and algebraic resolutions for SU (5) with 5 and 10 representation, and codimension 3 monodromy reduced e 6 fibers in agreement with [3]. The labels for box graphs are as in (2.8) and (2.9). The Coulomb phase labels are as in [4]. In parenthesis we write the phases, which are obtained by the same resolution by choosing the inverted labeling of the roots of su (5). The sectionsŷ, P, S, W are defined in (4.12). The first three resolutions are toric (realized as weighted blowups), the fourth standard algebraic, and the last is determinantal. 28 The map between each of these fiber face diagrams is a triangulation of the top cone, and thereby a weighted blowup following our general discussion. E.g. the second step corresponds to subdividing the cone y, ζ 0 ,ζ 1 byζ 1 . From this we can determine the scalings ( It remains to show that this reproduces the box graph splitting (4, III). We can either apply the weighted blowups, or use a slightly more elegant method, which will be proven in [16]. Here we provide the explicit resolutions for reference in the appendix B.
Box Graph (7, III) and (9, III) and (9,II) These are obtained as a standard resolution with unit weights: first we resolve in codimension one, which corresponds to introducing the subdivisions of the cone using ζ 1 and ζ 2 : We obtain the factored form (4.10), withζ i = 1 after these blowups. There are three distinct small resolutions of which correspond either to the fine triangulation , with blowups (y, ζ 1 ;ζ 1 ) , (y, ζ 2 ;ζ 2 ) , (4.17) or the fine triangulation , which can be reached by (y, ζ 2 ;ζ 2 ) , (y, ζ 1 ;ζ 1 ) . (4.18) These two resolutions were studied algebraically in [15,4] and from since these are fine triangulations and thus standard toric resolutions, they are exactly also those discussed in [12]. The two cases correspond to the phases (7, III) and (9, III) respectively.
Finally, we can use P in the resolution, which implies that the Tate model becomes a complete intersection (y, ζ 1 ;ζ 1 ) , (ŷ, P ; δ) . This corresponds to the triangulation , where the red node indicates that the nodeζ 2 is not part of the triangulation, but the non-toric resolution with (y, P ; δ) was applied. This resolution was studied from algebraic resolutions in [13,14], and corresponds to the box graph (9, II). Box Graph (9, III) and (11,III) There is an alternative resolution that results in the fiber corresponding to the box graph (9,III), which is in fact more amenable to the flop from (9, III) to (11,III). This furthermore prepares the flop to (11,IV) which is the subject of the next section. The sequence of blowups is  Applying all of these results in the fine triangulation corresponding to (9, III), which we discussed already. Again each arrow corresponds to a weighted blowup, which we have listed in table 1. Applying only the first three, results in a partial triangulation, which has the factored form (4.11) withζ 2 set to 1 (as we have not used this in the triangulation), We can now apply the blowup (W,ζ 1 ; δ), which is a algebraic blowup not realized purely in terms of homogeneous coordinates, with W , S as in (4.12). This yields the phase characterized by the box graph (11,III). The details of this resolution are provided for the reader's convenience in appendix B. The last case will be discussed in the next section and is also based on the above equation (4.20). 30

Flops and Codimension 3 Fibers
The box graphs have a simple realization of flops as single box sign changes. The algebraic resolutions that we constructed based on the fiber faces allow us to equally simply spot the flops. The flops among the fiber faces with fine triangulations is completely standard and explained around figure 7. The more interesting cases are the partially triangulated fiber face, where we have already seen the partial resolutions take one of the two forms (4.10) or (4.11), which also make the flop transitions manifest.
Finally, we can confirm by direct constructions the monodromy-reduce e 6 fibers obtained from the box graphs in [3]. Note that the monodromy-reduction arises due to the absence of an extra section, we refer the reader for details to [3].

Determinantal Blowups
In this section, we show how to obtain the resolution associated to the box graph (11, IV) (and thus by reversal of the ordering of simple roots, the box graph (6, I)) by a series of successive blowups. After a sequence of weighted algebraic blowups, we reach a singular space which sits in between phase (11, IV) and (11,III). Whereas a further standard algebraic blowup realizes (11, III), we need to blow up the ambient space along a determinantal ideal to reach phase (11,IV). This means that the Tate model corresponding to this phace is not a complete intersection. The box graphs (11, III) and (11, IV) are connected by a flop along a 5 curve, C 2 , which is above the codimension two locus ζ 0 = 0 and which is less manifest in the Tate formulation, and thus makes the construction of this phase more challenging. 31

Setup and determination of singular locus
First we introduce the subdivisions corresponding to a weighted blowup introducingζ 1  After this blowup, the geometry is a complete intersection This is not a completely resolve space yet as there are singularities remaining. Let us see this explicitely in order to guide us to the resolution realzing phase (11,IV). To do so, we go to a chart C 4 (for the fiber coordinates) spanned by x, y, δ and ω (or, equivalently,ζ 1 and compute the Jacobian matrix. As we already anticipate that we will find the singularity over x = y = δ = 0 we evaluate it there. The two homogenous coordinates ζ 0 and ζ 1 cannot vanish simultaneously with x and y, we can set them to unity. The Jacobian matrix then gives as a condition for a singularity over x = y = δ = 0. We can rewrite this condition as

32
As ω andζ 1 cannot vanish simultaneously, these equations can only have a common solution ifζ 1 = 0 and b 1 = 0 or all the resultants vanish, which implies that thus confirming that the curve is indeed a 5 curve. Besides this relation, [ω :ζ 1 ] are fixed by the above conditions, so that we find a singularity at codimension 5 − 2 = 3 after the blow down. We have hence learned that while (5.5) is smooth in codimension two (codimension one over the base), it is still singular in codimesion three (two in the base). There are two singular strata located along the 10 and 5 matter curves.
The singularities over x = y = δ = 0 may be easily resolved by performing the blowup 1]). This will change the anticanonical class of the ambient space by 2Dζ 2 , so that we obtain a crepant resolution after computing the proper transform of (5.5). It is not hard to see that this again realizes the box graph resolution (11,III). To find (11, IV), we hence need to use a different resolution. The resolution we are looking for must be similar to a (partially) flopped version of the resolution at x = y = δ = 0.

Resolution and Determinantal variety
We now construct the resolution that realizes the box graph (11,IV). To find the alternate locus to resolving x = y = δ = 0, we can resolve along, note that we can rewrite (5.5) as Observe that e.g. the divisor y = 0 is not irreducible. It splits into x = δ = 0 or as the simultanous solution of (5.8) with Similarly, δ = 0 splits in two components and x = 0 into three components.
Let us define new coordinates ρ x , ρ y and ρ δ as the determinants of the 2 × 2 matrices which are obtained from R by deleting the columns corresponding to x, y, δ (with an extra sign for ρ y ), i.e. In order to implement the blowup along x = ρ x = 0, we start with the following observations: by construction the coordinates ρ satisfy Furthemore, the ideal generated by (5.8) contains the polynomials: The meaning of this is not difficult to see: both the vectors (x, y, δ) and (ρ x , ρ y , ρ δ ) are orthogonal to both row vectors of R. Hence they must be parallel, as expressed in (5.12) above. As we have discussed above, there is a singularity when both vectors vanish. To perform the resolution, we hence introduce an auxiliary P 1 with coordinates [ξ 1 : ξ 2 ] subject to the relations so that [ξ 1 : ξ 2 ] measures the proportionality constant of the two parallel vectors (x, y, δ) and (ρ x , ρ y , ρ δ ).
We now show that (5.13) and (5.8) indeed describe a crepant resolution of the singularity at x = y = δ = ρ x = ρ y = ρ δ = 0. To see this, first note that the relations (5.13) uniquely specify a point on the auxiliary P 1 except when we are at the locus of the former singularity.
This means that we not only have a resolution, but that it is also small (i.e. there is no exceptional divisor), from which it follows that we have a crepant resolution as well.
The weight system of the ambient space is now The fiber components of the resolved phase just obtained can be written as intersections of (5.13) and (5.8) with F 0 : ζ 0 = 0 (5.14) Writing its defining relations out explicitely, we find that F 1 is given by the (non-independent) equations δ = ρ δ = 0 Our first task is to show that the splitting of the fiber components over the 10 curve is as expected for phase (11,IV). For this, it is sufficient to consider the component F 1 , which is expected to split into four components. Over b 1 = 0, (5.15) splits into the four irreducible components ζ 1 = 0 : (5. 16) where δ = 0 is understood for all of them. Note that in some cases, not all equations defining the ideal corresponding to the fiber component are independent. We recognize these as F 4 , F 3 restricted to b 1 = 0, as well as the two curves C + 1,5 and C − 2,3 , which is the splitting we expect over the 10 matter curve from the box graphs discussed in Section 2.2.2.
Finally, let us see that we have obtained the expected splitting over the 5 matter curve.
Over P = 0 in the base, there exist x, y, ω simultaneously solving ρ δ = ρ y = ρ x = 0, as well as R(x, y, 0) = 0. On the other hand we can also simultaneously solve x = y = δ = ρ δ = ρ y = ρ x = 0 using only ω when P = 0 as noticed before. Correspondingly, F 1 splits into two irreducible components defined by the intersection of (5.15) with C + 1 : From these splittings it follows that we have indeed realized a global three/fourfold description of phase (11, IV). Although we have no proof that this phase cannot be realized as a complete intersection, we find it amusing that we have to venture out of well-charted territory to realize this 'outlying' phase.
Finally, we can confirm the splitting along codimension three, which yields one of the e 6 monodromy reduced fibers. Setting b 2 = 0 in addition, we observe the splitting Again, using the projective relations, the intersections are readily obtained to be as shown in The exceptional set Z, which is equivalent to the Stanley-Reisner (SR) ideal in the ring of homogenous coordinates, is defined such that a collection of coordinates z I can only vanish simultaneously if the corresponding cones share a common cone in Σ. We write such relations as [z a , z b , · · · ] for a, b ∈ I. We will not be interested in cases with a non-trivial group G, so we omit its description from the discussion, see e.g. [27][28][29][30][31] for a nice exposition.
A toric variety T Σ only has orbifold singularities if all cones are simplicial 14 . A simplicial p-dimensional cone σ with generators v 1 , · · · , v p leads to a singularity at codimension p in T Σ if its generators fail to span the restriction of the lattice N to their supporting hyperplane. The singularity is then located at the locus z 1 = · · · = z p = 0. T Σ is hence smooth if all cones are simplicial and the generators of all of the n-dimensional cones span N .
A fan gives rise to a compact toric variety T Σ if the union of all cones spans N ⊗ R.
The vanishing loci of the homogeneous coordinates z i define toric (Weil) divisors D i . As these Divisors can only vanish simultaneously if they are in a common cone, we may think of higher-dimensional cones as corresponding to algebraic subvarieties of higher codimension.
The toric divisors obey the linear relations for every m in the dual lattice (usually called the M -lattice). This means that the class of any divisor D corresponding to the vanishing locus of a polynomial is specified by the weights s i of P under the C * actions.
To compute intersection numbers between divisors, we can first use the SR ideal to see if the intersection can be non-vanishing. Non-zero intersection numbers can be computed by using that, for a collection of n different v i , i ∈ I spanning an n-dimensional cone σ in Σ Here, Vol(σ) is the lattice volume of the cone σ, which is given by the determinant of its Using the projective relations we can determine the splittings of the Cartan divisors ζ i = 0 andζ i = 0. Along b 1 = 0, which is the 10 locus, the curves dual to the divisors ζ 0 = 0 and ζ 2 = 0 split into two three components, respectively. Computation of the intersections with Smooth [32] yields that this is precisely the splitting for the box graph 4 respectively 13 of the 10 matter. Likewise along P = b 2 1 b 6 − b 1 b 3 b 4 + b 2 b 2 3 againζ 2 = 0 splits into two components, consistently with the box graph III or II respectively. Box Graph (9, III) or (8,II) After the weighted blowups we again obtain the equation (4.9), as the corresponding fiber face has a fine triangulation. The projective relations are Along b 1 = 0, which is the 10 locus, the curves dual to the divisors ζ 2 = 0 andζ 1 = 0 split into two and three components respectively. The precise charges from Smooth [32] yields that this is precisely the splitting for the box graph 9 and 8, respectively. Along P = b 2 1 b 6 −b 1 b 3 b 4 +b 2 b 2 3 againζ 2 = 0 splits into two components, consistently with the box graph III or II respectively.
Thus showing that these are (9, III) and (8, II), depending on which ordering of the simple roots we choose.
Box Graph (11, III) or (6, II) Finally, we get to a resolution which corresponds to a partially triangulated fiber face. The As this resolution has not appeared anywhere so far in the literature, we will provide some more details. The curves associated to the simple roots are (we choose one of the orderings here, corresponding to (6, II), however trivially, the reverse ordering will give rise to the other 40 resolution (11, III)) F 0 : ζ 0 = 0 xω =ζ 1ζ2 y 2 δω = −b 1 y + x 2ζ 2 ζ 1 F 1 : ζ 1 = 0 xω =ζ 1ζ2 y 2 δω = −b 1 y F 2 :ζ 1 = 0 Along b 1 = 0, only F 4 splits, using the projective relations, which precisely correspond to the splitting F 4 → F 1 + F 2 + C + 3,4 + C − 1,5 . (B.9) Along the 5 locus P = 0 it is F 3 that splits into two components. Using the projective relations the fiber intersections are I * 1 and I 6 respectively. And thus confirming that these realize the box graphs (11, III) or (6, II).
Finally we can also determine the splitting and fiber along the codimension 3 locus b 1 = b 2 = 0, where from the above the further splitting is of C − 1,5 b 1 = b 2 = δ = xζ 2 = xω − yζ 1 (ζ 2 y + b 3 ζ 1 ζ 2 0 ) = 0 (B.10) which has three components and F 3 , which split as (B.11) Again these splittings are consistent with the box graphs. Intersections of the fiber components using the projective relations, yield precisely the monodromy reduced e 6 fiber shown in table 1.