Sharp degree bounds for fake weighted projective spaces

We give sharp upper bounds on the anticanonical degree of fake weighted projective spaces, only depending on the dimension and the Gorenstein index.


Introduction
A d-dimensional fake weighted projective space is a quotient X = (C d+1 \{0})/G by a diagonal action of G := C * × Γ, where Γ is a finite abelian group and the factor C * acts via positive weights.Any fake weighted projective space X is normal, Q-factorial, of Picard number one and is a Fano variety, i.e. its anticanonical divisor −K is ample.Apart from the classical projective spaces, all fake weighted projective spaces are singular, but have at most abelian quotient singularities.
Fake weighted projective spaces form an interesting example class for the general question of effectively bounding geometric data of a Fano variety in terms of its singularities.For instance, Kasprzyk [7] bounds the order of the torsion part of the divisor class group of a fake weighted projective space X provided that it has at most canonical singularities.Another invariant of the singularities is the Gorenstein index, i.e., the minimal positive integer ι such that ιK is Cartier.In the case of Gorenstein index ι = 1, Nill [8] provides a bound for the degree of a d-dimensional fake weighted projective space X, i.e., the self intersection number (−K) d of its anticanonical divisor.
In the present paper, we extend Nill's bound to higher Gorenstein indices.For any d ≥ 2 define a d + 1 tuple of positive integers by where s ι,1 := ι + 1.Our main result provides sharp upper bounds on the degree (−K) d in terms of the Gorenstein index and lists the cases attaining these bounds: Theorem 1.The anticanonical degree of any d-dimensional fake weighted projective space X of Gorenstein index ι is bounded according to the following table. 2(ι+1) Equality on the degree holds if and only if X is isomorphic to one of the weighted projective spaces in the last row of the table.
The article is organized as follows.Section 2 provides basic properties of fake weighted projective spaces.In Section 3 we assign to any d-dimensional fake weighted projective space of Gorenstein index ι a certain partition of 1/ι into d + 1 unit fractions and give a formula to compute the anticanonical degree in terms of the denominators of these unit fractions.Section 4 contains the number theoretic part of the proof of Theorem 1.In Section 5 we complete the proof of the main result.This amounts to constructing a weighted projective space of given dimension d and Gorenstein index ι whose unit fraction partition of 1/ι meets a maximality condition.

Fake weighted projective spaces
We recall basic properties of fake weighted projective spaces and fix our notation, see also [8,Sect. 3].The reader is assumed to be familiar with the very basics of toric geometry [4,5].Throughout the article N is a rank d lattice for some d ∈ Z ≥2 .Its dual lattice is denoted by are assumed to be full dimensional with 0 ∈ N R in their interior.The normalized volume of a d-dimensional polytope P is Vol(P ) = d!vol(P ), where vol(P ) denotes its euclidean volume.The dual of a polytope P ⊆ N R is the polytope For a facet F of P we denote by u F ∈ M R the unique linear form with u F , v = −1 for all v ∈ F .We have A lattice polytope P ⊆ N R is a polytope whose vertices are lattice points in N .We regard two lattice polytopes P ⊆ N R and P ′ ⊆ N ′ R as isomorphic if there is a lattice isomorphism ϕ : N → N ′ mapping P bijectively to P ′ .Proposition 2.1.The fake weighted projective spaces are precisely the toric varieties X = X(P ) associated to the face fan of a lattice simplex P ⊆ N R with primitive vertices.
Two fake weighted projective spaces are isomorphic if and only if their corresponding lattice simplices are isomorphic.The (true) weighted projective spaces among them correspond to lattice simplices whose vertices generate the lattice.Many geometric properties of a fake weighted projective space can be read off the corresponding lattice simplex.Here we focus our attention on the Gorenstein index and the anticanonical degree.Definition 2.2.The index of a lattice polytope P ⊆ N R is the positive integer Lemma 2.3.The Gorenstein index of any fake weighted projective space X = X(P ) equals the index ι P of the corresponding lattice simplex P ⊆ N R .
Any weighted projective space P(q 0 , . . ., q d ) is up to an isomorphism uniquely determined by its weights (q 0 , . . ., q d ).More generally we assign weights to any lattice simplex P ⊆ N R .Definition 2.5.(cf.[3,8]) A weight system Q (of length d) is a (d + 1)-tuple Q = (q 0 , . . ., q d ) of positive integers.We call the total weight, the factor and the reduction of Q.A weight system Q is called reduced if it coincides with its reduction and it is called well-formed if we have gcd(q j ; j = 0, . . ., d, j = i) = 1 for all i = 0, . . ., d.
Definition 2.6.(cf.[3,8]) To any lattice simplex P = conv(v 0 , . . ., v d ) ⊆ N R we associate a weight system by Q P := (q 0 , . . ., q d ), The weight systems of isomorphic lattice simplices coincide up to order.Denote by v 0 , . . ., v d ∈ N the vertices of the lattice simplex P ⊆ N R .The reduction (Q P ) red is the unique reduced weight system satisfying Moreover, if the vertices of P are primitive, then (Q P ) red is well-formed.Following the naming convention in [8] we call λ P := [N : N P ] the factor of the lattice simplex P ⊆ N R , where N P ⊆ N is the sublattice generated by the vertices of P .In [7] it is called the multiplicity of P .If P has primitive vertices then its factor λ P coincides with the order of the torsion part of Cl(X(P )).In particular X(P ) is a weighted projective space if and only if Q P is reduced.The following Theorem is a reformulation of [3, 4.5-4.7].Compare also [1,Thm. 5.4.5] and [2, Prop.2].
Theorem 2.8.To any well-formed weight system Q of length d there exists a ddimensional lattice simplex P Q ⊆ N R , unique up to an isomorphism, with Q PQ = Q.Any fake weighted projective space X = X(P ) with (Q P ) red = Q is isomorphic to the quotient of P(Q) by the action of the finite group N/N P corresponding to the inclusion N P ⊆ N .
As an immediate consequence of Theorem 2.8 we can relate the Gorenstein index and the anticanonical degree of a fake weighted projective space X(P ) to those of the weighted projective space P((Q P ) red ).
Corollary 2.9.Let X = X(P ) a d-dimensional fake weighted projective space and let X ′ = P((Q P ) red ) the corresponding weighted projective space.Then the Gorenstein index of X is a multiple of the Gorenstein index of X ′ .Moreover we have λ Proof.By Theorem 2.8 there is a square matrix H in a lattice basis of N with determinant λ P such that P = HP Q holds.Dualizing yields P * Q = H * P * .Now apply Lemma 2.3 and Lemma 2.4.

Unit fraction partitions
To any d-dimensional lattice simplex P ⊆ N R of index ι we assign a partition of 1/ι into a sum of d + 1 unit fractions.The main result of this section is Proposition 3.3 where give a formula to compute the normalized volume of the dual polytope P * in terms of the denominators of these unit fractions.
Proposition 3.2.Let P ⊆ N R a d-dimensional lattice simplex of index ι with weight system Q P = (q 0 , . . ., q d ).Then We call it the uf-partition of ι associated to P .
Proof.We show that A(P ) consists of positive integers.Let v 0 , . . ., v d ∈ N the vertices of P .For 0 ≤ i ≤ d let F i = conv(v 0 , . . ., vi , . . ., v d ) the i-th facet of P , where vi means that v i is omitted.We have By definition of ι we have ιu Fi ∈ M .Thus q i divides ι|Q P |, so A(P ) consists of positive integers.Now summing over the reciprocals of A(P ) we see that it is in fact a uf-partition of ι.
Proposition 3.3.For any d-dimensional lattice simplex P ⊆ N R with associated uf-partition A(P ) = (a 0 , . . ., a d ) of ι P we have a d ) .
Proposition 3.3 generalizes [8, Prop.4.5.5] to the case ι ≥ 2. For the proof of Proposition 3.3 and in preparation for the proof of Theorem 1 we extend Batyrev's correspondence between weight systems of reflexive polyhedra and uf-partitions of 1 given in [1,Thm. 5.4.3] to the case of higher indices.Definition 3.4.The index of a weight system Q = (q 0 , . . ., q d ) is the positive integer ι Q := min( k ∈ Z ≥1 ; q i | k|Q| for all i = 0, . . ., d ).

Definition 3.5. A tuple
the total weight, the factor and the reduction of A. A uf-partition A is called reduced if it coincides with its reduction and it is called well-formed if a i | lcm(a j ; j = i) holds for all i = 1, . . ., n Proposition 3.6.Let Q = (q 0 , . . ., q d ) any weight system of length d and index ι and let A = (a 0 , . . ., a d ) any uf-partition of length d + 1.Then the following hold: = A red and this correspondence respects well-formedness.
For the proof of Proposition 3.6 we need the following Lemma.
Lemma 3.7.For ι, a 1 , . . ., a n ∈ Z set Proof.We prove the Lemma by induction on n.The cases n = 1 and n = 2 are verified by direct computation.Let n ≥ 3. Subtracting the second to last row of G := G(ι; a 1 , . . ., a n ) from the last row, we obtain where G ′ = G(ι; a 1 , . . ., a n−1 ) and G ′′ = G(ι; a 1 , . . ., a n−2 , 0).By the induction hypothesis we have Proof of Proposition 3.6.We prove (i).The weight system Q is of index ι, so q i divides ι|Q|.Hence A(Q) consists of positive integers.Summing over the reciprocals of A(Q) shows that it is a uf-partition of ι.Assume A(Q) is not reduced and let Then A ′ is a uf-partition of some ι ′ < ι.Thus q i | ι ′ |Q| holds for all i = 0, . . ., d, contradicting the minimality of ι.So A(Q) is reduced.Item (ii) follows from the fact that t A is the least common multiple of a 0 , . . ., a d .We prove (iii).Let Q = (q 0 , . . ., q d ) a weight system of length d and index ι and write A(Q) = (a 0 , . . ., a d ).The matrix G = G(ι; a 0 , . . ., a d ) as defined in Lemma 3.7 has rank d.Both Q and Q(A(Q)) are contained in the kernel of G and the latter weight system is reduced.Thus it suffices to show that G has rank d.Its kernel is non-trivial, so it has at most rank d.The inequality yields det(G(ι; a 0 , . . ., a d−1 )) > 0. Hence the minor of G, obtained by deleting the last column and row, does not vanish, which yields rk(G) = d.Now let A = (a 0 , . . ., a d ) a uf-partition of ι of length d + 1. Write Q(A) = (q 0 , . . ., q d ) and let . This is a uf-partition of ι Q and we have a ′ i q i = ι Q |Q| for all i = 0, . . ., d.Note that ι Q divides ι.Write ι = λι Q .We obtain Hence A(Q) = λA ′ holds.As A ′ is reduced, this yields A(Q) red = A ′ .For the last assertion in (iii) let Q = (q 0 , . . ., q d ) a reduced weight system of length d and write A := A(Q) = (a 0 , . . ., a d ).We have q i = t A(Q) /a i .The well-formedness of Q is equivalent to saying that holds for all i = 0, . . ., d.This in turn is equivalent to the well-formedness of A(Q).Proof.The first assertion follows from the definitions of A(P ), A(Q P ) and Proposition 3.6 (iii).For the second assertion note that |(Q P ) red | = t A(QP ) /ι QP = t A(P ) /ι P holds.

Sharp bounds for uf-partitions
Following the naming convention in [8] we call syl ι,n the enlarged sylvester partition
-A is the enlarged sylvester partition syl ι,n .
This Proposition is a generalization of [8,Thm. 5.1.3].There Nill utilizes and expands the techniques of Izhboldin and Kurliandchik presented in [6].Here we modify Nill's arguments to incorporate the cases for ι ≥ 2. Let ι, n ∈ Z ≥1 .We denote by A n ι ⊆ R n the compact set of all tuples x ∈ R n with (A1)

Lemma 4.3. For any uf-partition
Proof.The tuple (1/a 1 , . . ., 1/a n ) fulfills conditions (A1) and (A2).For the third condition let 1 ≤ k ≤ n − 1.Then we have The numerator on the right hand side is a positive integer.In particular, it is at least one.
The main part of the proof of Proposition 4.2 is incorporated in the following Lemma, which extends [8,Lemma 5.6].
Case 3. Assume i 0 ≥ 3. Since y n−1 = y n holds, this case only appears for n ≥ 4. We have 1 ≤ r ≤ n − 3.By Lemma 4.6 we have y k = 1/s ι,k for all 1 ≤ k ≤ i 0 − 2. Similar to the second case we use (A2) and (A3) to express y i0−1 in terms of y n and determine an interval of possible values for y n : Again, we define the function f (y n ) := y 1 • • • y n−1 on that interval.It is monotone increasing up to some point and then it is monotone decreasing, so it attains its minimum at the boundary.We obtain: Comparing this to Lemma 4.5 for 1 ≤ r ≤ n − 3, this is only possible for r = 1 and y n = 1/(2t ι,n−1 ).Hence (1/y 1 , . . ., 1/y n ) = syl ι,n and in this case equality holds.

Proof of Proposition 4.2. Let
The first inequality is due to the fact that a n divides lcm(a 1 , . . ., a n ).By Lemma 4.3 the tuple x = (1/a 1 , . . ., 1/a n ) is contained in A n ι .The second inequality and the assertions thereafter now follow immediately from Lemma 4.4.

Proof of the main result
We state and prove the main result of the article.For d = 1 there is only one fake weighted projective space, namely P 1 , which has anticanonical degree −K = 2. Let d ≥ 2. In case ι = 1 and d = 2 the right hand side of the inequality is bounded from above by 9 and P 2 is the only Gorenstein fake weighted projective plane whose degree attains that value, see [8,Ex. 4 (iii) A = syl ι,d+1 .Note that the uf-partition in (i) is not reduced.In particular, there is no weighted projective plane X(P ) of Gorenstein index 2 with A(P ) = (6,6,6).The uf-partitions in (ii) and (iii) are reduced and well-formed.By Theorem 2.8 and Proposition 3.6 the uf-partition A = (2, 6, 6, 6) corresponds to the threedimensional Gorenstein weighted projective space X = P(3, 1, 1, 1) and the ufpartition A = syl ι,d+1 corresponds to the d-dimensional weighted projective space X = P(Q ι,d ).

Corollary 3 . 8 .
For any d-dimensional lattice simplex P ⊆ N R we have A(P ) red = A(Q P ) and ι P |Q P | = λ P t A(P ) .

Definition 5.1. For
any d ≥ 2 and any ι ∈ Z ≥1 we denote by Q ι The anticanonical degree of any d-dimensional fake weighted projective space X of Gorenstein index ι is bounded according to the following table.Equality on the degree holds if and only if X is isomorphic to one of the weighted projective spaces in the last row of the table.