Extensions of toric line bundles

For any two nef line bundles L+:=OX(Δ+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}^+ :={{\mathcal {O}}}_X(\Delta ^+)$$\end{document} and L-:=OX(Δ-)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}^- :={{\mathcal {O}}}_X(\Delta ^-)$$\end{document} on a toric variety X represented by lattice polyhedra Δ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^+$$\end{document} respectively Δ-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^-$$\end{document}, we present the universal equivariant extension of L-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}^-$$\end{document} by L+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {L}}}^+$$\end{document} under use of the connected components of the set theoretic difference Δ-\Δ+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta ^-\setminus \Delta ^+$$\end{document}.

By [ABKW20, Thm.III.6] or [AP20] we can describe their homogeneous component of degree m ∈ M by where H i−1 (Z, C) on the right hand side denotes the reduced singular cohomology of a topological space Z with complex coefficients, (∆ + − m) denotes the polytope ∆ + shifted by the lattice point −m ∈ M and ∆ − \ (∆ + − m) denotes the set-theoretic difference of the two polytopes.Recall the (−1)-st reduced singular cohomology: The 0-th reduced cohomology is the quotient H 0 (Z, C) = H 0 (Z, C) H 0 ({•}, C), and thus, its dimension is the number of connected components of Z minus 1.
In this paper we will take another point of view.The partition of ∆ (0,2) into two connected components induces the following "exact sequences of polytopes": (1.6) 0 → → ⊕ → → 0.
In section 3 we show that this corresponds to an exact sequence of sheaves We obtain an extension of O F 1 (0, 2) by O F 1 (1, 0), that is, an element of the group which we know to be one-dimensional by [ABKW20] and [AP20].We will show that the short exact extension sequence (1.7) induced by the "exact sequences of polytopes" (1.6) represents this one-dimensional vector space Ext 1 O F 1 (0, 2), O F 1 (1, 0) and, moreover, that this concept works in general.
1.3.Acknowledgements.We would like to thank Christian Haase for helpful discussions throughout the work on this paper, in particular for the triangulation argument in the proof of Proposition 13.Moreover, we thank the anonymous referee for the careful reading and many valuable hints and remarks.The second author was supported by the BMS ("Berlin Mathematical School") and the third one by the DFG and Exzellenzcluster "Mathematik Münster".1

Toric Geometry
We introduce some basics of toric geometry central to this paper.Readers not familiar with toric geometry can take a look at one of the numerous introductory texts, for example [CLS11], [Ful93], [Dan78], or [Dem70].
Let M ∼ = Z r be a lattice and N = Hom Z (M, Z) ∼ = Z r its dual lattice.There is a natural pairing •, • : M × N → Z.We consider the algebraic torus T = Spec C[M].The isomorphism M ∼ = Z r induces an isomorphism T ∼ = (C * ) r .The lattice M can be recovered as the character lattice Hom(T, C * ).We denote the character of M ∋ m → (a 1 , . . ., a r ) ∈ Z r by χ m : T → C * , (t 1 , . . ., t r ) → t a 1 1 • • • t ar r .The dual lattice N corresponds to the group of 1-parameter subgroups Hom(C * , T ).Here N ∋ n → (b 1 , . . ., b r ) ∈ Z r corresponds to λ n : C * → T, t → (t b 1 , . . ., t br ).A toric variety is an irreducible variety containing an algebraic torus T ∼ = (C * ) r as an open dense subset, such that the action of the torus on itself by multiplication extends to an algebraic action on the whole variety [CLS11, Def.3.1.1].We sketch how normal toric varieties can be constructed from cones and fans in We write τ σ whenever τ is a face of σ.A polyhedron ∆ in M R is the intersection of finitely many closed half spaces where for some inward pointing normal vector v i ∈ N R and some scalar λ i ∈ R for i ∈ {1, . . ., s}.A compact polyhedron is called a polytope.A polyhedron ∆ can be written as the Minkowski sum ∆ = ∇+δ of a polytope ∇ and its tail cone To a full-dimensional lattice polyhedron ∆ we associate its inner normal fan whose maximal cones σ m are given by σ This yields a semiprojective toric variety P(∆) := TV(N (∆)).We will assume all polyhedra to have at least one vertex.The normal fan of such a polyhedron will have convex support of full dimension.
2.1.Divisors on Toric Varieties.Let Σ be a fan in N R with convex support of full dimension r = dim N R , for example, the normal fan of a full-dimensional lattice polyhedron in M R .Let X := TV(Σ) be the toric variety given by Σ.
Every lattice polyhedron ∆ ⊆ M R with tail cone |Σ| ∨ and whose normal fan N (∆) is refined by Σ gives rise to a nef Cartier divisor D ∆ on X by the following construction: so the vertex m σ of ∆ encodes the local sections of the line bundle O X (∆) over the affine open U σ .The line bundle O X (∆) is ample if and only if N (∆) = Σ.
We denote the set of lattice polyhedra with prescribed tail cone δ by Pol + δ and the set of lattice polyhedra compatible with Σ, that is, with tail cone |Σ| ∨ and whose normal fan is refined by Σ, by Pol + (Σ).The set Pol + δ forms a semigroup with respect to Minkowski addition.For a fan Σ the set Pol + (Σ) forms a finitely generated subsemigroup.The union over the Pol + (Σ) with |Σ| ∨ = δ is the semigroup Pol + δ .These semigroups are cancellative, which is caused by the presence of the prescribed common tail cone δ: It makes sure that a polyhedron ∆ ∈ Pol + δ is uniquely determined by the values of min ∆, v with v running through δ ∨ .Hence, these semigroups embed into their respective Grothendieck groups of formal differences: On a quasiprojective toric variety X = TV(Σ), every Cartier divisor D can be written (non-uniquely) as a difference D = D + − D − with both D + and D − nef Cartier divisors [CLS11,Thm. 6.3.22].In particular every Cartier divisor on X can be represented by a pair of lattice polyhedra (∆ + , ∆ − ) compatible with the fan Σ, that is, an element of the Grothendieck group Pol(Σ).

From Complexes of Polyhedra To Complexes of Sheaves
3.1.The Koszul Complex of Polyhedra.In this section we construct "exact sequences of polyhedra" that induce exact sequences of split vector bundles on a toric variety X = TV(Σ) over C. We start out on the polyhedral side and let Σ be a fan in N R with convex support of full dimension.Let Pol + (Σ) denote the set of lattice polyhedra in M R compatible with Σ.Also include the empty set in Pol + (Σ).
satisfying the following conditions: (1) i∈I ∇ i =: ∇ ∈ Pol + (Σ) is a lattice polyhedron compatible with Σ; (2) all intersections ∇ I ′ := i∈I ′ ∇ i with ∅ = I ′ ⊆ I are either empty or compatible with Σ, that is, We consider two categories and a functor associated to a Σ-family between them.
Definition 2. Let 2 I be the poset category associated to the power set of the finite set I, that is, objects are subsets of I and there exists a unique morphism from I ′ to I ′′ whenever I ′ ⊆ I ′′ .Let Pol + (Σ) also denote the category of lattice polyhedra compatible with the fan Σ, that is, objects are compatible lattice polyhedra as defined above or the empty set and for two polyhedra ∆ 1 and ∆ 2 in Pol + (Σ) we define Given an Σ-family of polyhedra S we can define a contravariant functor In general, given any contravariant functor F : 2 I → Pol + (Σ), we define a subcomplex of the Koszul complex • C I as follows.
Definition 3.For a contravariant functor F : 2 I → Pol + (Σ) and p ∈ N let Here e I ′ := i∈I ′ e i , where {e i | i ∈ I}, denotes the canonical basis of C I .
The map d : C F p+1 → C F p is defined as for the Koszul complex • C I .For a fixed total order on I, say I = {1 < • • • < #I}, and where |i| refers to the index j of i = i j in I ′ .
Recall that the polyhedra F (I ′ ) ∈ Pol + (Σ) are contained in M R .For each m ∈ M R we define the evaluation subcomplex ("at m") as m) as boundary maps.
Lemma 4. For a Σ-family S set For statement (iii) first note that we have ) are all subspaces of C F p spanned by a subset of the prescribed basis {e I ′ | I ′ ⊆ I, #I ′ = p, F (I ′ ) = ∅} of C F p .Now, we switch to a slightly more general setup.Assume that C i • ⊆ C F • (i = 1, . . ., k) are complexes such that all their mutual intersections are exact and The generalization of the setup is needed to ensure that the induction hypothesis implies that, besides the central term, the left most complex is exact, too.
Finally, we apply the previous claim to the complexes where m ∈ ∇∩M replaces i = 1, . . ., k.Note that even though the set ∇ ∩ M may be infinite, the induction ends after finitely many steps since the Koszul complex Example 5. We revisit an example already encountered in the Introduction (1.2).Let Σ be the fan of the first Hirzebruch surface F 1 = P(∆) = TV(Σ).
The fan is complete, so compatible polyhedra will be polytopes.Define the Σ-family There is a particularly nice case in which exactness of the evaluation subcomplexes for all lattice points m ∈ M is equivalent to exactness for all m ∈ M R .Lemma 6. Assume that Σ consists of a single cone σ and its faces, with σ ∨ smooth and full-dimensional.Then for any contravariant functor F : 2 I → Pol + (Σ), the complexes C F • (m) are exact for all m ∈ M if and only they are exact for all m ∈ M R .Remark 7.This situation occurs naturally when we consider smooth affine toric varieties TV(σ) or an affine open subset U σ ⊆ TV(Σ) for a cone σ ∈ Σ.In the latter case the polyhedra F (I ′ ) are changed to F (I ′ ) + σ ∨ when considering the affine open U σ (see section (3.2)).In the case of a functor F S associated to a Σ-family S the ∇ i become ∇ i + σ ∨ and the ∇ I ′ for I ′ ⊆ I become ∇ I ′ + σ ∨ .
Proof of the Lemma.Because σ ∨ is smooth, we may choose coordinates such that has only one vertex and can be written as where ⌊m⌋ and the relation ≥ are meant componentwise.Hence, Example 8. We give a counterexample for the global situation, that is, for Σ consisting of more than a cone σ.Let Σ = N (∆) be the fan of Let I := {0, 1} and define the functor F : 2 The only non-trivial evaluation subcomplexes at lattice points are at m = 0, m = 1: These are exact.But at m = 1 2 ∈ M R we have the non-exact evaluation subcomplex 3.2.Localization of the Koszul Complex.We use the notation of section (3.1).
In section (4) we will think about the polyhedra ∇ I ′ as representing some nef line bundles on some toric variety P(∆) = TV(N (∆)) over C. Then the complexes constructed in subsection (3.1) describe maps between the global sections of them, and thus among the sheaves themselves.To understand the local behavior of these complexes, we will look at the affine charts TV(σ) ⊆ P(∆) for cones σ ∈ N (∆).
Definition 9.For a contravariant functor F : 2 I → Pol + (Σ), where |Σ| is convex and full-dimensional in N R , and a cone σ ∈ Σ define the functor Proposition 10.For a polyhedron ∆ and a finite set I, let F : 2 I → Pol + (N (∆)) be a contravariant functor satisfying the conclusions (i) -(iii) of Lemma 4. Then the complexes C F σ • (m) are exact for all m ∈ M R and for all σ ∈ N (∆).Remark 11.Moreover, this claim remains true if we consider any polyedra compatible with the fan N (∆), not just lattice polyhedra.This generalization will be important for the induction performed in the upcoming proof.
Proof.Step 1. Assume that r = rk M = 1.Then ∆ ⊆ R is a subset of the real line and its normal fan N (∆) can contain the origin σ = {0}, the rays σ = R ≥0 and σ = R ≤0 and the real line σ = R as cones.For σ = R and σ = {0} the claim is clear from the assumptions.We deal with the case be the sets of start and end points, respectively.Since I is finite, A(F ) ⊆ R and B(F ) ⊆ R ∪ {∞} are finite.
We understand the local complex C F σ • in terms of global evaluation subcomplexes: We make three observations: (1) For m ∈ R satisfying m < b for all b ∈ B(F ): and denote its cokernel by C • .This yields an exact sequence for the original functor F , also with cokernel C • .This gives the short exact sequence 0 in which the first two compexes are exact by assumption, so C • is also exact.By observation (1) above we can start from the exact complex Step 2. Consider a general lattice M of rank r ∈ N ≥1 and let C ⊆ M R be any ray.Replacing the polyhedra F (I ′ ) by the Minkowski sums F (I ′ ) + C yields the functor The complex C (F +C)∩(m+(C−C)) • inherits this property by step 1 and we obtain exactness of Step 3. If σ ∨ ∈ N (∆) is an arbitrary (non-trivial) polyhedral cone, then we may apply step 2 successively to all its fundamental rays.
3.3.Exactness of the Sequence Associated to a Σ-Family.
Theorem 12. Let ∆ be a full-dimensional lattice polyhedron in M R with normal fan Σ := N (∆) and X := P(∆) the toric variety given by ∆.
Summing the C F S • (m) for lattice points m ∈ M yields a sequence of global sections: The sheaves O X (∇ I ), O X (∇ i ), i ∈ I, and O X (∇) are globally generated.By T -equivariance, they are subsheaves of j * O T for j : T ֒→ X the inclusion of the torus.Hence, the sequence of global sections (3.11) determines a sequence of sheaves: For a cone σ ∈ Σ and a lattice point m ∈ M, the evaluation subcomplex C . The sequence of sections over U σ ⊆ X therefore corresponds to the direct sum of the evaluation subcomplexes C (m) for m ∈ M. By Proposition 10 this sequence is exact for each σ ∈ Σ.Hence, the restriction of the sequence of sheaves to each affine chart of the covering {U σ } σ∈Σ is exact and consequently the constructed sequence is exact.

Displaying Ext 1
Throughout this section let X := P(∆) be the toric variety over C associated to the lattice polyhedron ∆ ⊆ M R , whose normal fan Σ := N (∆) has convex support of full dimension.The (possibly trivial) tail cone of ∆ is δ := tail(∆) = |Σ| ∨ .Let ∆ + , ∆ − ∈ Pol + (Σ) be lattice polyhedra compatible with Σ, that is, their normal fans are refined by Σ.These polyhedra correspond to T -invariant nef Cartier divisors We study the space of extensions that is, extensions of the line bundle O X (∆ − ) by the line bundle O X (∆ + ).More specifically, we study T -equivariant extension sequences.These are elements in Ext(∆ − , ∆ + ) 0 .To understand this space we start with specific extension sequences induced by inclusion/exclusion sequences of polyhedra, such as the sequence considered in the Introduction (1.2).For an n-dimensional Ext ∆ − , ∆ + 0 all extensions are encoded in a single sequence of the form 0 We will show that this is the universal extension sequence for Ext ∆ − , ∆ + 0 .First, we show that it can be constructed from the exact sequence of sheaves associated to a Σ-family (Theorem 12).We then trace the sequence along the identifications: Note that all of the above groups are M-graded and all identifications respect these M-gradings.Since we consider equivariant extensions, we will obtain an n-tuple of elements in H 1 (X, O X (∆ + − ∆ − )) 0 ∼ = H 0 (∆ − \ ∆ + ), without an integral shift.
4.1.Inclusion of Polyhedra.Since we are relating Ext O X (∆ − ), O X (∆ + ) 0 to the reduced singular cohomology group H 0 (∆ − \∆ + ), the easiest case is to assume ∆ + to be contained in ∆ − .The inclusion/exclusion sequence of polyhedra is obtained by covering ∆ − by certain polyhedra ∇ i that intersect (pairwise) in ∆ + .These ∇ i are obtained by taking the unions of the connected components C 0 , . . ., C n of the set-theoretic difference ∆ − \ ∆ + with ∆ + , that is, Proposition 13.For two lattice polyhedra Proof.It is easy to see that ∇ is closed.We first show that ∇ is convex.Assume that x, y ∈ ∇.We know that xy ⊆ ∆ − , and this line segment might touch ∆ + or not.Next we realize that vertices of ∇ are vertices of ∆ + or of ∆ − .We start with the case of ∆ + and ∆ − being compact.Choose a generic regular triangulation induced from some map ω : vert(∆ + ) → R and then extend it generically to vert(∆ − ) \ vert(∆ + ) with sufficiently independent heights.This yields a triangulation ∆ − = i∈I ∆ i that restricts to a triangulation ∆ + = j∈J⊆I ∆ j that uses only the vertices of ∆ − and ∆ + .In particular all ∆ i are lattice simplices.The union ∇ = C ∪ ∆ + is a union of lattice simplices in I.In particular all of its vertices are lattice points.
If ∆ + and ∆ − are not compact but have the same tail cone δ we choose a half-space H such that all vertices of ∆ + and ∆ − are contained in the interior of H. Then we can write ∆ + = P + + δ and ∆ − = P − + δ with The previous discussion applied to the polytopes P + and P − , all of whose relevant vertices are integral (not necessarily those lying on the boundary of H, but they do not yield vertices of C ∪ ∆ + ), yields that all vertices of ∇ = C ∪ ∆ + are integral.
family in the sense of Definition 1 for some refinement Σ ′ ≤ Σ.
Proof.By the previous Proposition 13 the ∇ i are lattice polyhedra.The tail cone of each ∇ i is |Σ| ∨ .Hence, each normal fan N (∇ i ) has the same support as Σ.Since we are only dealing with finitely many ∇ i there is a common refinement Σ ′ of Σ and all N (∇ i ).Then ∇ i ∈ Pol + (Σ ′ ).The conditions for a Σ ′ -family follow from the assumptions on Σ ′ and ∇ I ′ = ∆ + ∈ Pol + (Σ ′ ) for any I ′ ⊆ I with #I ′ ≥ 2.
4.1.1.Refining the Fan.In many cases the lattice polyhedra ∇ i built from the components of ∆ − \ ∆ + will already be compatible with the fan Σ we started with.We believe that if ∆ + and ∆ − are sufficiently ample, this will always be the case.However, in general, if the ∇ i are not compatible with Σ, we can refine Σ to a fan Σ ′ with the same support such that all ∇ i are compatible with Σ ′ , as in Corollary 14.This induces a proper birational toric morphism π : The projection formula implies that π * π * F = F for a locally free sheaf F on X.For a short exact sequence of sheaves on X ′ of the form derived pushforward Rπ * yields a long exact sequence of sheaves on X: which turns out to be a short exact sequence by the derived version of the projection formula.In particular, the vanishing R ν π * π * F j = 0 for ν ≥ 1 implies R ν π * E = 0, too.Hence, using both vanishings, we have for all i ≥ 0 and for j = 1, 2 the isomorphisms In the study of Ext(∆ we can therefore pull back and push foward along π without impacting the extension or cohomology classes and groups.In the following, we will assume for simplicity that the fan Σ is already refined enough so that all ∇ i are compatible with it.
4.1.2.Two Components.We dive into Theorem 12 in the case where ∆ − \ ∆ + consists of two components C 0 and C 1 and we assume ∇ 0 = C 0 ∪ ∆ + and ∇ 1 = C 1 ∪ ∆ + to be compatible with the fan Σ.The complex Proof.We follow the steps mentioned at the start of section 4. We first tensor the sequence (4.6) with O X (∆ − ) −1 and set L := O X (∆ + − ∆ − ) and E i := O X (∇ i − ∆ − ) for i = 0, 1 to obtain an extension sequence Consider the associated long exact sequence in cohomology: The image of the extension sequence (4.6) in H 1 (X, L) under identification (2) is the image η of 1 ∈ Γ(X, O X ) under d.Note that η ∈ H 1 (X, L) 0 is of degree 0.
To understand identification (3) we translate η ∈ H 1 (X, L) to an element in the Čech cohomology group Ȟ1 (U, L) for toric affine open covering of X.The long exact sequence in cohomology (4.8) corresponds to a long exact sequence of Čech cohomology groups Order the maximal cones σ ∈ Σ max so that 1 ∈ E 0 (U i ) for i ∈ {0, . . ., l} and 1 ∈ E 1 (U i ) for i ∈ {l + 1, . . ., m}.This is possible because the map of sheaves Understanding η in terms of Čech cohomology is convenient because by [AP20] the degree 0 part of the Čech complex C • (U, L) giving Ȟ• (U, L) is the same as a Čech complex C • (∆ − , S) defined in [AP20, (3.4)] giving the relative singular cohomology H ) can be lifted to H 0 (∆ − \ ∆ + ) by considering the long exact sequence of the pair (∆ − , ∆ − \ ∆ + ).In terms of Čech complexes the boundary morphism H ) is given by the snake lemma.One shows that η can be lifted to the cocycle which is 1 on the component C 1 of ∆ − \ ∆ + and 0 on the component C 0 .It can also be lifted to the cocycle which is 0 on the component C 1 and −1 on the component C 0 .Modulo H 0 (∆ − ) these cocycles are equivalent and we have found the image of sequence (4.6) to be [ 4.1.3.More than Two Components.We now deal with the case where ∆ − \ ∆ + consists of n+1 connected components C 0 , . . ., C n for some n ≥ 2. Set ∇ i := C i ∪∆ + for i = 0, . . ., n and note that n i=0 ∇ i = ∆ − and ∇ I ′ = i∈I ′ ∇ i = ∆ + for I ′ ⊆ I of cardinality k with 2 ≤ k ≤ n + 1.By subsection (4.1.1)we may assume that the set S := {∇ i | i ∈ {0, . . ., n}} forms a Σ-family.The exact sequence of sheaves induced from this Σ-family as in Theorem 12 looks as follows: We will replace this sequence (4.11) by a quasi-isomorphic short exact sequence.To this end, consider the exact Koszul complex The green part is identical to part of sequence (4.11).We obtain a quasi-isomorphism from sequence (4.11): Denoting K := ker( d1 ) = ker(d 1 ) ⊗ O X (∆ + ), we obtain a short exact sequence We now choose the set {e i − e 0 | i ∈ {1, . . ., n}} with respect to the standard basis {e 0 , . . ., e n } for C n+1 as a basis for ker(d 1 ).This induces an isomorphism K ∼ = O X (∆ + ) n under which sequence (4.13) corresponds to the short exact sequence where the maps can be thought of as of reduced singular 0-th cohomology classes.
Proof.The steps in this case are very similar to the case of two components.The extension sequence ) n under the long exact sequence in cohomology associated to the short exact extension sequence.
In our case Again, this image is understood using the sequence of Čech complexes, only looking at degree 0: Recall that Ext 1 (O X (∆ − ), −) is a covariant functor, where a map of extensions is induced by a pushout.

Via the identification Ext
Proof.This follows from functoriality of Ext(O X (∆ − ), −) and naturality of the isomorphisms we identify along.The i-th projection
4.1.4.The Universal Extension.For two O X -modules F and G and their (finitedimensional) space of extensions E := Ext(F , G), a universal extension is a short exact sequence of the form 0 , where E ∨ = Hom C (E, C), such that for any t ∈ E the sequence induced by the pushout along Theorem 19.The extension sequence (4.13 We now check the universal property on the basis . By definition of the dual and double dual, corresponds to the projection to the i-th coordinate pr i : H

By Corollary 17 the pushout along the i-th projection homomorphism gives an extension in Ext
4.1.5.The Cremona Example.Let X be the graph of the Cremona transformation P 2 −→ P 2 .In toric language, X = P(H) = TV(Σ), where Σ = N (H): Here, the ray ρ i of the inner normal fan Σ corresponds to the facet F i of H. Let D i := orb(ρ i ).The Minkowski summands A and B of H correspond to the divisors There is no other shift m such that (2B − m) \ A, or equivalently 2B \ (A + m), has non-trivial reduced 0-th cohomology.Hence, H 1 (X, O X (∆ + −∆ − )) is 2-dimensional, sitting completely in degree m = 0.

Denote the connected components of ∆
From the facet presentations of ∇ 0 , ∇ 1 and ∇ 2 we can read off the associated divisors In this case, sequence (4.14) in subsection (4.1.3)is the sequence The pushout construction described in Corollary 17 yields two extension sequences 0 ) and the negative of the Baer sum of the two sequences above is represented by 4.2.General Position of Polyhedra.In the previous section we dealt with the special case of one polyhedron ∆ + being contained in the other polyhedron ∆ − .In this section we deal with ∆ + and ∆ − lying in general position.
4.2.1.Observing Two Problems.To illustrate the problem for the general case, we start with an example.Let Σ 0 be the 2-dimensional, singular fan made from the rays spanned by (1, 0), (0, 1), (−2, −1), and (0, −1), respectively (displayed in black below).Adding the rays spanned by (−1, 0) and (−1, −1) (displayed in blue below), we obtain the smooth subdivision Σ.However, since ∆ + is not contained in ∆ − , there is no map from O X (∆ + ) into the sheaves of the two components, which are subsheaves of O X (∆ − ).The solution to this problem is to replace ∆ + by (∆ + ∩ ∆ − ), then proceed as in section (4.1), and, finally, to use functoriality of Ext along the embedding But here one can spot the second problem.As in the example, the intersection ∆ + ∩∆ − is not necessarily a lattice polyhedron.This will be overcome by refining the lattice.We replace the lattice M by a larger lattice M ⊇ M such that ∆ + ∩ ∆ − is a lattice polyhedron with respect to M .Dually, this means to consider some sublattice N ⊆ N of finite index and the induced finite covering p : TV(Σ, N ) → TV(Σ, N).

4.2.2.
The Intersection is a Lattice Polyhedron.We start with the case where the intersection ∆ + ∩ ∆ − is a lattice polyhedron with respect to the lattice M. Suppose that As before, we can assume without loss of generality that the fan Σ is refined enough so that ∆ + , ∆ − , ∆ + ∩ ∆ − and ∇ i for i ∈ {0, . . ., n} are compatible with Σ and define nef Cartier divisors on X = TV(Σ).
Consider the following pushout diagram: Here the upper exact sequence is constructed as in subsection (4.1 The sheaf H is the pushout of the left square and the universal property of the pushout induces a map H → O X (∆ − ), that makes the lower sequence exact and the right square commutative.All together we obtain a map of complexes from the upper to the lower exact sequence.This is functoriality of Ext(O X (∆ − ), −).
the short exact extension sequence where is also the pushout of the left square.The map of complexes between the short exact sequences induces a map of complexes between the long exact sequences in cohomology.In particular we obtain a commuting square where η and µ denote the images of 1 ∈ Γ X, O X 0 under the differential d.By Theorem 16 the isomorphism from H 1 X, O X (∆ + −∆ − ) n to reduced singular cohomol- 4.2.3.The Intersection is Not a Lattice Polyhedron.We now deal with the case where ∆ + ∩ ∆ − ⊆ M R is not a lattice polyhedron with respect to the lattice M. Take any refinement M ⊇ M such that M is a sublattice of finite index in M and Construction 22. Back to our general case, the refinement M ⊇ M was chosen so that ∆ + ∩ ∆ − is a lattice polyhedron with respect to M and we are in the case of section (4.2.2), however on the space X.We have the short exact extension sequence by Theorem 16 and Proposition 20.Since p : X → X is affine, the pushfoward p * is exact and we obtain a short exact extension sequence of sheaves on X: Recall that in the first step of the identification of Ext O X (∆ − ), O X (∆ + ) n 0 with H 0 ∆ − \ ∆ + n we tensor sequence (4.24) with O X (∆ − ) −1 .Its pushforward is By the projection formula this is equal to sequence (4.25) tensored with O X (∆ − ) −1 .For an affine morphism p : X → X of noetherian separated schemes and a short exact sequence 0 → F → G → H → 0 of quasi-coherent sheaves on X there is a commuting diagram in which all vertical morphisms are isomorphisms: ) and O X (∆ − ), respectively, the latter are a direct summand: Furthermore, O X (∆ ± ) corresponds precisely to the G-invariants of p * p * O X (∆ ± ).
Example 23.For O X (∆ ± ) and the pullback p * O X (∆ ± ) = O X (∆ ± ) we have where {m ± σ } σ∈Σ is the Cartier data of we can view the above as an inclusion of C[σ ∨ ∩ M]-modules.Any M-graded module obtains an M-grading by adding zeros in degrees m ∈ M \M.
For the sheaves in the short exact sequence (4.25) we have the M -graded subsheaves O X (∆ + ) of p * O X (∆ + ) and O X (∆ − ) of p * O X (∆ − ).The inclusions are isomorphisms when restricted to M ⊆ M .Because the morphisms in sequence (4.25) are homogeneous of degree 0, taking the M-graded part of p * H yields a subsheaf Proposition 24.Given the commuting diagram (4.29), the upper sequence the lower sequence inducing the desired n-tuple by Construction 22.We obtain a commutative diagram of long exact sequences (4.32) The homomorphisms (1) and (2) restrict to isomorphisms in degree 0 ∈ M, so Corollary 25.Sequence (4.30) induces n extensions in Ext O X (∆ − ), O X (∆ + ) 0 , one for each i = 1, . . ., n: Furthermore, sequence (4.30) is a universal extension for Ext O X (∆ − ), O X (∆ + ) 0 .

Using Klyachko's Description of Toric Reflexive Sheaves
We briefly show how to construct the universal extension sequence in the case where ∆ + ∩ ∆ − is not a lattice polyhedron in terms of Klyachko's description of toric reflexive sheaves (see [Kly90] and [Kly02]; a short summary can be found in [Pay08]; see [DJS14] and [KM] for more recent approaches).
5.1.Describing Sheaves via Filtrations.Consider a toric variety X = TV(Σ) given by a fan Σ in N R .Let 1 ∈ T ⊆ X denote the neutral element.Each O X -module E gives rise to a C-vector space E := E(1) := E 1 /m of the vector space E which are parametrized by the rays ρ ∈ Σ(1).Let v ρ denote the primitive generator of the ray ρ.The filtrations encode the sections of E on the T -invariant open subsets U ρ = TV(ρ) ⊆ X defined by ρ.Namely, for u ∈ M, Example 27.Let D ρ = orb(ρ) be the closure of the orbit defined by ρ ∈ Σ(1).For Example 28.The tangent sheaf T X corresponds to the filtration (5.4) Remark 29.Projective n-space P n is the toric variety associated to the normal fan N (∆ n ) of the standard n-simplex ∆ n .The fan N (∆ n ) has n + 1 rays ρ 0 , . . ., ρ n .The direct sum of invertible sheaves ⊕ n j=0 O P n (orb(ρ j )) corresponds to the filtrations (5.5) The canonical surjection π : E ։ N C , e ρ i → v ρ i , where v ρ i generates the ray ρ i , induces the filtrations of N C corresponding to T P n .On ker(π) ∼ = C the induced filtrations are those of the structure sheaf O P n .This yields the Euler sequence 5.3.Pullbacks under Toric Morphisms.Let E be a toric reflexive sheaf on X := TV(Σ, N).Similarly to subsection (4.2.3), let N ⊆ N be a sublattice of finite index and X := TV(Σ, N ).Let v ρ ∈ N denote the primitive generator of the ray ρ ∈ Σ(1) in N and let v ρ be its image in the quotient group G := N/ N .For is the primitive generator of ρ in N. Let p : X → X be the toric covering morphism.
Analogously to [Pay06, Prop.4.9] one can give the following description of the filtrations of a pullback of a toric reflexive sheaf on X along p.
Proposition 30.Suppose the toric reflexive sheaf E corresponds to the vector space E = E(1) with filtrations (E l ρ ) l∈Z for ρ ∈ Σ(1).Then the pullback F := p * E of E is a toric reflexive sheaf on X.It corresponds to the same vector space F = E and the filtration for ρ ∈ Σ(1) is given by F l ρ = E ⌈ l dρ ⌉ ρ , l ∈ Z, where ⌈•⌉ denotes the ceiling function, giving the smallest integer equal to or larger than the argument.This filtration (F l ρ ) l∈Z of F = E can be thought of as the filtration (E l ρ ) l∈Z stretched by the factor d ρ and we will refer to it as the the d ρ -th stretching of the filtration.

Constructing the Universal Extension using Klyachko's Description.
We are in the setting of subsection (4.2.3).Consider the two line bundles O X (∆ + ) and O X (∆ − ) on X.In Klyachko's description, let them be given by the C-vector spaces with decreasing Z-filtrations (E + , E • +,ρ ) ρ∈Σ(1) and (E − , E • −,ρ ) ρ∈Σ(1) , respectively.By Proposition 30 the pullback line bundles O X (∆ + ) = p * O X (∆ + ) and O X (∆ − ) = p * O X (∆ − ) on X correspond to the C-vector spaces and decreasing Z- In sequence (4.24) the outer two sheaves are the pullbacks of line bundles on X.
Lemma 31.The sheaf H in sequence (4.24) is a reflexive sheaf on X and corresponds to a C-vector space with filtrations ( H, H • ρ ) ρ∈Σ(1) , where the filtration H • ρ for the ray ρ ∈ Σ(1) is a d ρ -th stretching in the sense introduced above.
Proof.Applying the contravariant functor Hom (−, O X ) : Coh( X) → Coh( X) to the sequence (4.24) twice yields a short exact sequence of double duals (using that the outer two sheaves are locally free) with a canonical homomorphism from the original sequence.Reflexivity of the outer two sheaves and the five lemma give reflexivity of the sheaf H. Let H • ρ , F • +,ρ and F • −,ρ , ρ ∈ Σ(1), denote the filtrations corresponding to the reflexive sheaves H, O X (∆ + ) and O X (∆ − ), respectively.Since sequence (4.24) is T -equivariant, it induces a short exact sequence of filtrations 0 → F • +,ρ → H • ρ → F • −,ρ → 0 for each ρ ∈ Σ(1).Hence, the filtration H • ρ of H is determined by the filtrations F • +,ρ and F • −,ρ and is thus also d ρ -th stretching.Remark 32.A filtration of a vector space that is a d-th stretching can also be squished back: For a vector space F with filtration , l ∈ Z, is the d ρ -th squishing of H • ρ , ρ ∈ Σ(1), fits into a short exact sequence (5.10) 0 → O X (∆ + ) n → H → O X (∆ − ) → 0 that pulls back to the short exact sequence (4.24) under p : X → X.Sequence (5.10) is precisely the universal extension sequence (4.30).
Taking pushouts allows us to calculate the filtrations for H on X.In order to obtain the filtrations for the sheaf H we need to take the d ρ -squishings of the filtrations for H, whenever d ρ = 1.This is the case for ρ 1 , ρ 3 and ρ 5 , with d ρ = 2 in each case.
The following table depicts the resulting filtrations for H.
Note that the vector bundle H (and even the vector bundle H on X) does not split.This can be seen using a criterion from Klyachko, that a vector bundle splits if and only if the vector spaces in the filtrations of all the rays form a distributive lattice, or, equivalently are given by coordinate subspaces (compare [Kly89, Cor.2.2.3]).
. Spotting Cohomology.Consider a projective toric variety X = P(∆) corresponding to a lattice polytope ∆ ⊆ M R , M R = M ⊗ Z R for a lattice M, over C. A torus invariant Cartier divisor D on X can be represented by a pair (∆ + , ∆ − ) of lattice polytopes, where both polytopes encode nef divisors D ∆ + and D ∆ − on X and D = D ∆ + − D ∆ − .Denote the associated line bundle by O X (D) =: O X (∆ + − ∆ − ).It is well-known that the cohomology groups H i X, O X (D) of the line bundle O X (D) of a torus invariant Cartier divisor D are M-graded (compare for example [CLS11, §9.1] or [Ful93, Section 3.5]): A pointed cone σ ⊆ N R leads to an affine toric variety TV(σ) := Spec C[σ ∨ ∩ M].The inclusion of a face τ σ induces an open embedding TV(τ ) ֒→ TV(σ).In particular, the inclusion of the origin induces an open embedding of the torus T = Spec C[M] ֒→ TV(σ).A fan Σ in N R is a finite collection of pointed cones that is closed under taking faces and such that the intersection of two cones is a face of each.The affine toric varieties U σ := TV(σ) associated to the cones σ in a fan Σ glue together to a normal separated toric variety TV(Σ) with open affine charts the U σ [CLS11, Thm.3.1.5].
If not, then xy ⊆ (∆ − \ ∆ + ).Since xy is connected, it then has to be contained in a single connected component of ∆ − \ ∆ + .Since x ∈ ∇ = C ∪ ∆ + , we know that x ∈ C. Hence, xy ⊆ C. If xy ∩ ∆ + = ∅, then this set is a closed subsegment x ′ y ′ ⊆ xy.In particular, the half open ends xx ′ and y ′ y (excluding x ′ and y ′ ) belong to ∆ − \ ∆ + , and the same argument as in the first case applies again: Since x, y ∈ ∇ = C ∪ ∆ + , both of these half open ends xx ′ and y ′ y lie in C. The segment xy thus lies in ∇ = C ∪ ∆ + .

Σ 0 Σ
Denote X := TV(Σ) and f : X → X 0 := TV(Σ 0 ), the birational contraction.The Picard group Pic(X 0 ) is freely generated by (the sheaves represented by) the polytopes C and D displayed below.Pic(X) = Cl(X) has {A, B, C, D} as a basis: A B C D Define ∆ − := D and ∆ + := C. The difference ∆ − \ ∆ + splits into two components: r is a lattice polyhedron with respect to M .Dually, N ⊆ N ∼ = Z r is a sublattice of finite index.Let G := N/ N be the finite quotient group.For a toric variety X = TV(Σ, N) we can consider the fan Σ in N R = N R with respect to the lattice N and obtain a second toric variety X = TV(Σ, N ) realizing X as the geometric quotient X = X/G.The lattice inclusion ι : N ֒→ N induces the toric covering morphism p : X → X [CLS11, Prop.3.3.7].We pull the sheaves O X (∆ + ) and O X (∆ − ) back to X via p and use the results from section (4.2.2).Example 21.For M = Z r and its dual N = Z r take M to be ( 1 d Z) r and correspondingly N = (dZ) r .Then each ray generator v ρ ∈ N of a ray ρ ∈ Σ(1) is the d-multiple of the corresponding ray generator v ρ ∈ N. The coefficients of the pullback p * D of a Weil divisor D on X will be d-multiples of the coefficients of D.
filtrations (F + , F • +,ρ ) ρ∈Σ(1) and (F − , F • −,ρ ) ρ∈Σ(1) with F ± = E ± and F l ±,ρ = E ⌈ l dρ ⌉ ±,ρ .The filtration F •ρ for a ray ρ corresponding to a pullback sheaf F = p * E of E on X is a d ρ -th stretching in the sense that a proper inclusion can only occur every d ρ steps in the filtration: (5.8)• • • = F dk ⊇ F dk+1 = • • • = F d(k+1) ⊇ F d(k+1)+1 = . ..the d-th squishing of (F, F • ) is the vector space F with filtration (5.9)• • • ⊇ F d(k−1) ⊇ F dk ⊇ F d(k+1) ⊇ . . .Corollary 33.The reflexive sheaf H on X corresponding to (H, H • ρ ), where H := H and the filtration H • ρ given by H l ρ affine, and everything is T -equivariant.Note that this is not a dichotomy; it might happen that 1 ∈ E 0 (U i )∩E 1 (U i ) for some i.Using this order, the boundary homomorphism d the element (1, . . ., 1) ∈ i O X (U i ) i<j≤l , 1, . . ., 1 i≤l<j , 0, . . ., 0 l<i<j [Kly90] , where E 1 denotes the stalk of E at 1 ∈ X and m X,1 the maximal ideal of 1.If E is a T -equivariant, torsion free sheaf on X, the sections of E on the open, affine, T -invariant subsets TV(σ) ⊆ X with σ ∈ Σ are M-graded subsets of E ⊗ C C[M].If, in addition, E is reflexive, then E is already determined by its restriction to open subsets whose complements are of codimension equal or greater than two.Via Klyachko's description[Kly90], a toric reflexive sheaf E corresponds to a set of decreasing Z-filtrations [Kly90]lexive sheaf E defines a toric vector bundle if it is subject to Klyachko's compatibility condition[Kly90]: For each cone σ ∈ Σ there exists a decompositionE = [u]∈M/M ∩σ ⊥ E [u] so that E l ρ = u,vρ ≥l E [u]for each ρ ∈ σ(1).5.2.Line and Tangent Bundles.Line bundles and the tangent are examples of toric bundles with filtrations as follows (compare [Kly90, Example 2.3]).