The Anticanonical Complex for Non-degenerate Toric Complete Intersections

The anticanonical complex generalizes the Fano polytope from toric geometry and has been used to study Fano varieties with torus action so far. We provide an explicit description of the anticanonical complex for complete intersections in toric varieties defined by non-degenerate systems of Laurent polynomials. As an application, we classify the terminal Fano threefolds that are embedded into a fake weighted projective space via a general system of Laurent polynomials.


Introduction
The idea behind anticanonical complexes is to extend the features of the Fano polytopes from toric geometry to wider classes of varieties and thereby to provide combinatorial tools for the treatment of the singularities of the minimal model programme.These singularities are defined in terms of discrepancies that means the coefficients a(E) of the exceptional divisors E showing up in the ramification formula for a resolution π : X ′ → X of singularities of a Q-Gorenstein variety: The variety X has at most terminal, canonical or log terminal singularities if always a(E) > 0, a(E) ≥ 0 or a(E) > −1.For an n-dimensional toric Fano variety Z, one defines the Fano polytope to be the convex hull A ⊆ Q n over the primitive ray generators of the describing fan of Z.For any toric resolution π : Z ′ → Z of singularities, the exceptional divisors E ̺ are given by rays of the fan of Z ′ and one obtains the discrepancies as where v ̺ ∈ ̺ is the shortest non-zero lattice vector and v ′ ̺ ∈ ̺ is the intersection point of ̺ and the boundary ∂A.In particular, a toric Fano variety Z is always log terminal and Z has at most terminal (canonical) singularities if and only if its corresponding Fano polytope A contains no lattice points except the origin and its vertices (no lattice points in its interior except the origin).This allows the use of lattice polytope methods in the study of singular toric Fano varieties; see [6,20,21] for work in this direction.
This principle has been extended by replacing the Fano polytope with a suitable polyhedral complex, named anticanonical complex in the setting of varieties with a torus action of complexity one, which encodes discrepancies in full analogy to the toric Fano polytope; see [5].The more recent work [14] provides an existence result of anticanonical complexes for torus actions of higher complexity subject to conditions on a rational quotient.Applications to the study of singularities and Fano varieties can be found in [2,7,15].
In the present article, we provide an anticanonical complex for subvarieties of toric varieties arising from non-degenerate systems of Laurent polynomials in the sense of Khovanskii [23]; see also Definition 3.4.Even in the hypersurface case, the subvarieties obtained this way form an interesting example class of varieties which is actively studied by several authors; see for instance [4,11,17].
We briefly indicate the setting; see Section 3 for the details.Let F = (f 1 , . . ., f s ) be a non-degenerate system of Laurent polynomials in n variables and let Σ be any fan in Z n refining the normal fan of the Minkowski sum B 1 + . . .+ B s of the Newton polytopes B j of f j .Moreover, denote by Z the toric variety associated with Σ.We are interested in the closure X ⊆ Z of the zero set of V (F ) ⊆ T n .The union Z X ⊆ Z of all torus orbits intersecting X turns out to be open in Z and thus the corresponding cones form a subfan Σ X ⊆ Σ and the support of Σ X equals the tropical variety of V (F ) ⊆ T n ; see Theorem 4.4 for all this.Suppose that Z X is Q-Gorenstein.Then, for every σ ∈ Σ X , we have a linear form u σ ∈ Q n evaluating to −1 on every primitive ray generator v ̺ , where ̺ is an extremal ray of σ.We set Theorem 1.1.Let F = (f 1 , . . ., f s ) be a non-degenerate system of Laurent polynomials in K[T ±1  1 , . . ., T ±1 n ] such that V (F ) ⊆ T n irreducible.Then its closure X ⊆ Z admits ambient toric resolutions.Moreover, if Z X is Q-Gorenstein, then X is so and X has an anticanonical complex That means that for all ambient toric modifications Z ′ → Z the discrepancy of any exceptional divisor E X ′ ⊆ X ′ is given in terms of the defining ray ̺ ∈ Σ ′ of its host E Z ′ ⊆ Z ′ , the primitive generator v ̺ ∈ ̺ and the intersection point v ′ ̺ of ̺ and the boundary ∂A X as Observe that in the above setting, each vertex of A X is a primitive ray generator of the fan Σ.Thus, in the non-degenerate hypersurface case, all vertices of the anticanonical complex are integral vectors; this does definitely not hold in other situations, see [5,14].The following consequence of Theorem 1.1 yields in particular Bertini type statements on terminal and canonical singularities.Corollary 1.2.Consider a subvariety X ⊆ Z as in Theorem 1.1 and the associated anticanonical complex A X .
(i) X has at most log-terminal singularities.(ii) X has at most terminal singularities if and only if A X contains no lattice points except the origin and its vertices.(iii) X has at most canonical singularities if and only if A X contains no interior lattice points except the origin.
Moreover, X has at most terminal (canonical) singularities if and only if its ambient toric variety Z X has at most terminal (canonical) singularities.
As an application, we classify the terminal Fano hypersurface threefolds defined by a general Laurent polynomial in a four-dimensional fake weighted projective space.Recall that a fake weighted projective space is an n-dimensional toric variety arising from a complete fan with n + 1 rays.Any fake weighted projective space Z is uniquely determined up to isomorphism by its degree matrix Q, having as its columns the divisor classes [D i ] ∈ Cl(Z) of the toric prime divisors D 1 , . . ., D n+1 of Z. Theorem 1.3.Any non-toric terminal Fano hypersurface threefold X ⊆ Z in a four-dimensional fake weighted projective space Z arising from a general nondegenerate Laurent polynomial is isomorphic to precisely one of the following, specified by its degree µ with respect to the Cl(Z)-grading.

No.
Cl(Z) weighted projective space are precisely the three-dimensional terminal fake weighted projective spaces of which there are eight [22].
We would like to thank Victor Batyrev for stimulating seminar talks and discussions drawing our attention to the class of varieties defined by non-degenerate systems of Laurent polynomials.

Background on toric varieties
In this section, we gather the necessary concepts and results from toric geometry and thereby fix our notation.We briefly touch some of the fundamental definitions but nevertheless assume the reader to be familiar with the foundations of the theory of toric varieties.We refer to [9,10,12] as introductory texts.
Our ground field K is algebraically closed and of characteristic zero.We write T n for the standard n-torus, that means the n-fold direct product of the multiplicative group K * .By a torus we mean an affine algebraic group T isomorphic to some T n .A toric variety is a normal algebraic variety Z containing a torus T as a dense open subset such that the multiplication on T extends to an action of T on Z.
Toric varieties are in covariant categorical equivalence with lattice fans.In this context, a lattice is a free Z-module of finite dimension.Moreover, a quasifan (a fan) in a lattice N is a finite collection Σ of (pointed) convex polyhedral cones σ in the rational vector space N Q = Q ⊗ Z N such that given σ ∈ Σ, we have τ ∈ Σ for all faces τ σ and for any two σ, σ ′ ∈ Σ, the intersection σ ∩ σ ′ is a face of both, σ and σ ′ .The toric variety Z and its acting torus T associated with a fan Σ in N are constructed as follows: where M is the dual lattice of N and σ ∨ ⊆ M Q is the dual cone of σ ⊆ N Q .The inclusion T ⊆ Z of the acting torus is given by the inclusion of semigroup algebras arising from the inclusions σ ∨ ∩ M ⊆ M of additive semigroups.In practice, we will mostly deal with N = Z n = M , where Z n is identified with its dual via the standard bilinear form u, v = u 1 v 1 + . . .+ u n v n .In this setting, we have Moreover, given a lattice homomorphism F : N → N ′ , we write as well F : N Q → N ′ Q for the associated vector space homomorphism.We briefly recall Cox's quotient construction p : Ẑ → Z of a toric variety Z given by a fan Σ in Z n from [8].We denote by v 1 , . . ., v r ∈ Z n the primitive generators of Σ, that means the shortest non-zero integral vectors of the rays ̺ 1 , . . ., ̺ r ∈ Σ.We will always assume that v 1 , . . ., v r span Q n as a vector space; geometrically this means that Z has no torus factor.By D i ⊆ Z we denote the toric prime divisor corresponding to ̺ i ∈ Σ.Throughout the article, we will make free use of the notation introduced around Cox's quotient presentation.Construction 2.1.Let Σ be a fan in Z n and Z the associated toric variety.Consider the linear map P : Z r → Z n sending the i-th canonical basis vector e i ∈ Z r to the i-th primitive generator v i ∈ Z n of Σ, denote by δ = Q r ≥0 the positive orthant and define a fan Σ in Z r by Σ := {δ 0 δ; P (δ 0 ) ⊆ σ for some σ ∈ Σ}.
As Σ consists of faces of the orthant δ, the toric variety Ẑ defined by Σ is an open T r -invariant subset of Z = K r .We also regard the linear map P : Z r → Z n as an n × r matrix P = (p ij ) and then speak about the generator matrix of Σ.The generator matrix P defines a homomorphism of tori: ).This homomorphism extends to a morphism p : Ẑ → Z of toric varieties, which in fact is a good quotient for the action of the quasitorus H = ker(p) on Ẑ.Let P * be the transpose of P , set K := Z r / im(P * ) and let Q : Z r → K be the projection.Then deg(T i ) := Q(e i ) ∈ K defines a K-graded polynomial ring There is an isomorphism K → Cl(Z) from the grading group K onto the divisor class group Cl(Z) sending Q(e i ) ∈ K to the class [D i ] ∈ Cl(Z) of the toric prime divisor D i ⊆ Z defined by the ray ̺ i through v i .Moreover, the K-graded polynomial ring R(Z) is the Cox ring of Z; see [3,Sec. 2.1.3].
We now explain the correspondence between effective Weil divisors on a toric variety Z and the K-homogeneous elements in the polynomial ring R(Z).For any variety X, we denote by X reg ⊆ X the open subset of its smooth points and by WDiv(X) its group of Weil divisors.We need the following pull back construction of Weil divisors with respect to morphisms ϕ : X → Y : Given a Weil divisor D having ϕ(X) not inside its support, restrict D to a Cartier divisor on Y reg , apply the usual pull back and turn the result into a Weil divisor on X by replacing its prime components with their closures in X.

Definition 2.2. Consider a toric variety Z and its quotient presentation
We list the basic properties of describing polynomials, which in fact hold in the much more general framework of Cox rings; see [3, Prop.1.6.2.1 and Cor 1.6.4.6].
Proposition 2.4.Let Z be a toric variety with quotient presentation p : Ẑ → Z as in Construction 2.1 and let D be any effective Weil divisor on Z.
(i) There exist describing polynomials for D and any two of them differ by a non-zero scalar factor.(ii) If g is a describing polynomial for D, then, identifying K and Cl(Z) under the isomorphism presented in Construction 2.1, we have (iii) For every K-homogeneous element g ∈ R(Z), the divisor p * (div(g)) is effective and has g as a describing polynomial.
Let us see how base points of effective divisors on toric varieties are detected in terms of fans and homogeneous polynomials.Recall that each cone σ ∈ Σ defines a distinguished point z σ ∈ Z and the toric variety Z is the disjoint union over the orbits T n • z σ , where σ ∈ Σ. Proposition 2.5.Let Z be the toric variety arising from a fan Σ in Z n and D an effective Weil divisor on Z. Then the base locus of In the later construction and study of non-degenerate subvarieties of toric varieties, we make essential use of the normal fan of a lattice polytope and the correspondence between polytopes and divisors for toric varieties.Let us briefly recall the necessary background and notation.Note the slight abuse of notation: the normal fan Σ(B) is a fan in the strict sense only if the polytope B is of full dimension n, otherwise Σ(B) is a quasifan.Given quasifans Σ and Σ ′ in Z n , we speak of a refinement Σ ′ → Σ if Σ and Σ ′ have the same support and every cone of Σ ′ is contained in a cone of Σ.

Reminder 2.7. Let
The normal fan Σ(B) of B is the coarsest common refinement of the normal fans Σ(B i ) of the B i .The cones of Σ(B) are given as ). Reminder 2.8.Let B ⊆ Q n be an n-dimensional polytope with integral vertices and let Σ be any complete fan in Z n with generator matrix P = [v 1 , . . ., v r ].Define a vector a ∈ Z r by a := (a 1 , . . ., a r ), Observe that the a i are indeed integers, because B has integral vertices.For u ∈ B set a(u) := P * u + a and let B(u) B be the minimal face containing u. Then the entries of the vector a(u Proposition 2.9.Let B ⊆ Q n be a lattice polytope and Σ any complete fan in Z n with generator matrix With a ∈ Z r from Reminder 2.8, we define a divisor on the toric variety Z arising from Σ by Moreover, for every vector u ∈ B ∩ Z n , we have a(u) ∈ Z r as in Reminder 2.8 and obtain effective divisors D(u) on Z, all of the same class as D by If Σ refines the normal fan Σ(B), then D and all D(u) are base point free.If Σ equals the normal fan Σ(B), then the divisors D and D(u) are even ample.

Laurent systems and their Newton polytopes
We consider systems F of Laurent polynomials in n variables.Any such system F defines a Newton polytope B in Q n .The objects of interest are completions X ⊆ Z of the zero set V (F ) ⊆ T n in the toric varieties Z associated with refinements of ther normal fan of B. In Proposition 3.6, we interprete Khovanskii's non-degeneracy condition [23] in terms of Cox's quotient presentation of Z. Theorem 3.7 gathers complete intersection properties of the embedded varieties X ⊆ Z given by nondegenerate systems of Laurent polynomial.
We begin with recalling the basic notions around Laurent polynomials and Newton polytopes.Laurent polynomials are the elements of the Laurent polynomial algebra for which we will use the short notation Given a face B B(f ) of the Newton polytope, the associated face polynomial is defined as The pullback of f with respect to the homomorphism p : T r → T n defined by the generator matrix P = (p ij ) of Σ has a unique presentation as being coprime to each of the variables T 1 , . . ., T r .We call g the Σ-homogenization of f .Lemma 3.3.Consider a Laurent polynomial f ∈ LP(n) with Newton polytope B(f ) and a fan Σ in Z n with generator matrix P := [v 1 , . . ., v r ] and associated toric variety Z.Let a := (a 1 , . . ., a r ) be as in Reminder 2.8 and D ∈ WDiv(Z) the push forward of div(f ) ∈ WDiv(T n ).
(i) The Σ-homogenization g of f is a describing polynomial of D and with the homomorphism p : T r → T n given by P , we have (ii) The Newton polytope of g equals the image of the Newton polytope of f under the injection Q n → Q r sending u to a(u) := P * u + a, in other words (iii) Consider a face B B(f ) and the associated face polynomial f B .Then the corresponding face P * B + a B(g) has the face polynomial (iv) The degree deg(g) ∈ K of the Σ-homogenization g of f and the divisor class (i) We speak of F = (f 1 , . . ., f s ) as a system in LP(n) and define the Newton polytope of F to be the Minkowski sum (ii) The face system F ′ of F associated with a face B ′ B of the Newton polytope is the Laurent system . ., g s ), where g j is the Σ-homogenization of f j .(v) By an F -fan we mean a fan Σ in Z n that refines the normal fan Σ(B) of the Newton polytope B of F .
Note that Khovanskii's non-degeneracy Condition 3.4 (iii) is fullfilled for suitably general choices of F .Even more, it is a concrete condition in the sense that for every explicitly given Laurent system F , we can explicitly check non-degeneracy.
where Z is the toric variety associated with Σ and Z = K r .The quotient presentation p : Ẑ → Z gives rise to a commutative diagram where X := X ∩ Ẑ ⊆ Z as well as X ⊆ Z are closed subvarieties and p : X → X is a good quotient for the induced H-action on X.In particular, X = p( X).
The key step for our investigation of varieties X ⊆ Z defined by Laurent systems is to interprete the non-degeneracy condition of a system F in terms of its Σ-homogenization G.We work with distinguished points z σ ∈ Z.In terms of Cox's quotient presentation, z σ ∈ Z becomes explicit as z σ = p(z σ), where σ = cone(e i ; v i ∈ σ) ∈ Σ and the coordinates of the distinguished point z σ ∈ Ẑ are z σ,i = 0 if v i ∈ σ and z σ,i = 1 otherwise.Proposition 3.6.Let F = (f 1 , . . ., f s ) be a non-degenerate system in LP(n) and let Σ be an Proof.We care about (i) and on the way also prove (ii).Since g 1 , . . ., g s are Hhomogeneous, the set of points ẑ ∈ Ẑ with DG(ẑ) of rank strictly less than s is H-invariant and closed in Ẑ.Thus, as p : Ẑ → Z is a good quotient for the Haction, it suffices to show that for the points point ẑ ∈ X with a closed H-orbit in Ẑ, the differential DG(ẑ) is of rank s.That means that we only have to deal with the points ẑ ∈ X ∩ T r • z σ, where σ ∈ Σ.So, consider a point ẑ ∈ X ∩ T r • z σ, let σ ′ ∈ Σ(B) be the minimal cone with σ ⊆ σ ′ and let B ′ B be the face corresponding to σ ′ ∈ Σ(B).Then we have the Minkowski decomposition From Reminder 2.7 we infer that σ ′ j = σ(B ′ j ) is the minimal cone of the normal fan Σ(B j ) ′ with σ ⊆ σ ′ j .Let F ′ be the face system of F given by B ′ ⊆ B. Define , where g ′ j is the face polynomial of g j defined by P * B ′ j + a j P * B j + a j = B(g j ), g j = T aj p * f j .
Due to Lemma 3.3 (iii), the polynomials g ′ j only depend on the variables T i with v i / ∈ σ(B ′ j ).Using σ ⊆ σ(B ′ j ) and the fact that we have ẑi = 0 if and only if v i ∈ σ holds, we observe This reduces the proof of (i) to showing that DG ′ (ẑ) is of full rank s, and the latter proves (ii).Choose z ∈ T r such that zi = ẑi for all i with v i ∈ σ.Using again that the polynomials g ′ i only depend on We conclude that F ′ (p(z)) = 0 holds.Thus, the non-degeneracy condition on the Laurent system F ensures that DF ′ (p(z)) is of full rank.Moreover, we have Since T aj (z) = 0 holds for j = 1, . . ., s and p : T r → T n is a submersion, we finally obtain that DG ′ (ẑ) of full rank, as wanted.
The first application gathers complete intersection properties for the Σhomogenization and the variety defined by a non-degenerate Laurent system.Note that the codimension condition imposed on X \ X in the fourth assertion below allows computational verification for explicitly given systems of Laurent polynomials.
(i) The variety X = V (G) in Z = K r is a complete intersection of pure dimension r − s with vanishing ideal (ii) With the zero sets V (F ) ⊆ T n and V (G) ⊆ K r and the notation of Construction 3.5, we have In particular, the irreducible components of X ⊆ Z are the closures of the irreducible components of V (F ) ⊆ T n .(iii) The closed hypersurfaces X j = V (f j ) ⊆ Z, where j = 1, . . ., s, represent X as a locally complete scheme-theoretic intersection (iv) If X \ X is of codimension at least two in X, then X is irreducible and normal and, moreover, X is irreducible.
Proof.Assertion (i) is clear by Proposition 3.6 (i) and the Jacobian criterion for complete intersections.For (ii), we infer from Proposition 3.6 (ii) that, provided it is non-empty, the intersection X ∩ T r • z σ is of dimension r − s − dim(σ).In particular no irreducible component of V (G) is contained in X \ T r .The assertions follow.
We prove (iii).Each f j defines a divisor on Z having support X j and according to Lemma 3.3 (v) this divisor is base point free on Z. Thus, for every σ ∈ Σ, we find a monomial h σ,j of the same K-degree as g j without zeroes on the affine chart Ẑσ ⊆ Ẑ defined by σ ∈ Σ.We conclude that the invariant functions g 1 /h σ,1 , . . ., g s /h σ,s generate the vanishing ideal of X on the affine toric chart Z σ ⊆ Z.We turn to (iv).Proposition 3.6 and the asumption that X \ X is of codimension at least two in X allow us to apply Serre's criterion and we obtain that X is normal.In order to see that X is irreducible, note that H acts on Z with attractive fixed point 0 ∈ Z.This implies 0 ∈ X, Hence X is connected and thus, by normality, irreducible.
Remark 3.8.Let F be a non-degenerate system of Laurent polynomials with fulldimensional Newton polytope B = B(F ) and assume Σ = Σ(B).If B is a simplex or, equivalenty, Z a fake weighted projective space, then Theorem 3.7 (iv) applies.

Non-degenerate toric complete intersections
We take a closer look at the geometry of the varieties X ⊆ Z arising from nondegenerate Laurent systems.The main statements of the section are Theorem 4.2, showing that X ⊆ Z is always quasismooth and Theorem 4.4 giving details on how X sits inside Z.Using these, we can prove Theorem 1.1 which describes the anticanonical complex.First we give a name to our varieties X ⊆ Z, motivated by Theorem 4.4.Definition 4.1.By a non-degenerate toric complete intersection we mean a variety X ⊆ Z defined by a non-degenerate system F in LP(n) and an F -fan Σ in Z n .
An immediate but important property of non-degenerate toric complete intersections is quasismoothness.Note that the second statement in the theorem below is Khovanskii's resolution of singularities [23,Thm. 2.2].Observe that our proof works without any ingredients from the theory of holomorphic functions.Theorem 4.2.Let F be a non-degenerate system in LP(n) and Σ an F -fan in Z n .Then the variety X is normal and quasismooth in the sense that X is smooth.Moreover, X ∩ Z reg ⊆ X reg .In particular, if Z is smooth, then X is smooth.Proof.By Proposition 3.6 (i), the variety X is smooth.As smooth varieties are normal and the good quotient p : X → X preserves normality, we see that X is normal.Moreover, if Z is smooth, then the quasitorus H = ker(p) acts freely on p −1 (Z reg ), hence on X ∩ p −1 (Z reg ) and thus the quotient map p : X → X preserves smoothness over X ∩ Z reg .
The next aim is to provide details on the position of X inside the toric variety Z.The considerations elaborate the transversality statement on X and the torus orbits of Z made in [23] for the smooth case.Definition 4.3.Let Z be the toric variety arising from a fan Σ in Z n .Given a closed subvariety X ⊆ Z, we set Theorem 4.4.Consider a non-degenerate system F = (f 1 , . . ., f s ) in LP(n), an F -fan Σ in Z n and the associated toric complete intersection X ⊆ Z.
(i) For every σ ∈ Σ X , the scheme Proof.We prove (i).Given a cone σ ∈ Σ X consider σ ∈ Σ and the corresponding affine toric charts and the restricted quotient map: From Proposition 3.6 we infer that X(σ) = T r • z σ ∩ X is a reduced subscheme of pure codimension s in T r • z σ.The involved vanishing ideals on Z σ and Ẑσ satisfy We conclude that the left hand side ideal is radical.In order to see that X(σ) is of codimension s in T n • z σ , look at the restriction This is a geometric quotient for the H-action, it maps X(σ) onto X(σ) and, as X(σ) is H-invariant, it preserves codimensions.We prove (ii) and (iii).First note that, due to (i), for any σ ∈ Σ X we have dim(σ) ≤ n − s.We compare Σ X with trop(X).Tevelev's criterion [26] tells us that a cone σ ∈ Σ belongs to Σ X if and only if σ • ∩ trop(X) = ∅ holds.As Σ is complete, we conclude that trop(X) is covered by the cones of Σ X .
We show that the support of every cone of Σ X is contained in trop(X).The tropical structure theorem provides us with a balanced fan structure ∆ on trop(X) such that all maximal cones are of dimension n − s.Together with Tevelev's criterion, the latter yields that all maximal cones of Σ X are of dimension n − s.The balancy condition implies that every cone δ 0 ∈ ∆ of dimension n − s − 1 is a facet of at least two maximal cones of ∆.We conclude that every cone σ ∈ Σ X of dimension n − s must be covered by maximal cones of ∆.
Knowing that trop(X) is precisely the union of the cones of Σ X , we directly see that Σ X is a fan: Given σ ∈ Σ X , every face τ σ is contained in trop(X).In particular, τ • intersects trop(X).Using once more Tevelev's criterion, we obtain τ ∈ Σ X .Corollary 4.5.Let F = (f 1 , . . ., f s ) be a non-degenerate Laurent system in LP(n).Then F is a tropical basis if and only if each of its face systems F ′ generates a monomial free ideal in LP(n).
We approach the proof of Theorem 1.1.The remaining ingredients are the adjunction formula given in Proposition 4.7 and Proposition 4.8 providing canonical divisors which are suitable for the ramification formula.The following pull back construction relates divisors of Z to divisors on X. Remark 4.6.Let X ⊆ Z be an irreducible non-degenerate toric complete intersection.Denote by ı : X ∩ Z reg → X and  : X ∩ Z reg → Z reg the inclusions.Then Theorems 4.2 and 4.4 (ii), yield a well defined pull back homomorphism where we set T = T n for short.By Theorem 4.4 (i), this pull back sends any invariant prime divisor on Z to a sum of distinct prime divisors on X.Moreover, we obtain a well defined induced pullback homomorphism for divisor classes Proposition 4.7.Let X ⊆ Z be an irreducible non-degenerate toric complete intersection given by a system F = (f 1 , . . ., f s ) in LP(n).
(i) Let C j ∈ WDiv(Z) be the push forward of div(f j ) and K Z an invariant canonical divisor on Z. Then the canonical class of X is given by (ii) If the toric variety Z X is Q-Gorenstein, then also the variety X is Q-Gorenstein.
Proof.Due to Theorem 4.2 and Theorem 4.4 (ii) it suffices to have the desired canonical divisor on Z reg ∩ X ⊆ X reg .By Theorem 3.7, the classical adjunction formula applies, proving (i).For (ii), note that the divisors C j on Z are base point free by Lemma 3.3 (v).Thus, using (i), we see that any canonical divisor on X is Q-Cartier.
Proposition 4.8.Consider an irreducible non-degenerate system F in LP(n), a refinement Σ ′ → Σ of F -fans and the associated modifications π : Then there is a vertex u ∈ B of the Newton polytope B = B(F ) such that the maximal cone σ(u) ∈ Σ(B) contains σ.Write u = u 1 + . . .+ u s with vertices u j ∈ B(f j ).With the corresponding vertices a(u j ) = P * u j + a j of the Newton polytopes B(g j ), we define Let C j ∈ WDiv(Z) be the push forward of div(f j ).Propositions 2.5 and 2.9 together with Lemma 3.3 (v) tell us Also for the Σ ′ -homogenization G ′ of F , the vertices u j ∈ B(f j ) yield corresponding vertices a ′ (u j ) ∈ B(g ′ j ) and define divisors As above we have the push forwards C ′ j ∈ WDiv(Z ′ ) of div(f j ) and, by the same arguments, we obtain Take the invariant canonical divisors K Z on Z and K Z ′ in Z ′ with multiplicity −1 along all invariant prime divisors and set According to Proposition 4.7, these are canonical divisors on X and X ′ respectively.Properties (i) and (ii) are then clear by construction.
Proof of Theorem 1.1.First observe that A X is an anticanonical complex for the toric variety Z X .Now, choose any regular refinement Σ ′ → Σ of the defining Ffan Σ of the irreducible non-degenererate toric complete intersection X ⊆ Z.This gives us modifications π : Z ′ → Z and π : X ′ → X.Standard toric geometry and Theorem 4.2 yield that both are resolutions of singularities.Proposition 4.8 provides us with canonical divisors on X ′ and X.We use them to compute discrepancies.Over each X ∩Z σ , where σ ∈ Σ X , we obtain the discrepancy divisor as By Theorem 4.4 (i), every exceptional prime divisor E ′ X ⊆ X ′ admits a unique exceptional prime divisor

guarantees that the discrepancy of E ′
X with respect to π : X ′ → X and that of E ′ Z with respect to π : Z ′ → Z X coincide.

Fake weighted terminal Fano threefolds
Here we prove Theorem 1.3.The idea is to find suitable upper bounds on the specifying data of the ambient fake weighted projective space Z that reduce the task to working out a managable number of cases.The following lemmas provide first restricting combinatorial conditions on the specifying data of X ⊆ Z. Lemma 5.1.Let X ⊆ Z be a hypersurface in a fake weighted projective space and denote by w 1 , . . ., w r ∈ Cl(Z) the divisor classes of the torus invariant prime divisors in Z. Then the class [X] ∈ Cl(Z) is base point free if and only if for any i = 1, . . ., r there exists an integer Proof.This is a direct consequence of Proposition 2.5 and the fact that the maximal cones of Σ are given by cone(v i ; i = j) for j = 1, . . ., n + 1. Lemma 5.2.Let X ⊆ Z be an irreducible Q-Gorenstein hypersurface arising from a non-degenerate Laurent polynomial and denote by w 1 , . . ., w r ∈ Cl(Z) the classes of the torus invariant prime divisors in Z.If dim(Z) ≥ 3 holds and X has at most terminal singularities, then each r − 2 of w 1 , . . ., w r generate Cl(Z) as a group.
Proof.Corollary 1.2 says that Z X has at most terminal singularities.Thus the singular locus of Z X has at least codimension three in Z X ; see [18,Cor. 8.3.2] for instance.The same is true for the singular locus of Z since Z \ Z X is finite.We conclude that every toric orbit in Z of dimension r − 2 is smooth.This is equivalent to the assertion.Lemma 5.3.Situation as in Construction 2.1 and assume Z to be Q-Gorenstein.Let κ : K → K ′ be an epimorphism of abelian groups of the same rank and consider the coarsened degree map Q ′ = κ • Q.Moreover, let P ′ be a matrix having a basis for ker(Q ′ ) as its rows and let v ′ 1 , . . ., v ′ r be the columns of P ′ .For any cone ) contains a lattice point different from its vertices and the origin, then the same is true for A(σ).
Proof.Clearly, ker(Q) is a linear subspace of ker(Q ′ ).Since κ is surjective and K, K ′ have the same rank, ker(Q) and ker(Q ′ ) share the same rank as well.In this situation there is a basis (u 1 , . . ., u n ) of ker(Q ′ ) together with α 1 , . . ., α n ∈ Z ≥1 such that (α 1 u 1 , . . ., α n u n ) is a basis for ker(Q).By choosing suitable coordinates for the ambient lattice of v ′ 1 , . . ., v ′ r we achieve that u 1 , . . ., u n are exactly the rows of P ′ .
Furthermore, observe that the rows of P = [v 1 , . . ., v r ] form a basis for ker(Q) as well.After a applying a suitable coordinate change to the ambient lattice of Σ, we may assume that α 1 u 1 , . . ., α n u n are the rows of P .Now we have P = DP ′ with D = diag(α 1 , . . ., α n ), hence the cones σ, σ ′ are related by shows that the linear form u σ ′ = Du σ ∈ Q n evaluates to −1 on each v ′ i ∈ σ ′ .Altogether we conclude A(σ) = DA(σ ′ ).In particular every lattice point v ′ ∈ A(σ ′ ) induces a lattice point Dv ′ ∈ A(σ).If v ′ is different from the vertices v ′ i of σ ′ and the origin, then Dv ′ is different from the vertices v i of σ and the origin.
We call a hypersurface X ⊆ Z in a complete toric variety sincere, if the describing polynomial has no linear term.Non-sincere hypersurfaces turn out to be toric varieties themselves.Remark 5.4.Let X ⊆ Z be a non-sincere hypersurface in a complete toric variety.The Cl(Z)-grading on R(Z) is pointed, thus the describing polynomial g ∈ R(Z) is of the form g = T i + g 0 where g 0 does not depend on T i .The graded automorphism on R(Z) given by T i → T i − g 0 and T j → T j whenever j = i induces an automorphism ϕ : Z → Z such that ϕ(X) = D i where D i ∈ WDiv(Z) is the torus invariant prime divisor with deg(g In particular X is toric. Then B(D) is a full-dimensional lattice polytope and Z is the toric variety associated with the normal fan of B(D).Moreover, any Laurent polynomial f ∈ K[T ±1 1 , . . ., T ±1 n ] having B(D) as its Newton polytope gives rise to a hypersurface X ⊆ Z with [X] = µ ∈ Cl(Z).According to [23,Thm. 2] a general f is nondegenerate, in particular it is possible to choose a non-degenerate f .Proof of Theorem 1.3.The divisor class group Cl(Z) of the ambient fake weighted projective space Z is of the form Cl denote the classes of X regarded as a divisor and the torus invariant prime divisors on Z. Since Z is four dimensional, there are precisely five of the latter.As the grading is pointed we may assume µ 0 , x 1 , . . .x 5 > 0.Moreover, since X is non-toric, X ⊆ Z is a sincere hypersurface and therefore µ 0 is a proper multiple of each of the x i .
According to [25,Thm. 1] restricting Weil divisors on Z to X leads to an isomorphism Cl(X) ∼ = Cl(Z).Moreover, by Proposition 4.7, the anticanonical class −K X of X is given by − Write m := lcm(x 1 , . . ., x 5 ) for short.Lemma 5.1 yields that each x i divides µ 0 , hence µ 0 is a multiple of m.Altogether X being Fano amounts to We split the further discussion into the cases m = x 5 and m > x 5 .
We conclude As µ 0 is a proper multiple of x 5 = m, subtracting x 5 from both sides of Eq. (5.5.1) leads to (5.5.2) Taking x 1 , . . ., x 4 being coprime into account, this forces x 1 = 1, x 2 = 1 and we are left with the following two configurations With x 3 = 2 and x 4 = 3 inserting into Eq.(5.5.2) directly gives x 5 < 7, hence x 5 satisfies the claimed bound.We turn to x 3 = 1.Note x 5 = dx 4 for some d ∈ Z ≥1 .Plugging into Eq.(5.5.2) gives dx 4 < 3 + x 4 .We arrive at one of the following constellations, each of which is as in the claim: Case 2: Assume m > x 5 i.e. one has m = lx 5 for some l ∈ Z ≥2 .From Eq. (5.5.1) we deduce lx 5 = m < 5x 5 , hence l ≤ 4. In what follows we have to distinguish between x 4 = x 5 and x 4 = x 5 .
We conclude x 5 ≤ 11, in particular x 5 satisfies the claimed upper bound.
We show that none of the three unbounded constellations of (x 1 , . . ., x 5 ) comes from a terminal variety X for a > 4; they are According to Corollary 1.2 we have to show that A X contains a lattice point v different from its vertices and the origin.Applying Lemma 5.3 to the projection Cl(Z) → Z shows that it suffices to deal with Cl(Z) = Z.For each constellation of (x 1 , . . ., x 5 ) the subsequent table shows a generator matrix P i = [v 1 , . . ., v 5 ] for a fan Σ of Z which makes it easy to see that the component A(σ) ⊆ A X associated with σ = cone(v 1 , v 2 , v 3 ) ∈ Σ contains a lattice point v as desired.
The next step is to bound the order of the torsion part Z/t 1 Z × • • • × Z/t k Z of Cl(Z).A first constraint comes from Lemma 5.2: Since Cl(Z) is generated by three elements, we may assume k ≤ 2. Thus our task is to compute all four dimensional fake weighted projective spaces determined by their degree map admitting a µ ∈ Cl(Z) such that the following conditions are satisfied: (i) µ is a multiple of each w i , (ii) 2x 5 ≤ µ 0 < x 1 + • • • + x 5 , (iii) apart from the toric fix points, Z has at most terminal singularities At first, one computes all quintuples (w 1 , . . ., w 5 ) ∈ Z 5 ≥1 with w 5 = x 5 ≤ 41 satisfying conditions (i) to (iii), i.e. we treat the case Cl(Z) = Z.Lemma 5.3 guarantees that this already covers all admissible configurations of (x 1 , . . ., x 5 ) in the general case, where w i ∈ Cl(Z) = Z × Z/t 1 Z × Z/t 2 Z.It turns out that always x 1 = 1 holds.This enables us to find effective bounds on t 1 and t 2 .As x 1 = 1 holds a suitable coordinate change leads to y 11 = y 12 = 0.As µ 0 > 0 holds, Lemma 5.1 provides us with l i ∈ Z ≥1 such that always l i w i = µ.From µ = l 1 w 1 we now infer µ 1 = µ 2 = 0. Let 2 ≤ d < e ≤ 5. Lemma 5.2 tells us that w 1 , w e , w d generate Z × Z/t 1 Z × Z/t 2 Z as a group.Since y 1j = 0 holds, y dj , y ej must span Z/t j Z, where j = 1, 2. So α j y dj + β j y ej = 1 holds for some α j , β j ∈ Z. Besides, we have l d y dj = µ j = 0 and l e y ej = 0 as well.Altogether we obtain lcm(l d , l e ) = lcm(l d , l e )(α j y dj + β j y ej ) = α j lcm(l d , l e )y dj + β j lcm(l d , l e )y ej = 0.This means t j | lcm(l d , l e ).We obtain an effective bound on t j by (5.5.4) Now, for a fixed admissible configuration (1, x 2 , . . ., x 5 ) we can simply go through the finite number of degree maps Q with non-trivial torsion part and directly check (i) to (iii) for each of them.We have used a Magma program for this purpose.
To finish the proof, we have to show that each pair (Q, µ) as Theorem 1.3 indeed comes from a terminal Fano hypersurface in a four dimensional fake weighted projective space.Let Z be the fake weighted projective space associated with Q. Lemma 5.1 tells us that µ ∈ Cl(Z) is base point free due to condition (i).Remark 5.5 guarantees that the specifying datum (Q, µ) is realized by some hypersurface X ⊆ Z arising from a non-degenerate Laurent polynomial.Using Proposition 4.7 and [25,Thm. 1] shows that for general f , condition (ii) is equivalent to X being Fano.According to Corollary 1.2, condition (iii) ensures that X has at most terminal singularities.
Comparing the intersection numbers −K 3 X from Theorem 1.3 with those of threedimensional terminal fake weighted projective spaces [22] shows that none of the varieties from Theorem 1.3 is toric.Furthermore, it is clear from the invariants Cl(X) = Cl(Z) and −K 3 X that two members from different families from Theorem 1.3 are non-isomorphic, except for Numbers 18 and 24.Note that for X belonging to Number 18 or 24 the Cox ring of X is given by R(X) ∼ = R(Z)/ g where g is a describing polynomial for X; see [3,Rem. 4.1.1.4].This shows that a member from Number 18 is not isomorphic to a member from Number 24 as the respective Cox rings have different sets of generator degrees, which are unique as we have a positive Z-grading in both cases.

Remark 5 . 5 .
Situation as in Construction 2.1.Given a base point free and ample class µ ∈ Cl(Z) one considers an invariant representative D = a 1 D 1 + • • • + a r D r of µ and the associated polytope