Birational invariants of toric orbifold surfaces

We study birational invariants of toric orbifold surfaces.


Introduction
Let k be an algebraically closed field.Algebraic orbifolds over k are smooth separated irreducible Deligne-Mumford stacks of finite type over k, with trivial generic stabilizer.Birational invariants of algebraic orbifolds were introduced in [7]; they are invariant under birational projective morphisms of algebraic orbifolds.In this paper, we focus on toric Deligne-Mumford stacks, introduced in [3], and specialize to dimension 2, i.e., toric orbifold surfaces.The goal of this paper is to study their birational invariants over an algebraically closed field of characteristic 0.

Toric varieties
To fix notation, we recall basic definitions of toric varieties [6].We fix a base field k, which will later be taken to be algebraically closed of characteristic 0, and a dimension n, which will later be taken to be 2.There are lattices M := Z n and N := Hom(M, Z).
The elements of M are characters on the algebraic torus B Denis Levchenko denis.levchenko@math.uzh.ch We view the lattice N as sitting inside its extension over the real numbers we employ analogous notation, e.g., for an extension over Q.
Cones in N R are always strongly convex rational polyhedral cones: A fan in N is a finite collection of cones in N R , such that for every cone in the fan, its faces are also in the fan, and for every pair of cones in the fan, the intersection is a face of each.The support of a fan is the union of the cones.
A cone σ determines an affine toric variety where σ ∨ denotes the dual cone in M R .
The toric variety determined by a fan is the union of U σ over all σ ∈ , glued by identifying a common open subvariety U σ ∩τ of U σ and U τ , for all σ , τ ∈ .This yields a normal algebraic vareity X ( ) over k.Every normal algebraic variety over k with T -action and equivariant open immersion of T is the toric variety of a uniquely determined fan in N .
Each U σ is T -invariant and has a unique closed T -orbit, the image of where σ ⊥ ⊂ M R is defined by pairing to 0 with all elements of σ .The closure is a closed subvariety of X ( ), determined by σ .This is a point when σ is an n-dimensional cone, and has codimension 1 when σ is a ray (1-dimensional cone).
A cone is smooth if it is generated by part of a Z-basis of N .We have X ( ) smooth if and only if all of the cones of are smooth.
For X ( ) to be projective, it is necessary and sufficient for the support of to be equal to N R and to admit a piecewise linear convex support function.When n = 2, the second condition is automatic, so projective toric surfaces correspond to fans with support equal to N R .
As well, when n = 2 we profit from the elementary structure of low-dimensional cones.The cones are 0, rays R ≥0 v, and 2-dimensional cones R ≥0 v + R ≥0 w.The generator v ∈ N R of a ray may be uniquely specified by the requirements to lie in N and not be a multiple of another generator in N ; we call such v ∈ N the primitive ray generator.For a 2-dimensional cone, primitive generators v, w ∈ N are uniquely determined up to swapping.

Orbifold toric surfaces
Toric Deligne-Mumford stacks were introduced in [3].The idea is to consider only fans whose cones are simplicial, and in this case to glue the stack quotients of smooth affine toric varieties by finite groups to obtain a smooth separated irreducible Deligne-Mumford stack with trivial generic stabilizer, i.e., an orbifold.The construction requires the given fan , with simplicial cones, to be supplemented by the additional data of a generator in N of every ray of , to make a so-called stacky fan .
We continue to work over a base field k, take n = 2, and fix a prime number p, not equal to the characteristic of k.Let be a fan in N , such that every 2-dimensional cone is of the form σ = R ≥0 v + R ≥0 w with primitive generators v, w ∈ N such that the (well-defined) absolute value of the determinant of the 2 × 2 integer matrix of coordinates of v and w, with respect to a Z-basis of N , is equal to 1 or p.We consider a stacky fan , where the additional data of a choice of generator of each ray ∈ is subject to the following conditions: • The chosen generator is equal to or is p times the primitive generator of .
• When the generator is p times the primitive generator, the 2-dimensional cones containing are smooth.• No pair of rays, for which the chosen generator is p times the primitive generator, generates a cone of .The construction of the toric orbifold X ( ) of a stacky fan proceeds by suitably modifying the construction of X ( ); we only describe the construction in the present setting, where the conditions have been chosen to correspond precisely to toric orbifold surfaces X ( ), whose nontrivial stabilizer groups all have order p.For every 2-dimensional cone σ = R ≥0 v + R ≥0 w with primitive vectors v, w ∈ N and corresponding 2 × 2 integer matrix with determinant of absolute value p, we have an index p sublattice of N generated by v and w, and by duality, a lattice is used in place of U σ in the construction; just with this modification, the result would be a smooth orbifold surface with finitely many points with μ p -stabilizer.This is further modified by passing to the pth root stack [4, Sect.2] [1, App.B] of the union of divisors, that correspond to rays where p times the primitive ray generator is given in .The resulting orbifold X ( ) has locus with μ p -stabilizer equal to the union of finitely many points and finitely many pairwise disjoint rational curves.
The smooth Deligne-Mumford stack X ( ) has coarse moduli space X ( ).We describe an orbifold as projective, when it has projective coarse moduli space.But n = 2, so X ( ) is projective if and only if the support of is equal to N R .

Orbifold Burnside group
The Burnside group of stacks Burn n and its weight module B = j≥0 B j were introduced in [7].The weight module is an abelian group, defined by explicit generators and relations.
We suppose that the base field k is algebraically closed of characteristic 0. Then a projective orbifold X of dimension n determines a class in an abelian group Burn n , generated by pairs (K , α) with K a finitely generated field over k, of some transcendence degree d ≤ n, and α ∈ B n−d .With suitable relations, this class is invariant under birational projective morphisms of projective orbifolds.In other words, [X ] = [X ] in Burn n if there exists an orbifold Y with birational projective morphisms Y → X and Y → X .
Symbols with K = k generate a subgroup We describe the group B 2 , or rather a particular subgroup B [ p] 2 , for which we have in the subgroup (1).The locus with μ p -stabilizer of X ( ), as mentioned above, consists of finitely many divisors and isolated points, which we call μ p -divisors and μ p -points, respectively.Their complement is a quasi-projective variety.
The contribution from the μ p -points is the sum of individual contributions from each μ p -point.At a μ p -point, an identification of the stabilizer with μ p is made, so that one of the weights of the action on the tangent space is 1; then a is the second weight.But this involves a choice: if we reverse the roles of the two weights then the resulting weight would be a −1 .Thanks to the corresponding relation in 2 of a μ p -point is well-defined.The quotient group 2 / (triv, (0, 0)), (Z/ pZ, (0, 1)) ( is the group denoted by 2 ) in [7,Sect. 4].In [7, Lemma 5.3], an explicit isomorphism is given from this group to a certain quotient of the group H 1 (X 0 ( p) orb , Z), the first orbifold homology of the modular curve X 0 ( p).This group ( 2) is also mentioned as providing an obstruction to being linked by blow-ups of points to an orbifold surface without μ p -points.For the group (2) the following isomorphism types were found: p = 2 : 0, p = 3 : 0, p = 5 : Z/2Z, p = 7 : 0, p = 11 : Z.

Refined invariant
We continue to work over an algebraically closed base field k of characteristic 0. A natural class of birational projective morphisms of projective toric Deligne-Mumford stacks is the class of morphisms associated with the stacky star subdivisions (of toric Deligne-Mumford stacks) of [5].All toric stacks will be projective toric orbifold surfaces with μ p as the only possible nontrivial stabilizer group.We emphasize that the torus T is regarded as fixed.All stacky fans will satisfy the corresponding conditions, stated in Sect.3, and have support equal to N R .
A stacky star subdivision is an operation that, given a stacky fan and a chosen 2-dimensional cone σ ∈ , yields a new stacky fan σ and a T -equivariant birational projective morphism Let us write σ = R ≥0 v + R ≥0 w with primitive vectors v, w ∈ N ; then the stacky fan σ is constructed by deleting the cone σ , adding the cones and endowing the ray R ≥0 (v + w) with the choice of generator v + w.Geometrically, f σ is the blow-up of X ( ) at the point, corresponding to σ ; this is a point of the locus when σ is a smooth cone whose rays come with integer multiple 1, a μ p -point when σ is not smooth, a point on a μ p -divisor when σ is smooth and one of its rays comes with multiple p.
By the classical factorization of birational projective morphisms of smooth surfaces as compositions of blow-ups, for any T -equivariant birational projective morphism we obtain from by performing a finite sequence of stacky star subdivisions, and f is the composite of the corresponding T -equivariant birational projective morphisms.
Inspired by [2, Example 4.2] and [7, Section 4], we might ask: For a given pair of toric orbifold surfaces X ( ) and X ( ), does there exist a toric orbifold surface X ( ) with T -equivariant birational projective morphisms When the answer is affirmative, we say that X ( ) and X ( ) are equivalent under T -equivariant birational projective morphisms.
We record the observation, that in the toric setting the abelian group 2 admits a refinement.

Proposition 2
The assignment, to X ( of the class 2 , where in the sum we have v = (v 1 , v 2 ), w = (w 1 , w 2 ) ∈ N , and i ∈ {1, 2} is chosen so that p w i , gives rise to an invariant of toric orbifold surfaces with μ p as only possible nontrivial stabilizer group, under torus-equivariant birational projective morphisms.The assignment is compatible with the homomorphism (3).
In particular, we obtain a nontrivial refinement for p = 7 of the trivial obstruction group mentioned in Sect. 4.
The isomorphism of Theorem 3 is compatible with that of [7,Lemma 5.3].
Funding Open access funding provided by University of Zurich.

Declarations
Conflict of interest The author has no conflicts of interest to declare that are relevant to the content of this article.
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