SYMBOLS AND EQUIVARIANT BIRATIONAL GEOMETRY IN SMALL DIMENSIONS

BRENDAN HASSETT, ANDREW KRESCH, AND YURI TSCHINKEL

Abstract. We discuss the equivariant Burnside group and related new invariants in equivariant birational geometry, with a special emphasis on applications in low dimensions.

Let G be a finite group. Suppose that G acts birationally and generically freely on a variety over k, an algebraically closed field of characteristic zero. After resolving indeterminacy and singularities we may assume that G acts regularly on a smooth projective variety X. Classifying such actions

X G,

up to birational conjugation, especially in cases where X −−→^{∼} ℙ^{n}, is a long-standing problem. This entails understanding realizations of G in the Cremona group BirAut(ℙ^{n}). There is an enormous literature on this subject; we will summarize some key results in Section 1.

The focus of this survey is a new approach to the analysis of actions via new invariants extracted from the fixed points and stabilizer loci of X. This new approach has its origin in the study of specializations of birational type. Suppose a smooth projective variety specializes to a reduced normal crossing divisor – we seek to gain information about the general fiber from combinatorial structures associated with the special fiber [30, 21]. That circle of ideas, in turn, was inspired by motivic integration – another instance of this philosophy [26]. Actions of finite groups yield formally similar stratifications: we have the open subset, on which the group acts freely, and the locus with nontrivial stabilizers. The combinatorics of the resulting stratification – along with the representations of the stabilizers of the strata on the normal directions – sheds light on the original group action. In many cases, we can extract additional information on the birational type of these strata and the action of the normalizer of the stabilizer. We explain these constructions in detail, with many examples of small dimensions, referring to the original papers [20, 23]. Our goal is to illustrate how the layers of this new formalism reveal various levels of structure among G-birational types. In particular, we apply these new obstructions to cyclic actions on cubic fourfolds, including rational ones, and produce examples of nonlinearizable actions.

Date: September 21, 2020.

Here is the roadmap of the paper: After briefly recalling several classical results, we discuss the case of finite abelian groups G. We introduce the group of equivariant birational types B_{n}(G), as a quotient of a ℤ-module of certain symbols by explicit defining relations and find a simpler presentation of these relations (Proposition 2.1). In Section 3 we explain, in numerous examples, how to compute the invariants on surfaces. In Section 4 we show that all symbols in B_{n}(G) are represented by smooth projective varieties with G-action. In Section 5 we introduce and study refined invariants of abelian actions, taking into account not only the representation on the tangent bundle to the fixed point strata, but also birational types of these strata. In Section 6 we exhibit cyclic actions on cubic fourfolds that are not equivariantly birational to linear actions; our main goal is to highlight the applications of the different invariants in representative examples. Finally, in Section 7 we consider nonabelian groups, define the equivariant Burnside group, which encodes new obstructions to equivariant rationality, and show how these obstructions work in a striking example, due to Iskovskikh [19]: distinguishing two birational actions of 𝔖_{2} × 𝔖_{3} on rational surfaces.

Acknowledgments: The first author was partially supported by NSF grant 1701659 and Simons Foundation award 546235. The second author was partially supported by the Swiss National Science Foundation. The third author was partially supported by NSF grant 2000099.

1. Brief history of previous work

These questions were considered by Bertini, Geiser, De Jonquières, Kantor, etc. over a century ago but continue to inspire new work:

• Manin [27], [28] studied G-surfaces both in the arithmetic and geometric context, focusing on the induced G-action on the geometric Picard group, and on cohomological invariants of that lattice;

• Iskovskikh [18] laid the groundwork for the G-birational classification of surfaces and their linkings;

• Bayle, Beauville, Blanc, and de Fernex [4, 5, 13, 9, 10] classified actions of finite abelian G on surfaces;

• Dolgachev and Iskovskikh [14] largely completed the surface case; • Bogomolov and Prokhorov [11, 34] considered the stable conjugacy problem for the surface case using cohomological tools introduced by Manin;

• Prokhorov, Cheltsov, Shramov, and collaborators proved numerous theorems for threefolds – both concerning specific groups, such as simple groups [33, 12], as well as general structural properties [35].

Much of this work fits into the Minimal Model Program, using distinguished models to reduce the classification problem to an analysis of automorphisms of a restricted class of objects, e.g., del Pezzo surfaces. With a few exceptions – the application of the cohomology on the Néron-Severi group, by Manin and Bogomolov–Prokhorov, and the ‘normalized fixed curve with action (NFCA)’ invariant of Blanc [10] – invariants play a limited role.

A fundamental observation, recorded in [36, App. A], is that the presence of a point fixed by a given abelian subgroup H of G is a birational invariant of a smooth projective variety X with generically free G-action. Furthermore, Reichstein and Youssin showed that the determinant of the action of abelian stabilizers on the normal bundle to the locus with the given stabilizer, up to sign, is also a birational invariant [37]. However, for finite groups this is only useful when the minimal number of generators of the abelian group equals the dimension of the variety [36, Th. 1.1]. For cyclic groups, it is applicable only for curves.

The invariants defined in [20, 22, 23] record all eigenvalues for the action of abelian stabilizers, as well as additional information about the action on the components of the fixed locus, and on their function fields. These collections of data are turned into a G-birational invariant, via explicit blowup relations. The groups receiving these invariants, the equivariant Burnside groups, have an elaborate algebraic structure. And they led to new results in birational geometry, some of which will be discussed below.

2. Equivariant birational types

Here we restrict to the situation where G is abelian and consider only fixed points of X G. In general, there are no such fixed points and we obtain no information. However, large classes of actions do have fixed points, e.g., if G is cyclic and h^{i}(X, O_{X}) = 0, for each i > 0, then the Atiyah-Bott holomorphic Lefschetz formula [2, Cor. 4.13] yields a fixed point. The vanishing assumption holds for rational and rationally connected X. If G is an abelian p-group (p prime) acting on X without fixed points then every Chern number of X is divisible by p [17, Cor. 1.1.2].

To define an invariant of X G, we consider collections of weights for the action of G in the tangent bundle at G-fixed points in X. To formalize this, let

A = G^{∨} = Hom(G, 𝔾_{m})

be the character group of G, and n = dim X. Let

S_{n}(G)

be the free abelian group on symbols

[a_{1}, …, a_{n}], a_{j} ∈ A, ∀j,

subject to the conditions:

(G) Generation: {a_{1}, …, a_{n}} generate A, i.e.,

∑_{i=1}^{n} ℤa_{i} = A,

thus, n is at least the minimal number of generators of G;

(S) Symmetry: for each permutation σ ∈ 𝔖_{n} we have

[a_{σ(1)}, …, a_{σ(n)}] = [a_{1}, …, a_{n}].

Let

S_{n}(G) → B_{n}(G)  (2.1)

be the quotient, by relations, for all 2 ≤ r ≤ n:

(B_{r}) Blow-up: for all [a_{1}, …, a_{r}, b_{1}, …, b_{n−r}] ∈ S_{n}(G) one has

[a_{1}, …, a_{r}, b_{1}, …, b_{n−r}] =
∑_{1≤i≤r, a_{i}≠a_{i′} for i′<i}[a_{1} − a_{i}, …, a_{i}, …, a_{r} − a_{i},b_{1}, …, b_{n−r}].  (2.2)

These relations reflect the transformations of weights in tangent spaces to components of the fixed locus upon blowing up along a G-stable stratum.

From the definition, we have

B_{1}(G) = \begin{cases}ℤ^{ϕ(N)} & if G is cyclic of order N, \\ 0 & otherwise. \end{cases}

Proposition 2.1. [23, Prop. 8.1] For n ≥ 2, all relations (B_{r}) are implied by relation (B_{2}).

Thus, B_{n}(G) is obtained by imposing the relation:

(B) Blow-up: for all [a_{1}, a_{2}, b_{1}, …, b_{n−2}] ∈ S_{n}(G) one has

[a_{1}, a_{2}, b_{1}, …, b_{n−2}] =
[a_{1}, a_{2} − a_{1}, b_{1}, …, b_{n−2}] + [a_{1} − a_{2}, a_{2}, b_{1}, …, b_{n−2}], a_{1} ≠ a_{2}, (2.3)
[0, a_{1}, b_{1}, …, b_{n−2}],  a_{1} = a_{2}.

Proof of Proposition 2.1. We prove the result by induction on r. We first treat the case that a_{1}, a_{2}, …, a_{r} are pairwise distinct; we drop the entries b_{1}, …, b_{n−r} from the notation, as they do not take part in relations. Suppose r ≥ 3. Then:

[a_{1}, …, a_{r}] = (by (B_{r−1})) [a_{1}, a_{2} − a_{1}, …, a_{r−1} − a_{1}, a_{r}] + ⋯

+ [a_{1} − a_{r−1}, …, a_{r−2} − a_{r−1}, a_{r−1}, a_{r}]

= (by (S)) [a_{1}, a_{r}, a_{2} − a_{1}, …, a_{r−1} − a_{1}] + ⋯ + [a_{r−1}, a_{r}, a_{1} − a_{r−1}, …, a_{r−2} − a_{r−1}]

= (by (B_{2}) + (S), applied to each term) [a_{1}, a_{2} − a_{1}, …, a_{r} − a_{1}] + [a_{1} − a_{r}, a_{2} − a_{1}, …, a_{r−1} − a_{1}, a_{r}]

+ ⋯ +
[a_{1} − a_{r−1}, …, a_{r−2} − a_{r−1}, a_{r−1}, a_{r} − a_{r−1}]
+ [a_{1} − a_{r−1}, …, a_{r−2} − a_{r−1}, a_{r−1} − a_{r}, a_{r}].

The right-hand terms, taken together, are equal by (B_{r−1}) to

[a_{1} − a_{r}, …, a_{r−1} − a_{r}, a_{r}],

which together with the left-hand terms gives us what we want.

Next we treat the more general case a_{r} ∉ {a_{1}, …, a_{r−1}}. In that case, for every i with a_{i} ∈ {a_{1}, …, a_{i−1}} we omit the ith term on the right-hand side in the initial application of (B_{r−1}) and (S) and omit the ith line after the applications of (B_{2}) and (S). We conclude as before.

Finally we treat the case a_{r} ∈ {a_{1}, …, a_{r−1}}. We start in the same way, by applying (B_{r−1}) and (S) as above. Now, when we apply (B_{2}), we have to pay special attention to terms with a_{i} = a_{r}: in the corresponding line we should leave the left-hand term but omit the righthand term. Each of the remaining right-hand terms vanishes by an application of (B_{2}) to a symbol of the form [0, …], i.e., the vanishing of any symbol with two nonzero weights summing to 0. The left-hand terms give us directly what we want. □

2.1. Antisymmetry. Let

B_{n}(G) → B_{n}^{−}(G)

be the projection to the quotient by the additional relation

[−a_{1}, …, a_{n}] = −[a_{1}, …, a_{n}].  (2.4)

In particular, symbols of the type [0, a_{2}, …, a_{n}] are in the kernel of the projection. We write

[a_{1}, …, a_{n}]^{−}

for the image of a standard generator [a_{1}, …, a_{n}].

2.2. Multiplication and co-multiplication. Consider a short exact sequence

0 → G′ → G → G′′ → 0

and its dual

0 → A′′ → A → A′ → 0.

The multiplication

∇ : B_{n′}(G′) ⊗ B_{n′′}(G′′) → B_{n}(G), n′ + n′′ = n,

is defined by

[a′_{1}, …, a′_{n′}] ⊗ [a′_{1}, …, a′_{n′}] ↦ ∑ [a_{1}, …, a_{n′}, a′′_{1}, … a′′_{n′′}],

summing over all lifts a_{i} ∈ A of a′_{i} ∈ A′. It descends to a similar map on quotients by the relation (2.4). .

The co-multiplication is defined only on B_{n}^{−}(G):

Δ^{−} : B_{n}^{−}(G) → B_{n′}^{−}(G′) ⊗ B_{n′′}^{−}(G′′), n′ + n′′ = n.

On generators is takes the form

[a_{1}, …, a_{n}]^{−} ↦ ∑ [a_{I′} mod A′′]^{−} ⊗ [a_{I′′}]^{−},

where the sum is over subdivisions {1, …, n} = I′ ⊔ I′′ of cardinality n′, respectively n′′, such that

• a_{j} ∈ A′′, for all j ∈ I′′

• a_{j}, j ∈ I′′, generate A′′.

The correctness of this definition is proved as in [20, Prop. 11]. Here are the main steps: By [23, Prop. 8.1] (= Proposition 2.1), it suffices to check 2-term relations (B_{2}), i.e., the image of the relation

[a_{1}, a_{2}, …]^{−} = [a_{1} − a_{2}, a_{2}, …]^{−} + [a_{1}, a_{2} − a_{1}, …]^{−}

after applying co-multiplication. The only interesting part is when the first two arguments are distributed over different factors in the definition of co-multiplication.

The relation is the same relation as that for the M_{n}(G)-groups, introduced and studied in [20], unless, a_{1} = a_{2} – recall that

[a, a, …] = [a, 0, …,] ∈ B_{n}(G).

Since

0 = [a, −a, …]^{−} = −[a, a, …]^{−}

we have

[a, a, …]^{−} = [a, 0, …]^{−} = 0.

Now it suffices to repeat the argument in [20, Prop. 11]. Using the terminology of that paper, there are four cases, of which only (1) and (4) are relevant. In both cases, all terms are zero.

We return to the definition of an invariant for X G, dim(X) = n, and G abelian. Consider irreducible components of the fixed locus

X^{G} = ∐_{α} F_{α},

and write

β_{α} = [a_{1,α}, …, a_{n,α}]

for the unordered n-tuple of weights for the G-action on the tangent space T_{x_{α}}X, for some x_{α} ∈ F_{α} – this does not depend on the choice of x_{α}. The number of zero weights is dim(F_{α}). We express

β(X G) = ∑_{α} β_{α} ∈ B_{n}(G),  (2.5)

and write

β^{−}(X G) ∈ B_{n}^{−}(G),

for the image under the projection.

Theorem 2.2. [20, Th. 3] The class

β(X G) ∈ B_{n}(G)

is a G-birational invariant.

The proof relies on G-equivariant Weak Factorization, connecting Gbirational varieties via blow-ups and blow-downs of smooth G-stable subvarieties.

Proposition 2.3. Consider a linear, generically free, action of a cyclic group C_{N}, of order N, on ℙ^{n}, for n ≥ 2. Then

β^{−}(ℙ^{n} C_{N}) = 0 ∈ B_{n}^{−}(C_{N}).

Proof. We know that all such actions are equivariantly birational, see, e.g., [37, Th. 7.1]. Thus it suffices to consider one such action. Take an action with weights (1, 0, …, 0). It fixes a hyperplane, and a point, the corresponding class is

[1, 0, …, 0] + [−1, −1, …, −1] = [1, 0, …, 0] + [−1, 0, …, 0],

here, we repeatedly used relation (2.3) to transform the second term. □

Remark 2.4. We shall show in Section 3.3 that [1, 0] + [−1, 0] is torsion in B_{2}(C_{N}). See [20, Prop. 7 and Lem. 32] for the case of prime N. The element is nontrivial when

N = 7, 9, 10, 11, 13, 14, 15, 17, ….

3. Computation of invariants on surfaces

3.1. Sample computations of B_{2}(C_{p}). This group is generated by symbols [a_{1}, a_{2}], where

a_{1}, a_{2} ∈ Hom(C_{p}, 𝔾_{m}) ≅ ℤ/pℤ

are not both trivial. To simplify notation, we write a_{i} = 0, 1, …, p − 1. Note that

[a, a] = [0, a] and [a_{1}, a_{2}] = [a_{2}, a_{1}]

so that the symbols

[a_{1}, a_{2}] : 0 < a_{1} ≤ a_{2} < p

generate B_{2}(C_{p}). The other relation – for a_{1} < a_{2} – takes the form

[a_{1}, a_{2}] = [a_{1}, a_{2} − a_{1}] + [p + a_{1} − a_{2}, a_{2}].

(p = 2) We obtain relations

[1, 1] = [0, 1], [0, 1] = [0, 1] + [1, 1]
forcing B_{2}(C_{2}) = 0.

(p = 3) We obtain relations

[0, 1] = [0, 1] + [2, 1] = [0, 1] + [1, 2]
[0, 2] = [0, 2] + [1, 2]
[1, 1] = [0, 1]
[1, 2] = [1, 1] + [2, 2]
[2, 2] = [0, 2]

forcing

[1, 2] = 0 and [0, 1] = [1, 1] = −[2, 2] = −[0, 2].

We conclude that B_{2}(C_{3}) ≅ ℤ. For instance, consider the standard diagonal action on ℙ^{2}

(x : y : z) ↦ (x : ζ_{3}y : ζ_{3}^{2}z)

with the convention that ζ_{N} is a primitive Nth root of unity. This has invariant

β(ℙ^{2} C_{3}) = [1, 2] + [1, 2] + [1, 2] = 0.

On the other hand, the action of C_{3} on the cubic surface

X = {w^{3} = f_{3}(x, y, z)}, (w : x : y : z) ↦ (ζ_{3}w : x : y : z)

fixes the cubic curve {w = 0} and we find that

β(X C_{3}) = [2, 0] ≠ 0,

thus X is not G-equivariantly birational to ℙ^{2}, with linear action. Note that β does not allow to distinguish among these cubic surfaces. Nor does it distinguish the cubic surfaces from the degree one del Pezzo surfaces with C_{3}-action

Y = {w^{2} = z^{3} + f_{6}(x, y)} ⊂ ℙ(3, 1, 1, 2),

given by

(w : x : y : z) ↦ (w : x : y : ζ_{3}z).

We shall see in Section 5 that taking into account the fixed locus [F_{α}] gives a complete invariant.

(p = 5) We have relations

[0, 1] = [0, 1] + [4, 1]
[0, 2] = [0, 2] + [3, 2]
[0, 3] = [0, 3] + [2, 3]
[0, 4] = [0, 4] + [1, 4]

forcing [1, 4] = [2, 3] = 0. We also have

[1, 2] = [1, 1] + [4, 2]
[1, 3] = [1, 2] + [3, 3]
[1, 4] = [1, 3] + [2, 4]
[2, 3] = [2, 1] + [4, 3]
[2, 4] = [2, 2] + [3, 4]
[3, 4] = [3, 1] + [4, 4]

which shows that B_{2}(C_{5}) is freely generated by

β_{1} = [1, 1] and β_{2} = [1, 2]

with

[1, 3] = β_{1} − β_{2}, [2, 2] = 2β_{2} − β_{1}, [2, 4] = β_{2} − β_{1},

[3, 3] = β_{1} − 2β_{2}, [3, 4] = −β_{2}, [4, 4] = −β_{1}.

For example, \overline{M}_{0,5}, a del Pezzo surface of degree 5, has a natural action of C_{5} by permuting the coordinates with fixed points given by the roots of z^{5} − 1 and z^{5} + 1. We compute β(\overline{M}_{0,5} C_{5}):

β_{1}+β_{2}+(β_{1}−β_{2})+(2β_{2}−β_{1})+(β_{2}−β_{1})+(β_{1}−2β_{2})+(−β_{2})+(−β_{1}) = 0.

Indeed, this action is in fact conjugate [5] to a linear action on ℙ^{2}

(x : y : z) ↦ (x : ζ_{5}y : ζ_{5}^{4}z).

However, there is also a nontrivial action of C_{5} on a del Pezzo surface of degree 1:

X = {w^{2} = z^{3} + λx^{4}z + x(μx^{5} + y^{5})} ⊂ ℙ(3, 1, 1, 2),
(w : x : y : z) ↦ (w : x : ζ_{5}y : z),

with fixed locus an elliptic curve and invariant β(X C_{5}) = [4, 0].

Let us compute an example of nonprime order. The group B_{2}(C_{4}) has generators

[0, 1], [0, 3], [1, 1], [1, 2], [1, 3], [2, 3], [3, 3]

and relations

[0, 1] = [0, 1] + [3, 1]
[0, 3] = [0, 3] + [1, 3]
[1, 1] = [0, 1]
[1, 2] = [1, 1] + [3, 2]
[1, 3] = [1, 2] + [2, 3]
[2, 3] = [2, 1] + [3, 3]
[3, 3] = [0, 3]

whence

[1, 3] = 0, β_{1} := [1, 2] = −[2, 3], [0, 3] = [3, 3] = 2[2, 3] = −2β_{1},

[0, 1] = [1, 1] = 2[1, 2] = 2β_{1}

and B_{2}(C_{4}) ≅ ℤ.

Consider the del Pezzo surface of degree 1, given by

X = {w^{2} = z^{3} + zL_{2}(x^{2}, y^{2}) + xyM_{2}(x^{2}, y^{2})} ⊂ ℙ(3, 1, 1, 2),

where L_{2} and M_{2} are homogeneous of degree two. It admits a C_{4}-action

(w : x : y : z) ↦ (iw : x : −y : −z)

with a unique fixed point (0 : 1 : 0 : 0). The weights on the tangent bundle are [2, 3] whence

β(X C_{4}) ≠ 0.

Observe that X^{C_{2}} is a curve of genus four.

See [8, §10.1] for a classification of automorphisms of large finite order N on del Pezzo surfaces:

(1) The surface

X = {w^{2} = z^{3} + x(x^{5} + y^{5})} ⊂ ℙ(3, 1, 1, 2)

admits an automorphism of order 30

(w : x : y : z) ↦ (−w : x : ζ_{5}y : ζ_{3}z)

with fixed point (0 : 0 : 1 : 0) and with weights [3, 2], thus

β(X C_{30}) = [3, 2] ≠ 0 ∈ B_{2}(C_{30}) ⊗ ℚ,

by a computation in Sage [38]. This implies (see Remark 2.4) that this action is not conjugate to a linear action. Note that dim B_{2}(C_{30}) ⊗ ℚ = 33.

(2) The surface

X = {w^{2} = z^{3} + xy(x^{4} + y^{4})} ⊂ ℙ(3, 1, 1, 2)

admits an automorphism of order 24

(w : x : y : z) ↦ (ζ_{8}w : x : iy : −iζ_{3}z).

The fixed point is (0 : 1 : 0 : 0) with symbol [21, 22]. Computing via Sage we find

β(X C_{24}) ≠ 0 ∈ B_{2}(C_{24}) ⊗ ℚ.

Here dim B_{2}(C_{24}) ⊗ ℚ = 23.

There are good reasons why we obtain nonvanishing invariants only when a curve is fixed: If G = C_{N} is a cyclic group acting generically freely on a complex projective smooth rational surface X G then the following are equivalent [8, Th. 4]:

• No g ≠ 1 ∈ G fixes a curve in X of positive genus.

• The subgroup G is conjugate to a subgroup of Aut(ℙ^{2}). Even more is true: if G contains an element fixing a curve of positive genus then X is not even stably G-birational to projective space with a linear G-action, indeed, in this case H^{1}(G, Pic(X)) ≠ 0 [11].

3.2. Examples for noncyclic groups. If G is a noncyclic abelian group then B_{1}(G) = 0 by definition but there are actions on curves:

Example 3.1. Consider the action of C_{2} × C_{2} on ℙ^{1} by

g_{1} := (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) , g_{2} := (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) ,

with the elements

g_{1}^{2} = g_{1}g_{2}g_{1}^{−1}g_{2}^{−1} = (\begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix})

acting trivially. Thus we obtain

ℙ^{1} (C_{2} × C_{2}).

The group has no fixed points whence β(ℙ^{1} (C_{2} × C_{2})) = 0. The cyclic subgroups do have fixed points

(ℙ^{1})^{⟨g_{1}⟩} = {(1 : ±i)}, (ℙ^{1})^{⟨g_{2}⟩} = {(1 : 0), (0 : 1)}, (ℙ^{1})^{⟨g_{1}g_{2}⟩} = {(1 : ±1)}.

We return to this in Example 5.2. In Section 5.2, we will discuss how to incorporate information from all points with nontrivial stabilizer.

We compute B_{2}(C_{2} × C_{2}). Writing

(C_{2} × C_{2})^{∨} = {0, χ_{1}, χ_{2}, χ_{1} + χ_{2}},

the only admissible symbols are

[χ_{1}, χ_{2}], [χ_{1}, χ_{1} + χ_{2}], [χ_{1} + χ_{2}, χ_{2}]

with relations:

[χ_{1}, χ_{2}] = [χ_{1}, χ_{1} + χ_{2}] + [χ_{1} + χ_{2}, χ_{2}]
[χ_{1}, χ_{1} + χ_{2}] = [χ_{1}, χ_{2}] + [χ_{1} + χ_{2}, χ_{2}]
[χ_{1} + χ_{2}, χ_{2}] = [χ_{1}, χ_{1} + χ_{2}] + [χ_{1}, χ_{2}].

Thus we obtain the Klein four group again

B_{2}(C_{2} × C_{2}) = C_{2} × C_{2}.

The classification of finite abelian noncyclic actions on rational surfaces may be found in [8, §10.2]. Examples of actions of C_{2} × C_{2} on rational surfaces include:

On ℙ^{1} × ℙ^{1}:

(1) (x, y) ↦ (±x^{±1}, y), without fixed points;

(2) (x, y) ↦ (±x, ±y), with fixed points (0, 0), (0, ∞), (∞, 0), (∞, ∞), thus

β(ℙ^{1} × ℙ^{1} C_{2} × C_{2}) = 4[χ_{1}, χ_{2}] = 0;

(3) the diagonal action

(x, y) ↦ (−x, −y), (x, y) ↦ (x^{−1}, y^{−1})

has no fixed points so the symbol sum is empty;

On conic fibrations over ℙ^{1}:

(4) (x_{1} : x_{2}) × (y_{1} : y_{2}) ↦

(x_{1} : −x_{2}) × (y_{1} : y_{2}), respectively,

(x_{1} : x_{2}) × (y_{2}(x_{1} − bx_{2})(x_{1} + bx_{2}) : y_{1}(x_{1} − ax_{2})(x_{1} + ax_{2}),

which also has four fixed points with the same symbol;

On a degree two del Pezzo surface:

(5) (w : x : y : z) ↦ (±w : x : y : ±z) on

{w^{2} = L_{4}(x, y) + z^{2}L_{2}(x, y) + z^{4}},

with the involutions fixing a genus three curve and an elliptic curve meeting in four points whence β(X C_{2} × C_{2}) = 0;

On a degree one del Pezzo surface:

(6) (w : x : y : z) ↦ (±w : x : ±y : z) on

{w^{2} = z^{3} + zL_{2}(x^{2}, y^{2}) + L_{3}(x^{2}, y^{2})},

with the involutions fixing a genus four curve and a genus two curve meeting in six points whence β(X C_{2} × C_{2}) = 0.

None of these actions are distinguished by B_{2}(C_{2} × C_{2}). Case (1) is stably equivalent to the action on ℙ^{1} described above. The second action is equivalent to a linear action on ℙ^{2} – project from one of the fixed points. We return to these examples in Section 5.4.

3.3. Linear actions yield torsion classes. Let C_{N} act linearly and generically freely on ℙ^{n}. We saw in Proposition 2.3 that

β(ℙ^{n} C_{N}) = [a, 0, …, 0] + [−a, 0, …, 0]

for some a relatively prime to N. Remark 2.4 pointed out this is torsion for n ≥ 2; we offer a proof now:

Proposition 3.2. For N ∈ ℕ, an a with gcd(a, N) = 1, and n ≥ 2 the element

[a, 0, …, 0] + [−a, 0, …, 0] ∈ B_{n}(C_{N})

is torsion.

Proof. It suffices to consider n = 2; the argument we present goes through without changes for n > 2.

We will work in B_{2}(C_{N})⊗ℚ. We use generators for this space arising from the alternative symbol formalism M_{2}(C_{N}) introduced in [20] with the property that [20, Prop. 7]

B_{2}(C_{N}) ⊗ ℚ = M_{2}(C_{N}) ⊗ ℚ.

For a, b ∈ Hom(C_{N}, 𝔾_{m}) generating the group we set

⟨a, b⟩ = \begin{cases}[a, b] & if a, b ≠ 0 \\ \frac{1}{2}[a, 0] & if a ≠ 0, b = 0. \end{cases}

The advantage of these generators is that the relations are uniformly

⟨a, b⟩ = ⟨a, b − a⟩ + ⟨a − b, b⟩ (B),

even when a = b.

We follow the proof of [20, Th. 14]. For all a, b with gcd(a, b, N) = 1 we write

δ(a, b) := ⟨a, b⟩ + ⟨−a, b⟩ + ⟨a, −b⟩ + ⟨−a, −b⟩ .

We claim this is zero in B_{2}(C_{N}) ⊗ ℚ.

First, we check that δ(a, b) is invariant under SL_{2}(ℤ/Nℤ). This has generators

(\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) , (\begin{matrix}1 & −1 \\ 0 & 1 \end{matrix}) .

and δ(a, b) = δ(−b, a) by the symmetry of the underlying symbols. We also have

δ(a, b − a) = ⟨a, b − a⟩ + ⟨−a, b − a⟩ + ⟨a, a − b⟩ + ⟨−a, a − b⟩

applying B to second and third terms above

= ⟨a, b − a⟩ + ⟨−a, b⟩ + ⟨−b, b − a⟩

+ ⟨a, −b⟩ + ⟨b, a − b⟩ + ⟨−a, a − b⟩ applying B to get first and four terms below

= ⟨a, b⟩ + ⟨−a, b⟩ + ⟨a, −b⟩ + ⟨−a, −b⟩
=δ(a, b).

Average δ(a, b) over all pairs a, b with gcd(a, b, N) = 1 to obtain

S := ∑_{a,b} δ(a, b) = 4 ∑_{a,b} ⟨a, b⟩ .

Applying the blowup relation (B) to all terms one finds

S = 2S,

which implies that S = 0 ∈ B_{2}(C_{N}) ⊗ ℚ.

We may regard δ(a, b) and S as elements of B_{2}(C_{N}). It follows that δ(a, b) is torsion in B_{2}(C_{N}), annihilated by the number of summands in S. Substituting b = 0, we obtain that

δ(a, 0) = [a, 0] + [−a, 0] = 0 ∈ B_{2}(C_{N}) ⊗ ℚ.
□

The invariance of δ(a, 0) shows that [a, 0] + [−a, 0] is independent of the choice of a relatively prime to N.

3.4. Algebraic structure in dimension 2. For reference, we tabulate

dim B_{2}(G) ⊗ ℚ,

for G = C_{N} and small values of N:

N 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 0 1 1 2 2 3 3 5 4 6 7 8 7 13 10

For primes p ≥ 5 there is a closed formula [20, §11]:

dim B_{2}(C_{p}) ⊗ ℚ = \frac{p^{2} − 1}{24} + 1 = \frac{(p − 5)(p − 7)}{24} + \frac{p − 1}{2}, (3.1)

which strongly suggested a connection with the modular curve X_{1}(p)! We also have

dim B_{2}^{−}(C_{p}) ⊗ ℚ = \frac{(p − 5)(p − 7)}{24},  (3.2)

and, by [20, Prop. 30],

B_{1}^{−}(C_{p}) ⊗ ℚ = Ker(B_{2}(C_{p}) → B_{2}^{−}(C_{p})) ⊗ ℚ.

Computations in noncyclic cases have been performed by Zhijia Zhang^{1}; we summarize the results: for primes p ≥ 5 one has

dim B_{2}(C_{p} × C_{p}) ⊗ ℚ = \frac{(p − 1)(p^{3} + 6p^{2} − p + 6)}{24},

dim B_{2}^{−}(C_{p} × C_{p}) ⊗ ℚ = \frac{(p − 1)(p^{3} − p + 12)}{24}

For G = C_{N_{1}} × C_{N_{2}} and small values of N_{1}, N_{2}, we have:

\begin{matrix}N_{1} & 2 & 2 & 2 & 2 & 2 & 2 & 3 & 3 & 3 & 3 & 4 & 4 & 4 & 5 & 6 \\ N_{2} & 2 & 4 & 6 & 8 & 10 & 16 & 6 & 3 & 9 & 27 & 8 & 16 & 32 & 25 & 36 \\ d_{ℚ} & 0 & 2 & 3 & 6 & 7 & 21 & 15 & 7 & 37 & 235 & 33 & 105 & 353 & 702 & 577 \\ d_{ℚ}^{−} & 0 & 0 & 0 & 1 & 1 & 9 & 7 & 3 & 19 & 163 & 17 & 65 & 257 & 502 & 433 \\ d_{2} & 2 & 5 & 8 & 13 & 18 & 36 & 15 & 7 & 37 & 235 & 34 & 106 & 354 & 702 & 578 \\ d_{2}^{−} & 2 & 3 & 5 & 8 & 12 & 24 & 7 & 3 & 19 & 163 & 17 & 65 & 257 & 502 & 433 \end{matrix}

4. Reconstruction theorem

The examples offered so far might suggest that very few invariants in B_{n}(G) are actually realized geometrically by smooth projective varieties X G. If one allows nonrational examples far more invariants arise:

Proposition 4.1. Let p be a prime. Then B_{n}(C_{p}) is generated as an abelian group by β(X C_{p}), where X is smooth and projective.

Proof. We proceed by induction on n. The case of n = 1 follows from the Riemann existence theorem applied to cyclic branched covers of degree p with the prescribed ramification data. (See also Lemma 4.2 below.)

For the symbols [a_{1}, …, a_{n−1}, 0] we construct (n − 1)-dimensional varieties D with the prescribed invariants and D×ℙ^{1} with trivial action on the second factor. Since [a, a, a_{3}, …, a_{n}] = [0, a, a_{3}, …, a_{n}] we may focus on symbols [a_{1}, a_{2}, …, a_{n}], 0 < a_{1} < a_{2} < … < a_{n} < p. We are reduced to the following lemma:

1see https://cims.nyu.edu/~zz1753/ebgms/ebgms.html

Lemma 4.2. Any sum

∑ m_{[a_{1},a_{2},...,a_{n}]}[a_{1}, a_{2}, …, a_{n}]  (4.1)

of symbols

[a_{1}, a_{2}, …, a_{n}], 0 < a_{1} < a_{2} < … < a_{n} < p,

with nonnegative coefficients, is realized as β(X C_{p}), where X is smooth, projective, and irreducible.

For each symbol [a_{1}, a_{2}, …, a_{n}] appearing in the sum, take an ndimensional representation V_{[a_{1},a_{2},...,a_{n}]} with the prescribed weights and the direct sum

W_{[a_{1},a_{2},...,a_{n}]} = (V_{[a_{1},a_{2},...,a_{n}]} ⊕ ℂ)^{m_{[a_{1},a_{2},...,an]}}

where ℂ is the trivial representation of C_{p}. Write

W = ⊕W_{[a_{1},a_{2},...,a_{n}]},

where the index is over the terms appearing in (4.1), and consider the projectivization ℙ(W) and the n-planes

P_{[a_{1},a_{2},...,a_{n}],j} ⊂ ℙ(W_{[a_{1},a_{2},...,a_{n}]}), j = 1, …, m_{[a_{1},a_{2},...,a_{n}]}

associated with the summands of W_{[a_{1},a_{2},...,a_{n}]}, each with distinguished fixed point

p_{[a_{1},a_{2},...,a_{n}],j} = (0 : 0 : ⋯: 0 : 1).

The action on

T_{p_{[a_{1},a_{2},...,an]}}P_{[a_{1},a_{2},...,a_{n}],j}

coincides with the action on V_{[a_{1},a_{2},...,a_{n}]}. The fixed points of ℙ(W) correspond to the eigenspaces for the C_{p} action. Write M = ∑ m_{[a_{1},a_{2},...,a_{n}]} for the number of summands; each weight occurs at most M times. Thus the fixed point loci are projective spaces of dimension ≤ M − 1.

Choose a high-degree smooth complete intersection X of dimension n, invariant under the action of C_{p}, containing the p_{[a_{1},a_{2},...,a_{n}],j} and tangent to P_{[a_{1},a_{2},...,a_{n}],j} for each [a_{1}, a_{2}, …, a_{n}]. This complete intersection exists by polynomial interpolation applied to the quotient ℙ(W)/C_{p}; smoothness follows from Bertini’s Theorem. Since complete intersections of positive dimension are connected, the resulting X is irreducible.

It only remains to show that such a complete intersection need not have fixed points beyond those specified. Now X has codimension (M − 1)(n + 1), so we may assume it avoids the fixed point loci – away from the stipulated points p_{[a_{1},s_{2},...,a_{n}],j} – provided (M −1)(n+1) ≥ M. It only remains to consider the special case M = 1. Here we take

W = (V_{[a_{1},a_{2},...,a_{n}]} ⊕ ℂ)^{2},

imposing conditions at just one point (0 : 0 : ⋯: 0 : 1). Here X ⊂ ℙ(W) has codimension n + 1 and the fixed point loci are ℙ^{1}’s, so we may avoid extraneous points of intersection. □

5. Refined invariants

5.1. Encoding fixed points. Since B_{2}(C_{2}) = 0 this invariant says nothing about involutions of surfaces! Bertini, Geiser, and De Jonquières involutions are perhaps the most intricate parts of the classification, which relies on the study of fixed curves. This leads to a natural refinement of the invariants: For the symbols of type [a, 0], corresponding to curves F_{α} ⊂ X fixed by C_{N}, we keep track of the (stable) birational equivalence class of F_{α} and the element of B_{1}(C_{N}) associated with [a].

In general, [20] introduced a group combining the purely numbertheoretic information encoded in B_{n}(G) with geometric information encoded in the Burnside group of fields from [21]. Let

Bir_{n−1,m}(k), 0 ≤ m ≤ n − 1,

be the set of k-birational equivalence classes of (n − 1)-dimensional irreducible varieties over k, which are k-birational to products W ×𝔸^{m}, but not to W′ × 𝔸^{m+1}, for any W′, and put

B_{n}(G, k) := ⊕_{m=0}^{n−1} ⊕_{[Y]∈Bir_{n−1,m}(k)} B_{m+1}(G).  (5.1)

Let X be a smooth projective variety of dimension n with a regular, generically free, action of an abelian group G. Put

β_{k}(X G) := ∑_{α} β_{k,α},

where, as before, the sum is over components F_{α} ⊂ X^{G} of the G-fixed point locus, but in addition to the eigenvalues a_{1}, …, a_{n−dim(F_{α})} ∈ A in the tangent space T_{x_{α}}X one keeps information about the function field of the component F_{α}. Choosing m_{α} so that

[F_{α} × 𝔸^{n−1−dim(F_{α})}] ∈ Bir_{n−1,m_{α}}(k)

we set

β_{k,α} := [a_{1}, …, a_{n−dim(F_{α})}, \underbrace{0, …, 0}_{m_{α}+1−n+dim(F_{α})}] ∈ B_{m_{α}+1}(G),

identified with the summand of (5.1) indexed by [F_{α} × 𝔸^{n−1−dim(F_{α})}]. When F_{α} is not uniruled we get a symbol in B_{codim(F_{α})}(G).

Theorem 5.1. [20, Remark 5] The class

β_{k}(X G) ∈ B_{n}(G, k)

is a G-birational invariant.

5.2. Encoding points with nontrivial stabilizer. We continue to assume that G is a finite abelian group, acting regularly and generically freely on a smooth variety X. Let H ⊂ G arise as the stabilizer of some point of X, F ⊂ X^{H} an irreducible component of the fixed point locus with generic stabilizer H, and Y the minimal G-invariant subvariety containing G. In Section 7.2, we will explain how to blow up X so that Y is always a disjoint union of translates of F.

Additional information about the action of G on X is contained in the action of the quotient G/H, which could act on the function field of F, or by translating F in X. The paper [23] introduced the group

Burn_{n}(G)

as the quotient by certain relations of the free abelian group generated by symbols

(H, G/H K, β),  (5.2)

where K is a G/H-Galois algebra over a field of transcendence degree d ≤ n over k, up to isomorphism, and β is a faithful (n−d)-dimensional representation of H (see [23, Def. 4.2] for a precise formulation of conditions on K and relations).

Passing to a suitable G-equivariant smooth projective model X – as discussed in Section 7.2 – its class is defined by

[X G] := ∑_{H⊆G} ∑_{Y}(H, G/H K(Y), β_{Y} (X)) ∈ Burn_{n}(G), (5.3)

where the sum is over all G-invariant strata Y ⊂ X with generic stabilizer H as above, the symbol records the eigenvalues of H in the tangent bundle to x ∈ Y as well as the G/H-action on the total ring of fractions K(Y).

Example 5.2. We revisit Example 3.1 using the notational conventions χ_{1}(g_{2}) = χ_{2}(g_{1}) = 1 and G = C_{2} × C_{2}. Here we have

[ℙ^{1} G] = (⟨1⟩ , G k(t)) + (⟨g_{1}⟩ , G/ ⟨g_{1}⟩ {(1 : ±i)}, χ_{1})
+ (⟨g_{2}⟩ , G/ ⟨g_{2}⟩ {(1 : 0), (0 : 1)}, χ_{2})
+ (⟨g_{1}g_{2}⟩ , G/ ⟨g_{1}g_{2}⟩ {(1 : ±1)}, χ_{1} + χ_{2}).

The action on K(ℙ^{1}) = k(t) is by g_{1}(t) = −t and g_{2}(t) = −1/t.

Blowup relations ensure that [X G] is a well-defined G-birational invariant – see Section 5.3 for more details.

There is a distinguished subgroup

Burn_{n}^{triv}(G) ⊂ Burn_{n}(G)

generated by symbols (1, G K = K(X)). For ‘bootstrapping’ purposes – where we seek invariants of n-folds in terms of lower-dimensional strata with nontrivial stabilizers – we may suppress these tautological symbols to get a quotient

Burn_{n}(G) → Burn_{n}^{nontriv}(G).

And there is also a natural quotient group

Burn_{n}(G) → Burn_{n}^{G}(G)

obtained by suppressing all symbols (H, G/H K, β) where H is a proper subgroup of G. By [23, Prop. 8.1 and Prop. 8.2], there are natural surjective homomorphisms

Burn_{n}^{G}(G) → B_{n}(G, k) → B_{n}(G).

Example 5.3. The group Burn_{1}^{nontriv}(G) is freely generated by nontrivial subgroups H ⊂ G and injective characters a : H ↪ 𝔾_{m}, e.g. Burn_{1}^{nontriv}(C_{N}) ≅ ℤ^{N−1} and Burn_{1}^{nontriv}(C_{2} × C_{2}) = ℤ^{3}.

5.3. Examples of blowup relations. We illustrate the relations for fixed points of cyclic actions on surfaces. More computations are presented in Section 5.4; the reader may refer to Section 4 of [23] for the general formalism. All the key ideas are manifest in the surface case because the full set of blowup relations follows from those in codimension two – see [23, Prop. 8.1] and the special case Prop. 2.1 above.

Suppose that G = C_{N} acts on the surface X with fixed point 𝔭 and weights a_{1} and a_{2} that generate A = Hom(C_{N}, 𝔾_{m}). Let X̃ denote the blowup of X at 𝔭 and E ≃ ℙ^{1} the exceptional divisor. Let \overline{H} = ker(a_{1} −a_{2}) ⊂ G denote the generic stabilizer of E which acts faithfully on the normal bundle N_{E/X̃} via ā_{1} = ā_{2} ∈ \overline{A} := Hom(\overline{H}, 𝔾_{m}).

Assume first that a_{1} and a_{2} are both prime to N, so 𝔭 is an isolated fixed point of X. The quotient G/\overline{H} (when nontrivial) acts faithfully on E = ℙ^{1} with fixed points 𝔭_{1} and 𝔭_{2}. If \overline{H} = G then a_{1} = a_{2} and we get the relation

(G, triv k = K(𝔭), (a_{1}, a_{1})) = (G, triv k(t) = K(E), (a_{1})).

If triv ⊊ H′ ⊊ G then

(G, triv k = K(𝔭), (a_{1}, a_{2})) = (\overline{H}, G/\overline{H} k(t) = K(E), ā_{1} = ā_{2})
+ (G, triv k = K(𝔭_{2}), (a_{1}, a_{2} − a_{1}))
+ (G, triv k = K(𝔭_{1}), (a_{2}, a_{1} − a_{2}))  (5.4)

where G/\overline{H} acts on t by a primitive dth root of unity with d = |G/\overline{H}|. If \overline{H} is trivial then

(G, triv k = K(𝔭),(a_{1}, a_{2})) = (G, triv k = K(𝔭_{2}), (a_{1}, a_{2} − a_{1}))
+ (G, triv k = K(𝔭_{1}), (a_{2}, a_{1} − a_{2})). (5.5)

Assume now that a_{1} = m_{1}n_{1} and a_{2} = m_{2}n_{2} where n_{1}, n_{2}|N (and are relatively prime modulo N) and m_{1} and m_{2} are prime to N and each other. Then we have

𝔭 ∈ F_{1} ∩ F_{2}

for irreducible curves F_{1} and F_{2} with generic stabilizers C_{n_{1}} and C_{n_{2}} respectively. Let F̃_{1}, F̃_{2} ⊂ X̃ denote the proper transforms of F_{1} and F_{2} and 𝔭_{1}, 𝔭_{2} ∈ E their intersections with the exceptional divisors. Thus the contribution to the strata containing 𝔭 is

(G, triv K(𝔭),(a_{1}, a_{2}))
+(C_{n_{1}}, C_{N}/C_{n_{1}} K(F_{1}), a_{2}) + (C_{n_{2}}, C_{N}/C_{n_{2}} K(F_{2}), a_{1})

with the latter two terms appearing in the symbol sum on X̃, with the F_{i} replaced by the F̃_{i}. Note that \overline{H} = ker(a_{1}−a_{2}) ⊊ G because a_{1} ≢ a_{2}. Here the blowup formula takes the form (5.4) or (5.5) depending on whether H′ is trivial or not.

Now suppose that a_{2} = 0. Let F ⊂ X denote the irreducible component of the fixed locus containing 𝔭, so that a_{1} is the character by which G acts on N_{F/X}. Write F̃ ⊂ X̃ for the proper transform of F, 𝔭_{1} = F̃ ∩ E, and 𝔭_{2} ∈ E for the other fixed point. Here we get the relation

(G, G K(F), a_{1}) = (G, G K(F), a_{1}) + (G, G K(𝔭_{2}), (a_{1}, −a_{1})),

whence the latter term vanishes.

5.4. Examples. We now complement the computations in Section 3, for G = C_{N}, and small N. As before, we work over an algebraicallyclosed based field k of characteristic zero.

(N = 2)

• B_{2}(C_{2}) = 0.

• B_{2}(C_{2}, k) ≅ Burn_{2}^{C_{2}}(C_{2}); has a copy of B_{1}(C_{2}) = ℤ, for every isomorphism class of curves of positive genus.

• Burn_{2}(C_{2}) = Burn_{2}^{triv}(C_{2}) ⊕ Burn_{2}^{C_{2}}(C_{2}).

(N = 3)

• B_{2}(C_{3}) ≅ ℤ, generated by [1, 1],

[1, 2] = 0, [2, 2] = −[1, 1].

• B_{2}(C_{3}, k) ≅ Burn_{2}^{C_{3}}(C_{3}), is a direct sum of ℤ, corresponding to points and rational curves, and a copy of B_{1}(C_{3}) = ℤ^{2} for every isomorphism class of curves of positive genus.

• Burn_{2}(C_{3}) = Burn_{2}^{triv}(C_{3}) ⊕ Burn_{2}^{C_{3}}(C_{3}).

(N = 4)

• B_{2}(C_{4}) ≅ ℤ, generated by [1, 2],

[1, 1] = 2[1, 2], [1, 3] = 0, [2, 3] = −[1, 2], [3, 3] = −2[1, 2].

• B_{2}(C_{4}, k) is a direct sum of ℤ, corresponding to points and rational curves, and a copy of B_{1}(C_{4}) = ℤ^{2} for every isomorphism class of curves of positive genus.

• Burn_{2}(C_{4}) = Burn_{2}^{triv}(C_{4}) ⊕ Burn_{2}^{nontriv}(C_{4}): Burn_{2}^{nontriv}(C_{4}) has, for every isomorphism class of curves of positive genus, a copy of B_{1}(C_{4}) and a copy of B_{1}(C_{2}), with an additional copy of B_{1}(C_{2}) for every isomorphism class of curves of positive genus with involution.

We claim points and rational curves contribute

ℤ^{2} ⊂ Burn_{2}^{nontriv}(C_{4}),

generated by [1, 2] and [2, 3] where

[i, j] = (C_{4}, k, (i, j)), (i, j) = (1, 1), (1, 2), (1, 3), (2, 3), (3, 3).

Abusing notation, write

[1, 0] = (C_{4}, k(t), 1) [3, 0] = (C_{4}, k(t), 1).

We write down the blowup relations, both orbits of points with special stabilizers and orbits on one-dimensional strata with nontrivial stabilizer:

[0, 1] = [0, 1] + [1, 3]
[0, 3] = [0, 3] + [1, 3]
[1, 1] = [1, 0]
[1, 2] = [1, 1] + [2, 3]
[1, 3] = [1, 2] + [2, 3] + (C_{2}, C_{2} k(t), 1)
[2, 3] = [1, 2] + [3, 3]
[3, 3] = [3, 0]
(C_{2}, C_{2} k^{2}, (1, 1)) = (C_{2}, C_{2} k(t)^{2}, 1)
(C_{2}, C_{2} k(t), 1) = (C_{2}, C_{2} k(t), 1) + (C_{2}, C_{2} k(t)^{2}, 1)
(C_{2}, C_{2} k(t)^{2}, 1) = (C_{2}, C_{2} k(t)^{2}, 1) + (C_{2}, C_{2} k(t)^{2}, 1)

Thus we find

[1, 3] = 0,
[0, 3] = [3, 3] = −[1, 2] + [2, 3],
[0, 1] = [1, 1] = [1, 2] − [2, 3],
(C_{2}, C_{2} k(t), 1) = −[1, 2] − [2, 3],
(C_{2}, C_{2} k^{2}(t), 1) = 0,
(C_{2}, C_{2} k^{2}, (1, 1)) = 0.

Here k^{n} denotes the total ring of fractions for an orbit of length n and k^{n}(t) the total ring of fractions of the exceptional locus of the blowup of such an orbit. Furthermore, C_{2} k(t) is via t ↦ −t.

For example, the linear action on ℙ^{2}

(x : y : z) ↦ (x : iy : −iz)

has invariant

[1, 3] + [1, 2] + [2, 3] + (C_{2}, C_{2} k(y/z), 1) = 0 ∈ Burn_{2}^{nontriv}(C_{4}).

We close this section with a noncyclic example:

(C_{2} × C_{2}) Write

G = C_{2} × C_{2} = {1, g_{1}, g_{2}, g_{3} = g_{1}g_{2}}

and

G^{∨} = {0, χ_{1}, χ_{2}, χ_{3} = χ_{1}+χ_{2}}, χ_{1}(g_{1}) = χ_{2}(g_{2}) = −1, χ_{1}(g_{2}) = χ_{2}(g_{1}) = 1 as before.

• B_{2}(C_{2} × C_{2}) = {0, [χ_{1}, χ_{2}], [χ_{1}, χ_{3}], [χ_{2}, χ_{3}]} with the structure of the Klein four group presented in Section 3.

• B_{2}(C_{2} × C_{2}, k) ≅ B_{2}(C_{2} × C_{2}) as B_{1}(C_{2} × C_{2}) = 0.

• Burn_{2}(C_{2}×C_{2}) = Burn^{triv}(C_{2}×C_{2})⊕Burn_{2}^{nontriv}(C_{2}×C_{2}) where the second term is a direct sum of the subgroup R generated by points and rational curves, a copy of

Burn_{1}^{nontriv}(C_{2} × C_{2}) ≅ ℤ^{3}

for each curve of positive genus, and another copy for each curve of positive genus equipped with an involution.

The group R fits into an exact sequence

0 → Burn_{1}^{nontriv}(C_{2} × C_{2}) → R → B_{2}(C_{2} × C_{2}) → 0

obtained by computing generators and relations.

Generators: Here we take 1 ≤ i < j ≤ 3:

[χ_{i}, χ_{j}] := (G, triv k, (χ_{i}, χ_{j}))
e_{i} := (⟨g_{i}⟩ , G/ ⟨g_{i}⟩ k(t), χ̄_{i} = 1)
q_{i} := (⟨g_{i}⟩ , G/ ⟨g_{i}⟩ k^{2}, (χ̄_{i}, χ̄_{i}) = (1, 1))
f_{i} := (⟨g_{i}⟩ , G/ ⟨g_{i}⟩ k^{2}(t), χ̄_{i} = 1)

Relations: Here we choose h so that {h, i, j} = {1, 2, 3}:

[χ_{i}, χ_{j}] = e_{h} + [χ_{h}, χ_{i}] + [χ_{h}, χ_{j}] (blow up fixed point)
q_{i} = f_{i} (blow up orbit q_{i})
e_{i} = e_{i} + f_{i} (blow up general orbit of e_{i})

Thus the q_{i} and f_{i} are zero and we have

R/ ⟨e_{1}, e_{2}, e_{3}⟩ = B_{2}(C_{2} × C_{2}).

We revisit the actions of C_{2} × C_{2} on rational surfaces in Section 3 using these new techniques:

(1) the action (x, y) ↦ (±x^{±1}, y) on ℙ^{1} × ℙ^{1} has invariant

f_{1} + f_{2} + f_{3} = 0;

(2) the action (x, y) ↦ (±x, ±y) on ℙ^{1} × ℙ^{1} has invariant

2e_{1} + 2e_{2} + 4[χ_{1}, χ_{2}] = 0;

(3) the diagonal action on ℙ^{1} × ℙ^{1} has invariant

2(q_{1} + q_{2} + q_{3}) = 0;

(4) the action on the conic fibration admits an elliptic curve

F = {y_{1}^{2}(x_{1} − ax_{2})(x_{1} + ax_{2}) = y_{2}^{2}(x_{1} − bx_{2})(x_{1} + bx_{2})}

that is fixed by g_{2} and fibers over (x_{1} : x_{2}) = (0 : 1), (1 : 0) fixed by g_{1} and thus has invariant

4[χ_{1}, χ_{2}] + 2e_{1} + (⟨g_{2}⟩ , ⟨g_{1}⟩ K(F), 1) ≠ 0

where g_{1} acts on K(F) by x_{1}/x_{2} ↦ −x_{1}/x_{2};

(5) the action on the degree two del Pezzo surface has nontrivial invariant arising from the positive genus curves fixed by g_{1} and g_{2};

(6) the action on the degree one del Pezzo surface has nontrivial invariant arising from the positive genus curves fixed by the involution.

5.5. Limitation of the birational invariant. It is an elementary observation, recorded in [36, App. A], that the presence of a point fixed by a given abelian subgroup H of G is a birational invariant of a smooth projective variety X with generically free G-action. Two smooth ndimensional projective varieties with generically free G-action might be distinguished in this way but nevertheless have the same class in Burn_{n}^{nontriv}(G).

Indeed, letting C_{2} act on ℙ^{1}, we consider the corresponding product action of C_{2} × C_{2} on ℙ^{1} × ℙ^{1}. As well, the action of C_{2} × C_{2} on ℙ^{1} gives rise to a diagonal action on ℙ^{1} × ℙ^{1}. The former, but not the latter, has a point fixed by C_{2} × C_{2}, so the actions belong to two distinct birational classes. However, both actions give rise to a vanishing class in Burn_{2}^{nontriv}(C_{2} × C_{2}).

5.6. Reprise: Cyclic groups on rational surfaces. As already discussed in Section 3, the presence of higher genus curves in the fixed locus of the action of a cyclic group of prime order on a rational surface is an important invariant in the study of the plane Cremona group. For example, for G = C_{2}, these curves make up entirely the group Burn_{2}^{G}(G) and famously characterize birational involutions of the plane up to conjugation [4].

For more general cyclic groups acting on rational surfaces, we recover the NFCA invariant of Blanc [10], which governs his classification.

We recall the relevant definitions: Let g ∈ Cr_{2} be a nontrivial element of the plane Cremona group, of finite order m.

• Normalized fixed curve [13]:

NFC(g) := \begin{cases}∅ if no curve of positive genus is fixed by g, \\ isomorphism class of the normalization of fixed curve. \end{cases}

• Normalized fixed curve with action:

NFCA(g) := ((NFC(g^{r}), g |_{NFC(g^{r})})_{r=1}^{m−1} ,

where g |_{NFC(g^{r})} is the automorphism induced by g on NFC(g^{r}). One of the main results in this context is the following characterization:

Theorem 5.4. [10] Two cyclic subgroups G and H of order m of Cr_{2} are conjugate if and only if

NFCA(g) = NFCA(h),

for some generators g of G and h of H.

It follows immediately from the definition that the information encoded in Burn_{2}(G), for G = C_{m}, is equivalent to NFCA(g).

Remark 5.5. It would be interesting to use symbol invariants to organize the classification of cyclic group actions on rational threefolds.

6. Cubic fourfolds

In this section we discuss several illustrative examples, showing various aspects of the new invariants introduced above. Equivariant geometry in dimension ≤ 3 is, in principle, accessible via the Minimal Model Program, and there is an extensive literature on factorizations of equivariant birational maps. We focus on dimension four, and in particular, on cubic fourfolds.

Let X ⊂ ℙ^{5} be a smooth cubic fourfold. No examples are known to be irrational! Here we show that there are actions X G where G-equivariant rationality fails, including actions on rational cubic fourfolds.

We found it useful to consult the classification of possible abelian automorphism groups of cubic fourfolds in [29]. Here is a list of N > 1, such that the cyclic group C_{N} acts on a smooth cubic fourfold:

N = 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 18, 21, 24, 30, 32, 33, 36, 48.

Note that

B_{4}(C_{N}) ⊗ ℚ = 0, for all N < 27, N = 30, 32.

We record their dimensions d_{ℚ} in the remaining cases:

\begin{matrix}N & 33 & 36 & 48 \\ d_{ℚ} & 2 & 3 & 7 \end{matrix}

One can also work with finite coefficients: let

d_{p} = d_{p}(N) := dim B_{4}(C_{N}) ⊗ 𝔽_{p},

we have d_{2}, d_{3}, d_{5} = 0, for all N ≤ 15, and N = 18, 21. In the other cases, we find:

\begin{matrix}N & 16 & 24 & 30 & 32 & 33 & 36 & 48 \\ d_{2} & 1 & 5 & 10 & 12 & 3 & 19 & 50 \\ d_{3} & 0 & 0 & 0 & 0 & 2 & 3 & 7 \\ d_{5} & 0 & 0 & 0 & 0 & 2 & 3 & 7 \\ d_{7} & 0 & 0 & 0 & 0 & 2 & 3 & 7 \end{matrix}

Thus, to exhibit applications of B_{4}(C_{N}) we have to look at large N.

Using B_{n}(G): Consider X ⊂ ℙ^{5} given by

x_{1}^{2}x_{2} + x_{2}^{2}x_{3} + x_{3}^{2}x_{1} + x_{4}^{2}x_{5} + x_{5}^{2}x_{0} + x_{0}^{3} = 0.  (6.1)

It carries the action of G = C_{36}, with weights

(0, 4, 28, 16, 9, 18)

on the ambient projective space, and isolated G-fixed points

P_{1} = (0 : 1 : 0 : 0 : 0 : 0), …, P_{5} := (0 : 0 : 0 : 0 : 0 : 1).

Computing the weights in the corresponding tangent spaces, we find that β(X) equals

[4, 24, 31, 22]+[28, 24, 19, 10]+[24, 12, 7, 34]+[9, 5, 17, 29]+[14, 26, 2, 9].

Solving a system of 443557 linear equations in 82251 variables, we find, by computer, that

β(X) ≠ β(ℙ^{4} C_{36}) = 0 ∈ B_{4}(C_{36}) ⊗ 𝔽_{2} = 𝔽_{2}^{19}.

This implies that X is not G-equivariantly birational to ℙ^{4}.

Using co-multiplication: The fourfold X ⊂ ℙ^{5} given by

x_{2}^{2}x_{3} + x_{3}^{2}x_{4} + x_{4}^{2}x_{5} + x_{5}^{2}x_{0} + x_{0}^{3} + x_{1}^{3} = 0  (6.2)

carries an action of C_{48} with weights (0, 16, 3, −6, 12, −24), and isolated fixed points. We find that β(X) ∈ B_{4}(C_{48}) equals

[−3, 13, 9, −27] + [6, 22, 9, −18] + [−12, 4, −9, −18] + [40, 27, 18, 36].

Here, we apply the co-multiplication formula from Section 2.2 to the image β^{−}(X) of β(X) under the projection

B_{4}(C_{48}) → B_{4}^{−}(C_{48}).

Let

G′ := ℤ/3ℤ, G′′ := ℤ/16ℤ, n′ = 1 and n′′ = 3.

We have a homomorphism

∇^{−} : B_{4}^{−}(C_{48}) → B_{1}^{−}(C_{3}) ⊗ B_{3}^{−}(C_{16})

and we find that ∇^{−}(β^{−}(X)) equals

[1]^{−} ⊗ ([−1, 3, −9]^{−} + [2, 3, −6]^{−} + [−4, −3, −6]^{−} + [9, 6, 12]^{−}).

Now we are computing in B_{3}^{−}(C_{16}), a much smaller group. We have

dim B_{3}(C_{16}) ⊗ ℚ = 3, but dim B_{3}^{−}(C_{16}) ⊗ ℚ = 0,

however

dim B_{3}(C_{16}) ⊗ 𝔽_{2} = 8 and dim B_{3}^{−}(C_{16}) ⊗ 𝔽_{2} = 7.

We find, by computer, that

[−4, −3, −6] = [9, 6, 12] ∈ B_{3}(C_{16}) ⊗ 𝔽_{2},

so that the sum of these terms does not contribute to ∇^{−}(β^{−}(X)), and check that

[−1, 3, −9]^{−} + [2, 3, −6]^{−} = [1, 2, 10]^{−} ≠ 0 ∈ B_{3}^{−}(C_{16}).

It follows that

∇^{−}(β^{−}(X)) ≠ 0, thus β^{−}(X) ≠ 0 ∈ B_{4}^{−}(C_{48}),

and this action of C_{48} on the cubic fourfold is not equivariantly birational to a linear action on ℙ^{4}.

Using B_{n}(G, k): There is also another way to analyze the fourfold in (6.2): observe that the divisor Y ⊂ X, a smooth cubic threefold given by x_{1} = 0, is fixed by C_{3} ⊂ C_{48}. This divisor is irrational, and we get a nontrivial contribution to β_{k}(X) ∈ B_{4}(C_{3}, k), in the summand labeled by Y ∈ Bir_{1,0}; thus X is not even C_{3}-equivariantly birational to ℙ^{4}.

The fourfold X ⊂ ℙ^{5} given by

f_{3}(x_{0}, x_{1}, x_{2}) + x_{3}^{2}x_{4} + x_{4}^{2}x_{5} + x_{5}^{2}f_{1}(x_{0}, x_{1}, x_{2}) = 0

carries the action of G = C_{8}, with weights

(0, 0, 0, 1, 6, 4).

The fourfold is smooth, e.g., for f_{3} = x_{0}^{3} + x_{1}^{3} + x_{2}^{3} and f_{1} = x_{0}.

Here, there are no isolated fixed points, but we find information from fixed point loci in higher dimension. The G-fixed locus contains the degree 3 curve Y given by

x_{3} = x_{4} = x_{5} = f_{3}(x_{0}, x_{1}, x_{2}) = 0,

which is smooth for appropriate f_{3}. Thus we get a contribution to β_{k}(X) ∈ B_{4}(G, k), in the summand labeled by Y ∈ Bir_{3,2}:

[7, 2, 4] ≠ 0 ∈ B_{3}(C_{8}) ≅ 𝔽_{2}.

Here, we solve 289 linear equations in 120 variables. This implies that X is not G-equivariantly birational to ℙ^{4}. Of course, this also follows by observing that the fourth power of the generator fixes a cubic threefold.

Using Burn_{n}(G): Consider X ⊂ ℙ^{5} given by

x_{0}x_{1}^{2} + x_{0}^{2}x_{2} − x_{0}x_{2}^{2} − 4x_{0}x_{4}^{2} + x_{1}^{2}x_{2} + x_{3}^{2}x_{5} − x_{2}x_{4}^{2} − x_{5}^{3} = 0. (6.3)

It carries the action of G = C_{6}, which acts with weights

(0, 0, 0, 1, 3, 4).

The cubic fourfold X is rational, since it contains the disjoint planes

x_{0} = x_{1} − x_{4} = x_{3} − x_{5} = 0 and x_{2} = x_{1} − 2x_{4} = x_{3} + x_{5} = 0.

Noticing a cubic surface S ⊂ X, with C_{3}-stabilizer and scalar action on the normal bundle, the fact that its C_{2}-action fixes an elliptic curve lets us conclude, by [11], that the cubic surface is not stably C_{2}-equivariantly rational; the corresponding symbol

[C_{3}, C_{2} K(S), β] ≠ 0 ∈ Burn_{4}(C_{6}),

moreover, it does not interact with any other symbols in [X G], which implies that X is not G-birational to ℙ^{4} with linear action. In this case, no subgroup of C_{6} fixes a hyperplane section.

We discuss obstructions of such type in Section 7.4 below – formally, Burn_{4}(C_{6}) admits a projection to ℤ that distinguishes the equivariant birational class of X from that of ℙ^{4} with linear action.

Kuznetsov [24] conjectures which cubic fourfolds X are rational, in terms of their derived categories D^{b}(X). Consider the line bundles O_{X}, O_{X}(1), O_{X}(2) and the right orthogonal complement

A_{X} = ⟨O_{X}, O_{X}(1), O_{X}(2)⟩^{⊥} .  (6.4)

Conjecturally, X is rational if and only if A_{X} ≅ D^{b}(S), the bounded derived category of a K3 surface S. However, the K3 surface need not be canonically determined as there are many examples of derivedequivalent but nonisomorphic K3 surfaces. Indeed the following conditions on complex projective K3 surfaces S_{1} and S_{2} are equivalent [31]

• D^{b}(S_{1}) ≅ D^{b}(S_{2});

• the Mukai lattices

H̃(S_{1}, ℤ) ≅ H̃(S_{2}, ℤ)

as Hodge structures;

• the transcendental cohomology lattices

H_{tran}^{2}(S_{1}, ℤ) ≅ H_{tran}^{2}(S_{2}, ℤ)

as Hodge structures. There is an alternative Hodge-theoretic version of the conjecture: A

smooth cubic fourfold X is rational if and only if there exists a K3 surface S and an isomorphism of integral Hodge structures

H^{4}(X, ℤ)_{tran} ≅ H^{2}(S, ℤ)_{tran}(−1),  (6.5)

where tran denotes the orthogonal complement of the Hodge classes and (−1) designates the Tate twist. Work of Addington and Thomas [1], and recent extensions [3, Cor. 1.7], show that the conditions (6.4) and (6.5) are equivalent. In particular, both are stable under specialization, consisting of an explicit countable collection of divisors in moduli [15]. The main theorem of [21] – that rationality is stable under specializations of smooth projective varieties – gives the equivalence of Kuznetsov’s conjecture with the Hodge-theoretic statement.

Suppose then that X admits an action of a finite group G. If X is rational – and the conjectures are true – then G naturally acts on A_{X} and D^{b}(S), for each surface S arising in (6.4). There is an induced action on H̃^{∗}(S, ℤ) as well. It is natural to speculate that a G-equivariant birational map ℙ^{4} −−→^{∼} X should imply that we may choose S in its derived equivalence class so that the G-action on the Mukai lattice is induced by a G-action on S.

There are several possible obstructions to finding such an S: • if S exists then there exists a sublattice of algebraic classes

U := (\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}) = H^{2}(S, ℤ)^{⊥}

in the G-invariant part of the abstract Mukai lattice arising from A_{X};

• the action of G on Pic(S) preserves the ample cone of S.

The first condition fails when G permutes various derived equivalent K3 surfaces. The second condition fails if G includes a Picard-Lefschetz transformation associated with a smooth rational curve ℙ^{1} ⊂ S. Derived equivalent K3 surfaces might have very different automorphism groups [16, Ex. 23]; this paper discusses descent of derived equivalence in the presence of Galois actions.

We mention some results on when the group action can be lifted to the associated K3 surface [32, §8]:

• if G ≠ {1}, G acts on X symplectically, i.e., acts trivially on H^{1}(Ω_{X}^{3}), then S is unique;

• if X is the Klein cubic fourfold

x_{1}^{3} + x_{2}^{2}x_{3} + x_{3}^{2}x_{4} + x_{4}^{2}x_{5} + x_{5}^{2}x_{6} + x_{6}^{2}x_{2} = 0

then X admits a symplectic automorphism of order 11 and A_{X} ≅ D^{b}(S) for a unique K3 surface, which has no automorphism of order 11.

We speculate that the Klein example should not be C_{11}-equivariantly rational, even though

β(X C_{11}) = 0 ∈ B_{4}(C_{11}),

as the C_{11}-action has isolated fixed points and the target group is trivial [20, §8].

Question 6.1. Let X be a smooth cubic fourfold with the action of a (finite) group G. Suppose that A_{X} ≅ D^{b}(S) for a K3 surface S with G-action, compatible with the isomorphism. Does it follow that

[X G] = [ℙ^{4} G] ∈ Burn_{4}(G),

for some action of G on ℙ^{4}?

It is mysterious how the invariants in the Burnside groups interact with the actions on the Hodge structures on the middle cohomology of X. Obstructions to G-equivariant rationality arise from fixed loci in various dimensions but the Hodge theory encodes codimension-two cycles only. The example (6.3), which is rational but not C_{6}-rational, is particularly striking to us: How is the cubic surface in the fixed locus coupled with the associated K3 surfaces?

7. Nonabelian invariants

In this section, G is a finite group, not necessarily abelian.

7.1. The equivariant Burnside group. As in Section 5.2, it is defined as the quotient of the ℤ-module generated by symbols

(H, N_{G}(H)/H K, β),

similar to those in (5.2), by blow-up relations. The required relations are a bit complicated but similar in spirit to what was written above; precise definitions are in [23, Section 4]. The resulting group

Burn_{n}(G)

carries a rich combinatorial structure, that remains largely unexplored.

7.2. Resolution of singularities. The class of X G, a projective variety with a generically free G-action, is computed on a suitable model of the function field K(X). We explain how such a model may be found in practice. While this is a corollary of Bergh’s ‘destackification’ procedure [6], the approach here can be helpful in specific examples.

We first review the resolution process of [36, §3]. A variety with group action as above is in standard form with respect to a G-invariant divisor Y if

• X is smooth and Y is a normal crossings divisor;

• the G action on X \ Y is free;

• for every g ∈ G and irreducible component Z ⊂ Y either g(Z) =

Z or g(Z) ∩ Z = ∅. We recall several fundamental results. First, we can always put actions in standard form:

If X is smooth and Y is a G-invariant closed subset such that G acts freely on X \Y then there exists a resolution

π : X̃ → X

obtained as a sequence of blowups along smooth Ginvariant centers, such that X̃ is in standard form with respect to Exc(π) ∪ π^{−1}(Y) [36, Th. 3.2].

Using a canonical resolution procedure as in [7] – take an embedded resolution of Y ⊂ X and then blow up the proper transform of Y – may assume that

π : X̃ \ π^{−1}(Y) →^{∼} X \ Y.

Even if we start with Y as the complement of the open set where G acts freely, the resulting strict normal crossings divisor may properly contain the locus with nontrivial stabilizers.

An action in standard form has stabilizers of special type:

Assume that X is in standard form with respect to Y and x ∈ X lies on m irreducible components of Y then the stabilizer H of x is abelian with ≤ m generators [36, Th. 4.1].

The proof of [36, Th. 4.1] (see Remark 4.4) yields étale-local coordinates on X about x

x_{1}, …, x_{k}, y_{1}, …, y_{l}, z_{1}, …, z_{m}

such that • H acts diagonally on all the coordinates;

• y_{1} = ⋯ = y_{l} = z_{1} = ⋯ = z_{m} = 0 coincides with X^{H}, i.e., these are the coordinates on which H acts nontrivially;

• y_{1} ⋯ y_{m} = 0 coincides with Y and the associated characters of χ_{i} : H → 𝔾_{m} generate H′ = Hom(H, 𝔾_{m}) so the induced representation

(χ_{1}, …, χ_{m}) : H ↪ 𝔾_{m}^{m}  (7.1)

is injective. Suppose that X is in standard form with respect to Y with irre-

ducible components Y_{1}, …, Y_{s}. For each orbit of these under the action of G, consider the reduced divisor obtained by summing over the orbit. The resulting divisors D(1), …, D(r) have smooth support – by the definition of standard form – and

D(1) ∪ ⋯ ∪ D(r) = Y_{1} ∪ ⋯ ∪ Y_{s}.

The line bundles O_{X}(D(i)) are naturally G-linearized and thus descend to line bundles on the quotient stack [X/G] and we obtain

φ : [X/G] → B𝔾_{m} × ⋯ × B𝔾_{m} (r factors).

We claim φ is representable. It suffices to check this by showing that the induced homomorphism of stabilizers is injective at each point [39, Tag 04YY]. For x ∈ X, fix the indices i_{1}, …, i_{m} so that D(i_{1}), …, D(i_{m}) are the components of Y containing x, and consider the induced

φ_{x} : [X/G] → B𝔾_{m} × ⋯ × B𝔾_{m} (m factors).

The homomomorphism on stabilizers is given by (7.1) which is injective. Thus we have established:

Proposition 7.1. Let X be a smooth projective variety with a generically free action by a finite group G, in standard form with respect to a divisor Y. Then the Assumptions of [23] hold and the invariants constructed there may be evaluated on X.

We are specifically using the formulation of Assumption 2 in [23, Rem. 3.2iii]. This is an alternate perspective on [23, Prop. 3.3], which guarantees these Assumptions can be satisfied on a suitable blowup.

7.3. The class of X G. On a suitable model X, we consider each stratum F ⊂ X with nontrivial (abelian) stabilizer H ⊂ G, the action of the normalizer N_{G}(H)/H on (the orbit of) the stratum, and the induced action of H on its normal bundle, and record these in a symbol. Then we define

[X G] := ∑_{H⊆G} ∑_{F}(H, N_{G}(H)/H K(F), β) ∈ Burn_{n}(G), (7.2)

as (5.3). The proof that this is a G-birational invariant relies on Gequivariant Weak Factorization and combinatorics [23, Section 5].

7.4. Elementary observations. As we discussed, the presence of higher genus curves in the fixed locus of the action of a cyclic group of prime order on a rational surface is an important invariant in the study of the plane Cremona group; see, e.g., [10]. These make up entirely the group Burn_{2}^{ℤ/2ℤ}(ℤ/2ℤ) and entirely characterize birational involutions of the plane up to conjugation [4].

More generally, for any nontrivial cyclic subgroup H of G and birational class of an (n − 1)-dimensional variety Y, not birational to Z × ℙ^{1} for any variety Z of dimension n − 2, we have a projection from Burn_{n}(G) onto the free abelian group on the N_{G}(H)-conjugacy classes of pairs (H′, a), where H′ is a subgroup, H ⊂ H′ ⊂ N_{G}(H), and a ∈ H^{∨} is a primitive character. This sends

(H, Ind_{H′/H}^{N_{G}(H)/H}(K(Y)), a),

for any H′/H K(Y), to the generator indexed by the conjugacy class [(H′, a)] of the pair.

A more refined version of this observation might also relax the restriction on Y but take into account the action H′/H K(Y). We do not go into details, but only point out, for instance, that for n = 2 and Y = ℙ^{1} there is a projection

Burn_{2}(G) → ⨁_{[(H′,a)] H′/H not cyclic} ℤ.

Taking G to be the dihedral group of order 12 and H the center of G, we may distinguish between the two inclusions of G into the plane Cremona group considered in [19], see Section 7.6 below.

7.5. Dihedral group of order 12. We now compute Burn_{2}(G), for G = D_{6}, the dihedral group with generators

ρ, σ, with ρ^{6} = σ^{2} = ρσρσ = e_{D_{6}}.

We list abelian subgroups, up to conjugacy:

• order 6: C_{6} = ⟨ρ⟩

• order 4: D_{2} = ⟨ρ^{3}, σ⟩

• order 3: C_{3} = ⟨ρ^{2}⟩

• order 2: central C_{2} = ⟨ρ^{3}⟩, noncentral S := ⟨σ⟩, S′ := ⟨ρ^{3}σ⟩ • order 1: triv

The subgroup of order 4 and two noncentral subgroups of order 2 have normalizer D_{2}, the others are normal.

As before, we use K(X) to denote the function field of the underlying surface X, K to denote the algebra of functions of a one-dimensional stratum (perhaps reducible) with nontrivial stabilizer, and k^{n} to denote the algebra of functions of a zero-dimensional orbit of length n. When we blow up such an orbit, we use k^{n}(t) to denote the total ring of fractions of the exceptional locus.

Generators: (C_{6}, ⟨σ̄⟩ K, (1)) (C_{6}, ⟨σ̄⟩ k^{2}, (1, j)), j = 1, …, 4, (C_{6}, ⟨σ̄⟩ k^{2}, (2, 3)) (D_{2}, triv k, (a_{1}, a_{2})), a_{1}, a_{2} ∈ 𝔽_{2}^{2}, generating 𝔽_{2}^{2} (C_{3}, ⟨ρ̄, σ̄⟩ K, (1)) (C_{3}, ⟨ρ̄, σ̄⟩ k^{4}, (1, 1)) (C_{2}, ⟨ρ̄, σ̄⟩ K, (1)) (S, C_{2} K, (1)) (S′, C_{2} K, (1)) (triv, D_{6} K(X), triv)

Relations: (C_{6}, ⟨σ̄⟩ k^{2}, (1, 1)) = (C_{6}, ⟨σ̄⟩ k^{2}(t), (1)) (C_{6}, ⟨σ̄⟩ k^{2}, (1, 2)) = (C_{6}, ⟨σ̄⟩ k^{2}, (1, 1)) + (C_{6}, ⟨σ̄⟩ k^{2}, (1, 4))

(C_{6}, ⟨σ̄⟩ k^{2}, (1, 3)) = (C_{6}, ⟨σ̄⟩ k^{2}, (1, 2)) + (C_{6}, ⟨σ̄⟩ k^{2}, (2, 3))+

(C_{2}, ⟨ρ̄, σ̄⟩ k^{2}(t), (1)),

where ρ̄ acts by cube roots of unity on t

(C_{6}, ⟨σ̄⟩ k^{2}, (1, 4)) = (C_{6}, ⟨σ̄⟩ k^{2}, (1, 3)) + (C_{6}, ⟨σ̄⟩ k^{2}, (2, 3))+

(C_{3}, ⟨ρ̄, σ̄⟩ k^{2}(t), (1)), where ρ̄ acts by −1 on t

0 = 2(C_{6}, ⟨σ̄⟩ k^{2}, (1, 4)) + (C_{2}, ⟨ρ̄, σ̄⟩ k^{2}(t), (1)), where ρ̄ acts by cube roots of unity on t

(C_{6}, ⟨σ̄⟩ k^{2}, (2, 3)) = (C_{6}, ⟨σ̄⟩ k^{2}, (1, 2)) + (C_{6}, ⟨σ̄⟩ k^{2}, (1, 3))

(D_{2}, triv k, ((1, 0), (0, 1))) = (D_{2}, triv k, ((1, 0), (1, 1)))+ (D_{2}, triv k, ((0, 1), (1, 1))) + (S′, C_{2} k(t), (1))

(D_{2}, triv k, ((1, 0), (1, 1))) = (D_{2}, triv k, ((1, 0), (0, 1)))+

(D_{2}, triv k, ((0, 1), (1, 1))) + (C_{2}, ⟨ρ̄, σ̄⟩ k^{3}(t), (1)),

permutation action on k^{3} with σ̄ acting by −1 on t

(D_{2}, triv k, ((0, 1), (1, 1))) = (D_{2}, triv k, ((1, 0), (0, 1)))+ (D_{2}, triv k, ((1, 0), (1, 1))) + (S, C_{2} k(t), (1))

(C_{3}, ⟨ρ̄, σ̄⟩ k^{4}, (1, 1)) = (C_{3}, ⟨ρ̄, σ̄⟩ k^{4}(t), (1))

0 = 2(C_{3}, ⟨ρ̄, σ̄⟩ k^{4}, (1, 1)) 0 = (C_{2}, ⟨ρ̄, σ̄⟩ k^{6}(t), (1))

0 = (S, C_{2} k^{2}(t), (1)) 0 = (S′, C_{2} k^{2}(t), (1))

7.6. Embeddings of 𝔖_{3}×C_{2} into the Cremona group. Iskovskikh [19] exhibited two nonconjugate copies of G = 𝔖_{3} × C_{2} = D_{6} in BirAut(ℙ^{2}):

• the action on x_{1} + x_{2} + x_{3} = 0 by permutation and reversing signs, with model ℙ^{2};

• the action on y_{1}y_{2}y_{3} = 1 by permutation and taking inverses,

with model a sextic del Pezzo surface. To justify the interest in these particular actions we observe that G is the Weyl group of the exceptional Lie group G_{2}, which acts on the Lie algebra of the torus, respectively on the torus itself, and it is natural to ask whether or not these actions are equivariantly birational. It turns out that they are stably G-birational [25, Proposition 9.11], but not G-birational. The proof of failure of G-birationality in [19] relies on the classification of links, via the G-equivariant Sarkisov program.

Here we explain how to apply Burn_{2}(G) to this problem. Note that neither model above satisfies the stabilizer condition required in the Definition (7.2)! We need to replace the surfaces by appropriate models X and Y, in particular, to blow up points:

• (x_{1}, x_{2}, x_{3}) = (0, 0, 0), with G as stabilizer;

• (y_{1}, y_{2}, y_{3}) = (1, 1, 1), with G as stabilizer, and

(ω, ω, ω), (ω^{2}, ω^{2}, ω^{2}), ω = e^{2πi/3},

with 𝔖_{3} as stabilizer. We describe these actions in more detail, following closely [19]. The

action on ℙ^{2}, with coordinates (u_{0} : u_{1} : u_{2}) is given by

(\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}) , (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & −1 & −1 \end{matrix}) , ρ^{3} = (\begin{matrix}1 & 0 & 0 \\ 0 & −1 & 0 \\ 0 & 0 & −1 \end{matrix}) .

There is one fixed point, (1 : 0 : 0); after blowing up this point, the exceptional curve is stabilized by the central involution ρ^{3}, and comes with a nontrivial 𝔖_{3}-action, contributing the symbol

(C_{2}, 𝔖_{3} K(ℙ^{1}), (1))  (7.3)

to [X G]. Additionally, the line ℓ_{0} := {u_{0} = 0} has as stabilizer the central C_{2}, contributing the same symbol. There are also other contributing terms, of the shape:

• (C_{6}, C_{2} k^{2}, β)

• (D_{2}, triv k, β′)

for some weights β, β′.

A better model for the second action is the quadric

v_{0}v_{1} + v_{1}v_{2} + v_{2}v_{0} = 3w^{2},

where 𝔖_{3} permutes the coordinates (v_{0} : v_{1} : v_{2}) and the central involution exchanges the sign on w. There are no G-fixed points, but a conic R_{0} := {w = 0} with stabilizer the central C_{2} and a nontrivial action of 𝔖_{3}. There are also:

• a G-orbit of length 2:

{P_{1} := (1 : 1 : 1 : 1), P_{2} := (1 : 1 : 1 : −1)},

exchanged by the central involution, each point has stabilizer 𝔖_{3} – these points have to be blown up, yielding a pair of conjugated ℙ^{1}, with a nontrivial 𝔖_{3}-action;

• another curve R_{1} := {v_{0} + v_{1} + v_{2} = 0} with effective G-action;

• additional points with stabilizers C_{6} and D_{2} in R_{0} and R_{1}.

The essential difference is that the symbol (7.3) appears twice for the action on ℙ^{2}, and only once for the action on the quadric: the pair of conjugated ℙ^{1} with 𝔖_{3}-action has trivial stabilizer and does not contribute. Further blow-ups will not introduce new curves of this type. Formally, examining the relations in Section 7.5, we see that the symbol (7.3) is not equivalent to any combination of other symbols, i.e., it is independent of all other symbols. This implies that

[X G] ≠ [Y G] ∈ Burn_{2}(G),

thus X and Y are not G-equivariantly birational. Note, that X and Y are equivariantly birational for any proper subgroup of G.

Remark 7.2. One can view the symbol (7.3) as the analog of a curve of higher genus in the fixed locus of an element in the classification of abelian actions on surfaces, as discussed in Section 3.

References

[1] N. Addington and R. Thomas. Hodge theory and derived categories of cubic fourfolds. Duke Math. J., 163(10):1885–1927, 2014.

[2] M. F. Atiyah and R. Bott. A Lefschetz fixed point formula for elliptic complexes. II. Applications. Ann. of Math. (2), 88:451–491, 1968.

[3] A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry, and P. Stellari. Stability conditions in families, 2019. arXiv:1902.08184.

[4] L. Bayle and A. Beauville. Birational involutions of P^{2}. Asian J. Math., 4(1):11–17, 2000. Kodaira’s issue.

[5] A. Beauville and J. Blanc. On Cremona transformations of prime order. C. R. Math. Acad. Sci. Paris, 339(4):257–259, 2004.

[6] D. Bergh. Functorial destackification of tame stacks with abelian stabilisers. Compos. Math., 153(6):1257–1315, 2017.

[7] E. Bierstone and P. D. Milman. Canonical desingularization in characteristic zero by blowing up the maximum strata of a local invariant. Invent. Math., 128(2):207–302, 1997.

[8] J. Blanc. Finite abelian subgroups of the Cremona group of the plane. PhD thesis, Université de Genève, 2006. Thèse no. 3777, arXiv:0610368.

[9] J. Blanc. Linearisation of finite abelian subgroups of the Cremona group of the plane. Groups Geom. Dyn., 3(2):215–266, 2009.

[10] J. Blanc. Elements and cyclic subgroups of finite order of the Cremona group. Comment. Math. Helv., 86(2):469–497, 2011.

[11] F. Bogomolov and Yu. Prokhorov. On stable conjugacy of finite subgroups of the plane Cremona group, I. Cent. Eur. J. Math., 11(12):2099–2105, 2013.

[12] I. Cheltsov and C. Shramov. Three embeddings of the Klein simple group into the Cremona group of rank three. Transform. Groups, 17(2):303–350, 2012.

[13] T. de Fernex. On planar Cremona maps of prime order. Nagoya Math. J., 174:1–28, 2004.

[14] I. V. Dolgachev and V. A. Iskovskikh. Finite subgroups of the plane Cremona group. In Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, volume 269 of Progr. Math., pages 443–548. Birkhäuser Boston, Boston, MA, 2009.

[15] B. Hassett. Special cubic fourfolds. Compositio Math., 120(1):1–23, 2000.

[16] B. Hassett and Yu. Tschinkel. Rational points on K3 surfaces and derived equivalence. In Brauer groups and obstruction problems, volume 320 of Progr. Math., pages 87–113. Birkhäuser/Springer, Cham, 2017.

[17] O. Haution. Fixed point theorems involving numerical invariants. Compos. Math., 155(2):260–288, 2019.

[18] V. A. Iskovskih. Minimal models of rational surfaces over arbitrary fields. Izv. Akad. Nauk SSSR Ser. Mat., 43(1):19–43, 237, 1979.

[19] V. A. Iskovskikh. Two non-conjugate embeddings of S_{3} ×Z_{2} into the Cremona group. II. In Algebraic geometry in East Asia—Hanoi 2005, volume 50 of Adv. Stud. Pure Math., pages 251–267. Math. Soc. Japan, Tokyo, 2008.

[20] M. Kontsevich, V. Pestun, and Yu. Tschinkel. Equivariant birational geometry and modular symbols, 2019. arXiv:1902.09894, to appear in J. Eur. Math. Soc.

[21] M. Kontsevich and Yu. Tschinkel. Specialization of birational types. Invent. Math., 217(2):415–432, 2019.

[22] A. Kresch and Yu. Tschinkel. Birational types of algebraic orbifolds, 2019.

arXiv:1910.07922.

[23] A. Kresch and Yu. Tschinkel. Equivariant birational types and Burnside vol-

ume, 2020. arXiv:2007.12538.

[24] A. Kuznetsov. Derived categories of cubic fourfolds. In Cohomological and geometric approaches to rationality problems, volume 282 of Progr. Math., pages 219–243. Birkhäuser Boston, Inc., Boston, MA, 2010.

[25] N. Lemire, V. L. Popov, and Z. Reichstein. Cayley groups. J. Amer. Math. Soc., 19(4):921–967, 2006.

[26] E. Looijenga. Motivic measures. Number 276, pages 267–297. 2002. Séminaire Bourbaki, Vol. 1999/2000.

[27] Ju. I. Manin. Rational surfaces over perfect fields. Inst. Hautes Études Sci. Publ. Math., (30):55–113, 1966.

[28] Ju. I. Manin. Rational surfaces over perfect fields. II. Mat. Sb. (N.S.), 72 (114):161–192, 1967.

[29] E. Mayanskiy. Abelian automorphism groups of cubic fourfolds, 2013.

arXiv:1308.5150.

[30] J. Nicaise and E. Shinder. The motivic nearby fiber and degeneration of stable rationality. Invent. Math., 217(2):377–413, 2019.

[31] D. O. Orlov. Equivalences of derived categories and K3 surfaces. J. Math. Sci. (New York), 84(5):1361–1381, 1997. Algebraic geometry, 7.

[32] G. Ouchi. Automorphism groups of cubic fourfolds and K3 categories, 2019. arXiv:1909.11033 to appear in Algebraic Geometry.

[33] Yu. Prokhorov. Simple finite subgroups of the Cremona group of rank 3. J. Algebraic Geom., 21(3):563–600, 2012.

[34] Yu. Prokhorov. On stable conjugacy of finite subgroups of the plane Cremona group, II. Michigan Math. J., 64(2):293–318, 2015.

[35] Yu. Prokhorov and C. Shramov. Jordan property for Cremona groups. Amer. J. Math., 138(2):403–418, 2016.

[36] Z. Reichstein and B. Youssin. Essential dimensions of algebraic groups and a resolution theorem for G-varieties. Canad. J. Math., 52(5):1018–1056, 2000. With an appendix by János Kollár and Endre Szabó.

[37] Z. Reichstein and B. Youssin. A birational invariant for algebraic group actions. Pacific J. Math., 204(1):223–246, 2002.

[38] The Sage Developers. SageMath, the Sage Mathematics Software System (Ver-

  sion 9.0), 2020. https://www.sagemath.org.

[39] The  Stacks  Project  Authors.  Stacks  Project.

  https://stacks.math.columbia.edu, 2018.

Brown University, Box 1917, 151 Thayer Street, Providence, RI 02912, USA

E-mail address: bhassett@math.brown.edu

Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

E-mail address: andrew.kresch@math.uzh.ch

Courant Institute, 251 Mercer Street, New York, NY 10012, USA

E-mail address: tschinkel@cims.nyu.edu

Simons Foundation, 160 Fifth Avenue, New York, NY 10010, USA