ON THE NOTION OF CANONICAL DIMENSION FOR ALGEBRAIC GROUPS

G. BERHUY AND Z. REICHSTEIN

Abstract. We define and study a numerical invariant of an algebraic group action which we call the canonical dimension. We then apply the resulting theory to the problem of computing the minimal number of parameters required to define a generic hypersurface of degree d in ℙ^{n−1}.

To M. Artin on his 70th birthday

Contents

1. Introduction 2 Acknowledgements 4 2. Notation and preliminaries 4 3. The canonical dimension of a G-variety 7 4. First properties 8 5. A lower bound 12 6. A comparison lemma 15 7. The canonical dimension of a group 16 8. Splitting fields 17 9. Generic splitting fields 21 10. The canonical dimension of a functor 23 11. Groups of type A 26 12. Orthogonal and Spin groups 28 13. Groups of low canonical dimension 31 14. The functor of orbits and homogeneous forms 32 15. Essential dimensions of homogeneous forms I 34 16. Essential dimensions of homogeneous forms II 36 References 39

1991 Mathematics Subject Classification. 11E72, 14L30, 14J70.

Key words and phrases. algebraic group, G-variety, generic splitting field, essential dimension, canonical dimension, homogeneous forms.

Z. Reichstein was partially supported by an NSERC research grant.

1. Introduction

Many important objects in algebra can be parametrized by a non-abelian cohomology set of the form H^{1}(K, G), where K is a field and G is a linear algebraic group defined over K. For example, elements of H^{1}(K, O_{n}) can be identified with isomorphism classes of n-dimensional quadratic forms over K, elements of H^{1}(K, PGL_{n}) with isomorphism classes of central simple algebras of degree n, elements of H^{1}(K, G_{2}) with isomorphism classes of octonion algebras, etc.; cf. [Se_{3}] or [KMRT]. Recall that H^{1}(K, G) has a marked (split) element but usually no group structure. Thus, a priori there are only two types of elements in H^{1}(K, G), split and non-split. However, it is often intuitively clear that some non-split elements are closer to being split than others. This intuitive notion can be quantified by considering degrees or Galois groups of splitting field extensions L/K for α; see, e.g., [T], [RY_{2}]. Another “measure” of how far α is from being split is its essential dimension (here and in the sequel we assume that K is a finitely generated extension of an algebraically closed base field of characteristic zero, and G is defined over k); for details and further references, see Section 2.1 and the first paragraph of Section 14.

In this paper we introduce and study yet another numerical invariant that “measures” how far α is from being split. We call this new invariant the canonical dimension and denote it by cd(α). We give several equivalent descriptions of cd(α); one of them is that cd(α) = min trdeg_{K}(L), where the minimum is taken over all generic splitting fields L/K for α (see Section 9). Generic splitting fields have been the object of much research in the context of central simple algebras (i.e., for G = PGL_{n}; see, e.g., [A], [Ar], [Roq_{1}], [Roq_{2}]) and quadratic forms (i.e., for G = O_{n} or SO_{n}; see, e.g., [Kn_{1}], [Kn_{2}], [KS]); related results for Jordan pairs can be found in [Pe]. Kersten and Rehmann [KR], who, following on the work of Knebusch, studied generic splitting fields in a setting rather similar to ours (cf. Remark 9.5), remarked, on p. 61, that the question of determining the minimal possible transcendence degree of a generic splitting field (or cd(α), in our language) appears to be difficult in general. Much of this paper may be viewed as an attempt to address this question from a geometric point of view.

Recall that we are assuming k to be an algebraically closed base field of characteristic zero, and K/k to be a finitely generated field extension. In this context every α ∈ H^{1}(K, G) is represented by a (unique, up to birational isomorphism) generically free G-variety X, with k(X)^{G} = K; see e.g., [Po, (1.3.3)]. We will often work with X, rather than α, writing cd(X, G) instead of cd(α) and using the language of invariant theory, rather than Galois cohomology. An advantage of this approach is that cd(X, G) is well defined for G-varieties X that are not necessarily generically free (see Definition 3.5), and the interplay between generically free and nongenerically free varieties can sometimes be used to gain insight into their canonical dimensions; cf., e.g., Lemma 6.1. If S is the stabilizer in general position for a G-variety X, then cd(X, G) can be related to the essential dimension of S. This connection is explored in Sections 5–6.

In Sections 7–13 we study canonical dimensions of generically free Gvarieties or, equivalently, of classes α ∈ H^{1}(K, G). We will be particularly interested in the maximal possible value of cd(α) for a given group G; we call this number the canonical dimension of G and denote it by cd(G). The canonical dimension cd(G), like the essential dimension ed(G), is a numerical invariant of G; if G is connected, both measure, in different ways, how far G is from being “special” (for the definition and a brief discussion of special groups, see Section 2.8 below). While cd(G) and ed(G) share some common properties (note, in particular, the similarity between the results of Section 7 in this paper and those of [R, Sections 3.1, 3.2]), their numerical values do not appear to be related to each other. For example, since cd(G) = 0 for every finite group G (see Lemma 7.5(b)), the rich theory of essential dimension for finite groups (see [BR], [BR_{2}], [JLY, Section 8]) has no counterpart in the setting of canonical dimension. On the other hand, our classification of simple groups of canonical dimension 1 in Section 13 has no counterpart in the context of essential dimension, because connected groups of essential dimension 1 do not exist; see [R, Corollary 5.7].

In Section 8 we prove a strong necessary condition for α ∈ H^{1}(K, G) to be of canonical dimension ≤ 2. A key ingredient in our proof is the Enriques-Manin-Iskovskih classification of minimal models for rational surfaces; see the proof of Proposition 8.2. In Sections 11 and 12 we study canonical dimensions of the groups GL_{n} /μ_{d}, SL_{n} /μ_{e}, SO_{n} and Spin_{n}. Our arguments there heavily rely on the recent results of Karpenko and Merkurjev [K_{1}], [KM], [M_{2}].

Our definition of canonical dimension naturally extends to the setting of functors F from the category of field extensions of k to the category of pointed sets; cd(G) is then a special case of cd(F), with F = H^{1}(_{−}, G) (see Section 10). A similar notion in the context of essential dimension is due to Merkurjev [M_{1}]; see also [BF_{2}] and the beginning of Section 14.

In the Sections 14 – 16 we apply our results on canonical dimension to the problem of computing the minimal number ed [H_{n,d}] of independent parameters, required to define the general degree d hypersurface in ℙ^{n−1}. (For a precise statement of the problem, see Section 14.) We show that if d ≥ 3 and (n, d) ≠ (2, 3), (2, 4) or (3, 3), our problem reduces to that of computing the canonical dimension of the group SL_{n} /μ_{gcd(n, d)}. In particular, combining Theorem 15.1 with Corollary 11.5, we obtain following theorem.

1.1. Theorem. Let n and d be positive integers such that d ≥ 3 and (n, d) ≠ (2, 3), (2, 4) or (3, 3). Suppose gcd(n, d) is a prime power p^{j} for some j ≥ 0. Then

ed(H_{n,d}) = (\begin{matrix}n + d − 1 \\ d \end{matrix}) − n^{2} + \begin{cases}0, if j = 0 \\ p^{i} − 1, if j ≥ 1. \end{cases} ,

where p^{i} is the highest power of p dividing n.

If d ≤ 2 or (n, d) = (2, 3), (2, 4), (3, 3), then our problem reduces to computing canonical dimensions for certain group actions that are not generically free; this is done in Section 16. Related results for (n, d) = (2, 3) and (3, 3) can be found in [BF_{1}].

Acknowledgements

We are grateful to J.-L. Colliot-Thélène, S. Garibaldi, D. W. Hoffmann, N. Karpenko, B. Kunyavskii, A. Merkurjev, A. Quéguiner-Mathieu, and J-P. Serre for helpful comments.

2. Notation and preliminaries

Throughout this paper we will work over an algebraically closed base field k of characteristic zero. Unless otherwise specified, all algebraic varieties, algebraic groups, group actions, fields and all maps between them are assumed to be defined over k, all algebraic groups are assumed to be linear (but not necessarily connected), and all fields are assumed to be finitely generated over k.

By a G-variety we shall mean an algebraic variety X with a (regular) action of an algebraic group G. We will usually assume that X is irreducible and focus on properties of X that are preserved by (G-equivariant) birational isomorphisms. In particular, we will call a subgroup S ⊂ G a stabilizer in general position for X if Stab(x) is conjugate to S for x ∈ X in general position; cf. [PV, Section 7]. As usual, if S = {1}, i.e., G acts freely on a dense open subset of X, then we will say that the G-variety X (or equivalently, the G-action on X) is generically free.

2.1. Essential dimension. Let X be a generically free G-variety. The essential dimension ed(X, G) of X is the minimal value of dim(Y)−dim(G), where the minimum is taken over all dominant rational maps X −−→ Y of G-varieties with Y generically free. For a given algebraic group G, ed(X, G) attains its maximal value in the case where X = V is a (generically free) linear representation of G. This value is called the essential dimension of G and is denoted by ed(G) (it is independent of the choice of V). For details, see [R, Section 3].

2.2. Rational quotients. A rational quotient for a G-variety X is an algebraic variety Y such that k(Y) = k(X)^{G}. The inclusion k(Y) ↪ k(X) then induces a rational quotient map π: X −−→ Y. Note that Y and π are only defined up to birational isomorphism; one usually writes X/G in place of Y. We shall say that G-orbits in X are separated by regular invariants if π is a regular map and π^{−1}(y) is a single G-orbit for every k-point y ∈ Y. By a theorem of Rosenlicht, X has a G-invariant dense open subset U, where G-orbits are separated by regular invariants. For a detailed discussion of the rational quotient and Rosenlicht’s theorem, see [PV, Section 2.4].

2.3. Generically free actions and Galois cohomology. Let X be a generically free variety. Then X may be viewed as a torsor over the generic point of X/G via the rational quotient map X −−→ X/G. Let α be the class of this torsor in H^{1}(K, G), where K = k(X)^{G}. This class is explicitly constructed in [Po, (1.3.1)]. Moreover, every α ∈ H^{1}(K, G) can be obtained in this way, and the G-variety X can be uniquely reconstructed from α, up to a (G-equivariant) birational isomorphism; see [Po, (1.3.2) and (1.3.3)]. In the sequel we shall say that α ∈ H^{1}(K, G) represents the generically free G-variety X.

2.4. Split generically free varieties. Let X be a generically free Gvariety, where the G-orbits are separated by regular invariants. We will call a rational map s: X/G −−→ X a rational section for π if s ∘ π = id on X/G. (Note that since the fibers of π are precisely the G-orbits in X, G·s(X/G) is dense in X. Consequently, some translate of s will “survive” if X is replaced by a birationally equivalent G-variety.) We shall say that X is split if one of the following equivalent conditions holds:

(i) X is birationally isomorphic to G × X/G,

(ii) π has a rational section,

(iii) X represents the trivial class in H^{1}(K, G),

(iv) ed(X, G) = 0. For a proof of equivalence of these four conditions, see [Po, (1.4.1)] and [R, Lemma 5.2].

2.5. The groups GL_{n} /μ_{d} and SL_{n} /μ_{e}. In this section we will review known results about the Galois cohomology sets H^{1}(K, G), where G = GL_{n} /μ_{d} or SL_{n} /μ_{e}, μ_{d} is the unique central cyclic subgroup of GL_{n} of order d, and e divides n.

2.6. Lemma. Let G = GL_{n} /μ_{d} (respectively, G = SL_{n} /μ_{e}), f: G ⟶ PGL_{n} be the canonical projection, and K/k be a field extension. Then

(a) The map f_{∗}: H^{1}(K, G) → H^{1}(K, PGL_{n}) has trivial kernel.

(b) The image of f_{∗} consists of those classes which represent central simple algebras of degree n and exponent dividing d (respectively, dividing e).

Lemma 2.6 can be deduced from [Sal_{1}, Theorem 3.2]; for the sake of completeness, we supply a direct proof below.

Proof. (a) The exact sequence 1 ⟶ Ker(f) ↪^{i} G ⟶^{f} PGL_{n} ⟶ 1 of algebraic groups, gives rise to an exact sequence

H^{1}(K, Ker(f)) ⟶^{i_{∗}} H^{1}(K, G) ⟶^{f_{∗}} H^{1}(K, PGL_{n})

of pointed sets; cf. [Se_{2}, pp. 123 - 126]. It is thus enough to show that i_{∗} is the trivial map (i.e., its image is {1}). If G = GL_{n} /μ_{d} this is an immediate consequence of the fact that Ker(f) = 𝔾_{m}/μ_{d} is isomorphic to 𝔾_{m} and thus H^{1}(K, Ker(f)) = \{1\}. If G = SL_{n} /μ_{e} then Ker(f) = μ_{\frac{n}{e}}, and the commutative diagram

(2.7)  [figure: 1; μ_{n}; × e; SL_{n}; PGL_{n}; 1; 1; μ_{\frac{n}{e}}; i; SL_{n} /μ_{e}; _{PGL_{n}}; 1,]

of group homomorphisms induces the commutative diagram

\xymatrix{H^{1}(K, μ_{n}) \ar[r] \ar[d] & H^{1}(K, SL_{n}) \ar[d] \\ H^{1}(K, μ_{\frac{n}{e}}) \ar[r]^{i_{∗}} & H^{1}(K, SL_{n} /μ_{e})}

of maps of pointed sets. Since the left vertical map is surjective (it is the natural projection K^{×}/(K^{×})^{n} ⟶ K^{×}/(K^{×})^{\frac{n}{e}}), and H^{1}(K, SL_{n}) = \{1\} (see [Se_{2}, p. 151]), we see that the image of i_{∗} is trivial, as claimed.

(b) We will assume G = SL_{n} /μ_{e}; the case G = GL_{n} /μ_{d} is similar and will be left to the reader. We now focus on the connecting maps

[figure: H^{1}(K, PGL_{n}); δ; H^{2}(K, μ_{n}); × e; H^{1}(K, SL_{n} /μ_{e}); f_{∗}; H^{1}(K, PGL_{n}); δ′; H^{2}(K, μ_{\frac{n}{e}})]

induced by the diagram (2.7). It is well known that H^{2}(K, μ_{n}) is the ntorsion part of the Brauer group of K and δ sends a central simple algebra A to its Brauer class [A]; see [Se_{2}, Section X.5]. Hence, δ′(A) = e · [A], and Im(f_{∗}) = Ker(δ′) consists of algebras A of degree n and exponent dividing e, as claimed. □

2.8. Special groups. An algebraic group G is called special if H^{1}(E, G) = {1} for every field extension E/k. Equivalently, G is special if every generically free G-variety is split. Special groups were introduced by Serre [Se_{1}] and classified by Grothendieck [Gro, Theorem 3] as follows: G is special if and only if its maximal semisimple subgroup is a direct product of simply connected groups of type SL or Sp; cf. also [PV, Theorem 2.8]. The following lemma can be easily deduced from Grothendieck’s classification; we will instead give a proof based on Lemma 2.6.

2.9. Lemma. GL_{n} /μ_{d} is special if and only if gcd(n, d) = 1.

Proof. If n and d are relatively prime then every central simple algebra of degree d and exponent dividing n is split. By Lemma 2.6, f_{∗} has trivial image and trivial kernel, showing that H^{1}(E, GL_{n} /μ_{d}) = {1} for every E, i.e., GL_{n} /μ_{d} is special. Conversely, suppose e = gcd(n, d) > 1. Let E = k(a, b), where a and b are algebraically independent variables over k, and D = (a, b)_{e} = generic symbol algebra of degree e. Then A = M_{\frac{n}{e}} (D) is a central simple algebra of degree n and exponent e, with center E. This algebra defines a class in H^{1}(E, PGL_{n}); since e divides d, Lemma 2.6 tells us that this class is the image of some α ∈ H^{1}(E, GL_{n} /μ_{d}). Since A is not split, α ≠ 1, and hence GL_{n} /μ_{d} is not special, as claimed. □

3. The canonical dimension of a G-variety

3.1. Definition. Let X be an irreducible G-variety (not necessarily generically free). We shall say that a rational map F: X −−→ X is a canonical form map if F (x) = f(x) · x for some rational map f: X −−→ G. Here we think of F (x) as a “canonical form” of x. Note that F and f will usually not be G-equivariant.

3.2. Remark. If the G-action on X is generically free and F: X −−→ X is a rational map then the following conditions are equivalent:

(a) F is a canonical form map,

(b) F (x) ∈ G · x for x ∈ X in general position,

(c) π ∘ F = π, where π: X −−→ X/G is the rational quotient map. The equivalence of (a) and (b) follows from the fact that the rational quotient map π: X −−→ X/G is a G-torsor over a dense open subset of X/G; cf. Section 2.3. The equivalence of (b) and (c) is a consequence of the theorem of Rosenlicht mentioned in Section 2.2.

3.3. Remark. If the G-action on X is generically free then the argument we used to prove that (a) ⇔ (b) in Remark 3.2 also shows that the rational map f: X −−→ G in Definition 3.1 is uniquely determined by F. On the other hand, if the G-action on X is not generically free then this may no longer be the case. For example, if the G-action on X is trivial then every f: X −−→ G gives rise to the trivial canonical form map F = id_{X}: X −−→ X. This means that in working with canonical form maps, we will want to keep track of both F and f. In the sequel we will often say that f: X −−→ G induces F: X −−→ X if F (x) = f(x) · x for x ∈ X in general position.

3.4. Example. Let X = M_{n}, with the conjugation action of G = GL_{n}. We claim that the rational map F: M_{n} −−→ M_{n}, taking A to its companion matrix

F (A) = (\begin{matrix}0 & 0 & … & 0 & −c_{n} \\ 1 & 0 & … & 0 & −c_{n−1} \\ 0 & 1 & … & 0 & −c_{n−2} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ 0 & 0 & … & 1 & −c_{1} \end{matrix}) ,

is a canonical form map. Here t^{n} + c_{1}t^{n−1} + ⋯ + c_{n} = det(tI − A) is the characteristic polynomial of A.

To prove the claim, fix a non-zero column vector v ∈ k^{n} and define f(A) as the matrix whose columns are v, Av, …, A^{n−1}v. It is now easy to see that f: A ↦ f(A) is a rational map M_{n} −−→ GL_{n}, and f(A) · A = f(A)^{−1}Af(A) is the companion matrix F (A). □

Our definition of a canonical form map is quite general; for example it includes the trivial case, where f(x) = 1_{G} and thus F (x) = x for every x ∈ X. Usually we would like to choose f so that the canonical form of every element lies in some subvariety of X of small dimension. With this in mind, we give the following:

3.5. Definition. The canonical dimension cd(X, G) of a G-variety X is defined as

cd(X, G) = min {dim F (X) − dim(X/G)},

where the minimum is taken over all canonical form maps F: X −−→ X. If the G-action on X is generically free, and X represents α ∈ H^{1}(K, G) (see Section 2.3), we will also write cd(α) in place of cd(X, G).

Note that the symbol cd does not stand for and should not be confused with cohomological dimension.

3.6. Lemma. The integer cd(X, G) is the minimal value of dim F (G·x) for x ∈ X in general position. Here the minimum is taken over all canonical form maps F: X −−→ X.

Proof. Let π: X −−→ X/G be the rational quotient map for the G-action on X. Then for any canonical form map F: X −−→ X, we have π = π ∘ F. In particular, π F (X) is dense in X/G. Applying the fiber dimension theorem to

π_{| F (X)} : F (X) −−→ X/G ,

we see that

dim F (X) − dim X/G = dim \overline{F (X)} ∩ G · x = dim F (G · x)

for x ∈ X in general position. By Definition 3.5, cd(X, G) is the minimal value of this quantity, as F ranges over all canonical form maps F: X −−→ X. □

3.7. Example. Let X = M_{n}, with the conjugation action of G = GL_{n}. Then the canonical form map F constructed in Example 3.4 takes every orbit in X to a single point. This shows that cd(M_{n}, GL_{n}) = 0. The same argument shows that cd(M_{n}, PGL_{n}) = 0; cf. also Lemma 4.10 below.

4. First properties

4.1. Subgroups.

4.2. Lemma. If X is a G-variety and H is a closed subgroup of G then

cd(X, G) + dim X/G ≤ cd(X, H) + dim X/H .

Proof. The left hand side is the minimal value of dim F (X), as F ranges over canonical form maps F: X −−→ X induced by f: X −−→ G. The right hand side is the same, except that f is only allowed to range over rational maps X −−→ H. Since there are more rational maps from X to G than from X to H, the inequality follows. □

4.3. Connected components.

4.4. Lemma. Let X be a G-variety and let G^{0} be the connected component of G. Then cd(X, G^{0}) = cd(X, G).

Proof. The inequality cd(X, G) ≤ cd(X, G^{0}) follows from Lemma 4.2, with H = G^{0}. To prove the opposite inequality, let F: X −−→ X be a canonical form map such that dim F (G · x) = cd(X, G) for x ∈ X in general position. Suppose F is induced by a rational map f: X −−→ G, as in Definition 3.1. Since X is irreducible, the image of f lies in some irreducible component of G. Let g be an element of this component. Then we can replace f by f′: X −−→ G^{0}, where f′(x) = g^{−1}f(x), and F by F′: X −−→ X given by F′(x) = f′(x) · x = g^{−1} · F (x). (Note that here g is independent of x.) Since F′(G · x) is a translate of F (G · x), we conclude that cd(X, G^{0}) ≤ dim F′(G^{0} · x) ≤ dim F′(G · x) = dim F (G · x) = cd(X, G). □

4.5. Direct products.

4.6. Lemma. Let X_{i} be a G_{i}-variety for i = 1, 2, G = G_{1} × G_{2} and X = X_{1} × X_{2}. Then cd(X, G) ≤ cd(X_{1}, G_{1}) + cd(X_{2}, G_{2}).

Proof. If F_{i}: X_{i} −−→ X_{i} are canonical form maps induced by f_{i}: X_{i} −−→ G_{i} (for i = 1, 2) then F = (F_{1}, F_{2}): X −−→ X is a canonical form map induced by f = (f_{1}, f_{2}): X = X_{1} × X_{2} −−→ G_{1} × G_{2}. Clearly,

F (G · x) = F_{1}(G_{1} · x_{1}) × F_{2}(G_{2} · x_{2})

for any x = (x_{1}, x_{2}) and thus dim(F · x) = dim(F_{1} · x_{1}) + dim(F_{2} · x_{2}). The desired inequality now follows from Lemma 3.6. □

4.7. Split varieties.

4.8. Lemma. Let X be a generically free G-variety and let π: X −−→ X/G be the rational quotient map.

(a) If X is split (cf. Section 2.4) then cd(X, G) = 0.

(b) Suppose G is connected. Then the converse to part (a) holds as well.

Proof. (a) Since X is split, we may assume X = G × X_{0}, where X_{0} = X/G; see Section 2.4(i). The map F: X ⟶ X, given by F: (g, x_{0}) ↦ (1_{G}, x_{0}) is clearly a canonical form map (see Remark 3.2), with dim F (X) = dim(X/G), and the desired equality follows.

(b) After replacing X be a G-invariant dense open subset, we may assume that the G-orbits in X are separated by regular invariants. Suppose cd(X, G) = 0, i.e., dim F (X) = dim(X/G) for some canonical form map F: X −−→ X. It is enough to show that π_{|F (X)} : F (X) −−→ X/G is a birational isomorphism. Indeed, if we can prove this then

π_{|F (X)}^{−1} : X/G −−→^{≃} F (X) ↪ X

will be a rational section (as defined in Section 2.4).

To prove that π_{|F (X)} is a birational isomorphism, consider the fibers of this map. If x ∈ X is a point in general position and y = π(x) ∈ X/G then π_{|F (X)}^{−1}(y) = F (G · x). Since G is connected, G · x is irreducible, and so is F (G·x). On the other hand, since cd(X, G) = 0, π_{|F (X)}^{−1}(y) is 0-dimensional. We thus conclude that π_{|F (X)}^{−1}(y) is a single k-point for y ∈ X/G in general position. Hence, π_{|F (X)} is a birational isomorphism (cf., e.g., [Hu, Section I.4.6]), and the proof is complete. □

4.9. Normal subgroups

4.10. Lemma. Let α: G ⟶ \overline{G} be a surjective map of algebraic groups and H = Ker(α). Suppose X is a \overline{G}-variety or, equivalently, a G-variety with H acting trivially. Then

(a) cd(X, G) ≥ cd(X, \overline{G}).

(b) If H is special then cd(X, G) = cd(X, \overline{G}).

Proof. Part (a) follows from Definition 3.5, because f: X −−→ G and

\overline{f} = f (mod H): X −−→ \overline{G}

give rise to the same canonical form map F: X −−→ X.

(b) Reversing the argument of part (a), it suffices to show that every rational map \overline{f}: X −−→ \overline{G} can be lifted to f: X −−→ G.

Since α: G ⟶ \overline{G} separates the orbits for the right H-action on G, it is a rational quotient map for this action: cf, [PV, Lemma 2.1]. If H is special then α has a rational section β: \overline{G} −−→ G. (Note that β is a rational map of varieties but not necessarily a group homomorphism.) Moreover, for any g_{0} ∈ G, the map β_{g_{0}} : \overline{G} −−→ G, given by β_{g_{0}} : \overline{g} ↦ g_{0}^{−1}β(α(g_{0})\overline{g}) is also a rational section of α. After replacing β by β_{g_{0}}, for a suitable g_{0} ∈ G, we may assume that \overline{f}(X) does not lie entirely in the indeterminacy locus of β. Now f = β ∘ \overline{f}: X −−→ G is the desired lifting of \overline{f}: X −−→ \overline{G}. □

4.11. Proposition. Let X be a G-variety and H be a closed normal subgroup of G. If H is special and the (restricted) H-action on X is generically free then cd(X, G) = cd(X/H, G) = cd(X/H, G/H).

Strictly speaking, the rational quotient variety X/H is only defined up to birational isomorphism. However, there exists a birational model Y of X/H, such that the G-action on X descends to a (regular) G/H-action (or equivalently, a regular G-action) on Y; see [PV, Proposition 2.6 and Corollary to Theorem 1.1]. The symbol X/H in the statement of Proposition 4.11 denotes Y as above; any two such models will are birationally isomorphic as G-varieties.

Proof. The equality cd(X/H, G) = cd(X/H, G/H) follows from Lemma 4.6(b); we shall thus focus on proving that cd(X, G) = cd(X/H, G).

After replacing X by an G-invariant open subset, we may assume that the quotient map π: X ⟶ X/H is regular. If π′: X/H −−→ Z is a rational quotient map for the G-action on X/H then π′ ∘ π: X −−→ Z is a rational quotient map for the G-action on X. In particular, X/G and Z = (X/H)/G have the same dimension. By Definition 3.5 we only need to show that

(i) given a canonical form map F: X −−→ X there exists a canonical form map F′: X/H −−→ X/H such that dim F (X) ≥ dim F′(X) and conversely, that

(ii) given a canonical form map F′: X/H −−→ X/H, there exists a canonical form map F: X −−→ X such that dim F (X) = dim F′(X).

Since H is special and its action on X is generically free, we can choose a rational section α: X/H −−→ X for the quotient map π: X ⟶ X/H.

We now proceed to prove (i). Suppose F is induced by a rational map f: X −−→ G. After translating α by an element of H, we may assume that the image of α meets the domain of f. Now let F′: X/H −−→ X/H be the canonical form map induced by f′ = f ∘ α: X/H −−→ G. Then the diagram

(4.12)  \xymatrix{X \ar@{-->}[r]^{F} \ar[d]^{π} & X \ar[d]^{π} \\ X/H \ar@{-->}[r]^{F′} & X/H}

commutes, and (i) follows.

To prove (ii), choose a Zariski open subset U ⊂ X/H, such that α is regular in U and π: π^{−1}(U) ⟶ U is an H-torsor. In particular, there is a morphism s: π^{−1}(U) ⟶ H such that s(x) · x = α(π(x)). Suppose the canonical form map F′: X/H −−→ X/H is induced by f′: X/H −−→ G. After replacing f′ by gf′ for a suitable g ∈ G, we may assume without loss of generality (and without changing dim F′(X/H)) that F′(\overline{x}) ∈ U for \overline{x} ∈ X/H in general position.

We will now construct the canonical form map F: X −−→ X, whose existence is asserted by (ii). To motivate this construction, we remark that it is easy to define a canonical form map F so that the diagram (4.12) commutes; such a map is induced by f = f′ ∘ π: X −−→ G. However, this diagram only shows that dim F (X) ≥ dim F′(X/H); equality will not hold in general. On the other hand, we are free to modify f(x) by multiplying f′(π(x)) by any element of H on the left (this element of H may even depend on x); the resulting canonical form map F will still give rise to a commutative diagram (4.12). With this in mind, we define F: X −−→ X as the canonical form map induced by f: X −−→ G, where

f(x) = s(f′(π(x)) · x)f′(π(x))

for x ∈ X in general position. As we mentioned above, the diagram (4.12) commutes; moreover, by our choice of s, F (X) ⊂ α(X/H). Hence, π restricts to an isomorphism between F (X) and π(F (X)) = F′(X/H). This proves (ii). □

4.13. Remark. Lemma 4.10 and Proposition 4.11 may fail if H is not special; see Example 5.9. If H is special, Proposition 5.9 may still fail if the H-action on X is not generically free; see Remark 15.5.

5. A lower bound

5.1. Definition. Let S be an algebraic group and Y be a generically free S-variety. We define e(Y, S) as the smallest integer e with the following property: given a point y ∈ Y in general position, there is an S-equivariant rational map f: Y −−→ Y such that f(Y) contains y and dim f(Y) ≤ e + dim(S).

5.2. Remark. Note that this definition is similar to the definition of the essential dimension ed(Y, S) of Y; cf. Section 2.1. The difference is that ed(Y, S) is the minimal value of dim f(Y) − dim(S), where f is allowed to range over a wider class of rational S-equivariant maps. In particular, e(Y, S) ≥ ed(Y, S). Note also that e(Y, S) depends only on the birational class of Y, as an S-variety.

5.3. Remark. In the sequel we will be particularly interested in the case where Y is itself an algebraic group, S is a closed subgroup of Y, and the S-action on Y is given by translations (say, by right translations, to be precise). In this situation, e(Y, S) is simply the minimal possible value of dim f(Y) − dim(S), where f ranges over all S-equivariant rational maps Y −−→ Y. Indeed, after composing f with a suitable left translation g: Y ⟶ Y, we may assume that f(Y) contains any given y ∈ Y.

5.4. Lemma. Let Y be a generically free S-variety.

(a) If Y is split (cf. Section 2.4) then e(Y, S) = 0.

(b) Suppose there exists a dominant rational S-equivariant map α: V −−→ Y, where V is a vector space with a linear S-action. Then e(Y, S) = ed(S).

(c) If Y = G is a special algebraic group, S is a subgroup of G and the S-action on Y is given by translations then e(Y, S) = ed(S).

Note that the condition of part (a) is always satisfied if S is a special group.

Proof. (a) If Y is split, it is birationally isomorphic to S × Z, where S acts by translations on the first factor and trivially on the second. In fact, we may assume without loss of generality that Y = S × Z. Now for any z_{0} ∈ Z consider f_{z_{0}} : S × Z −−→ S × Z, given by (s, z) ↦ (s, z_{0}). As z_{0} ranges over Z, the images of f_{z_{0}} cover Y. Each of these images has the same dimension as S; this yields e(Y, S) = 0.

(b) Let β: Y −−→ Y_{0} be the dominant S-equivariant rational map from Y to a generically free S-variety Y_{0} of minimal possible dimension, ed(S) + dim(S); cf. Remark 5.3. Then for any v ∈ V, there is a rational Gequivariant map γ: Y_{0} −−→ V such that v lies in the image of γ; see [R, Proposition 7.1]. Taking f = α ∘ γ ∘ β: Y −−→ Y in Definition 5.1 and varying v over V, we see that e(Y, S) ≤ dim(Y_{0})−dim(S) = ed(S). The opposite inequality was noted in Remark 5.2.

(c) Let V be a generically free linear representation of G (and thus of S). Since G is special, V is split; cf. Section 2.4. Consequently, there is a dominant rational map V −−→ G of G-varieties (and hence, of S-varieties). The desired conclusion now follows from part (b). □

5.5. Proposition. Let G be a connected group and X be an irreducible Gvariety with a stabilizer S in general position. Then

(a) cd(X, G) ≥ e(G, S), where S acts on G by translations. In particular,

(b) cd(X, G) ≥ ed(G, S), and

(c) if G is special then cd(X, G) ≥ ed(S).

Proof. (b) and (c) follow from (a) by Remark 5.2 and Lemma 5.4(c) respectively.

To prove part (a), choose a canonical form map F: X −−→ X such that dim F (G · x) = cd(X, G) for x in general position; cf. Lemma 3.6. Suppose that F is induced by a rational map f: X −−→ G, as in Definition 3.1, and consider the commutative diagram

\xymatrix{G \ar@{-->}[r]^{F′} \ar[d]^{ϕ} & G \ar[d]^{ϕ} \\ G · x \ar@{-->}[r]^{F} & G · x}

of rational maps, where ϕ: G ⟶ G · x is the orbit map, ϕ(g) = g · x, and F′(g) = f(g · x) · x.

Now set S = Stab_{G}(x) and observe that F′(gs^{−1}) = F′(g)s^{−1} for every s ∈ S. In view of Remark 5.3, this implies

dim F′(G) − dim S ≥ e(G, S) .

On the other hand, F′(G) is an S-invariant subvariety of G and (because the above diagram in commutative) F (G · x) = ϕ(F′(G)). Finally, since for any g ∈ G, ϕ(gs^{−1}) = ϕ(g) if and only if s ∈ S, we see that the fibers of the map ϕ_{|F′(X)} : F′(X) ⟶ F (G · x) are precisely the S-orbits in F′(X). Consequently,

cd(X, G) = dim F (G · x) = dim F′(G) − dim S ≥ e(G, S) ,

as claimed. □

5.6. Remark. Proposition 5.5 assumes that the G-action on X has a stabilizer in general position, i.e., there exists a subgroup S ⊂ G such that Stab(x) is conjugate to S for x ∈ X in general position. This condition is satisfied by many but not all group actions; see [PV, Section 7]. For an arbitrary G-action on X, our proof of part (a) shows that if e(G, Stab(x)) ≥ d for x in a Zariski dense open subset of X then cd(X, G) ≥ d.

Note also that if L_{x} is a Levi subgroup of Stab(x) then by a theorem of Richardson (see [Ri, Theorem 9.3.1] or [PV, Theorem 7.1]), there exists a non-empty Zariski open subset U ⊂ X such that L_{x} and L_{y} are conjugate in G. Since ed(L_{x}) = ed(Stab(x)) (this is an immediate consequence of [San, Lemma 1.13]; for a direct geometric proof, see [Ko]), ed(Stab(x)) assumes the same value for every x ∈ U. In particular, Proposition 5.5(c) remains valid for an arbitrary G-action, provided that we replace the inequality cd(X, G) ≥ ed(S) by cd(X, G) ≥ ed(Stab(x)) for x ∈ U.

5.7. Corollary. Let G be a connected group, S be a closed subgroup, and X = G/S be a homogeneous space. Then

(a) cd(X, G) = e(G, S), where S acts on G by translations.

(b) If G is special then cd(X, G) = ed(S).

Proof. Part (b) follows from part (a) and Lemma 5.4(c).

To prove (a), note that by Proposition 5.5, we only need to show that cd(X, G) ≤ e(G, S), i.e., to construct a canonical form map F: X −−→ X such that

(5.8)  dim F (G · x) = e(G, S)

for x in general position. We will define F by reversing the construction in Proposition 5.5. Let F′: G −−→ G be an S-equivariant rational map (with respect to the right translation action of S on G), such that dim F′(G) assumes its minimal possible value, e(G, S) + dim(S); cf. Remark 5.3. Then f′: G −−→ G given by f′(g) = F′(g)g^{−1} is S-invariant (with respect to the right translation action of S on G). Hence, f′ descends to f: G/S −−→ G. Thus we have a commutative diagram

G −−→^{F′} G
↓  ↓
G/S −−→^{F} G/S

where F (x) = f(x) · x. Here F is, by construction, a canonical form map, and

dim F (G/S) = dim F′(G) − dim S = e(G, S) ,

as desired. □

5.9. Example. Let G be a special group and H be a non-special closed normal subgroup of G. (For example, G = SL_{n} and H = μ_{n} is the center of G.)

(i) Let X = G/H. Then by Corollary 5.7(b), cd(X, G) = ed(H), which is ≥ 1; cf. [R, Proposition 5.3]. On the other hand, cd(X, G/H) = 0; (cf. Lemma 4.8(a)). This shows that the equality cd(X, G) = cd(X, G/H) in Lemma 4.10(b) may fail if H is not special.

(ii) Now let X = G (viewed as a G-variety with the translation action). Then cd(X, G) = 0 (cf. Lemma 4.8(a)) but, cd(X/H, G) = ed(H). This shows that the equality cd(X, G) = cd(X/H, G) in Proposition 4.11 may also fail if H is not special.

6. A comparison lemma

6.1. Lemma. Let α: X −−→ Y be a dominant rational map of irreducible G-varieties. Suppose dim(G · x) = d and dim(G · y) = e for x ∈ X and y ∈ Y in general position. Then cd(X, G) ≤ cd(Y, G) + d − e.

Proof. Let F: Y −−→ Y be a canonical form map such that dim F (G · y) = cd(Y, G) for Y in general position. Suppose F is induced by f: Y −−→ G, i.e., F (y) = f(y) · y, as in Definition 3.1.

Now consider f′ = f ∘ α: X −−→ G and the induced canonical form map F′: X −−→ X given by F′(x) = f′(x) · x. The relationship between F and F′ is illustrated by the following commutative diagram, where x is a point in general position in X and y = α(x) ∈ Y.

G · x −−→^{F′} F′(G · x)
α ↓  ↓ α
G · y −−→^{F} F (G · y) .

Each fiber of α: G · x ⟶ G · y has dimension d − e. Hence, each fiber of the right vertical map α_{|F′(G·x)} has dimension ≤ d − e. Applying the fiber dimension theorem to this map, we obtain

dim F′(G · x) ≤ dim F (G · y) + d − e = cd(Y, G) + d − e ,

and the proposition follows; cf. Lemma 3.6. □

Let X be a G-variety and H be a closed subgroup of G. Recall that an H-invariant (not necessarily irreducible) subvariety Y ⊂ X is called a (G, H)-section if (i) G · Y is dense in X and (ii) for y ∈ Y in general position, g · y ∈ Y ⇔ g ∈ H. Note that in some papers a (G, H)-section is called a relative section (cf. [PV, Section 2.8]) or a standard relative section with normalizer H (cf [Po, (1.7.6)]).

6.2. Corollary. Let X be an irreducible G-variety.

(a) If X has a (G, H)-section then cd(X, G) ≤ e(G, H) + d − dim(G) + dim(H), where d = dim(G · x) for x ∈ X in general position.

(b) If X has a stabilizer S in general position then

e(G, S) ≤ cd(X, G) ≤ e(G, N) − dim(S) + dim(N) ,

where N is the normalizer of S in G.

Proof. (a) The existence of a (G, H)-section is equivalent to the existence of a G-equivariant rational map X −−→ G/H; see [Po, Theorem 1.7.5]. Thus by Lemma 6.1, cd(X, G) ≤ cd(G/H, G) − d + dim(G/H). By Corollary 5.7(a) cd(G/H, G) = e(G, H), and part (a) follows.

(b) The inequality e(G, S) ≤ cd(X, G) follows from Proposition 5.5(a). To prove the inequality

(6.3)  cd(X, G) ≤ e(G, N) − dim(S) + dim(N) ,

note that by [Po, (1.7.8)], X has a (G, N)-section. Substituting H = N and d = dim(G) − dim(S) into the inequality of part (a), we obtain (6.3). □

7. The canonical dimension of a group

In this section we will define the canonical dimension of an algebraic group G. We begin with a simple lemma.

7.1. Lemma. Let X be an irreducible G-variety, and let Z be an irreducible variety with trivial action of G. Then cd(X × Z, G) = cd(X, G).

Proof. The inequality cd(X × Z, G) ≤ cd(X, G) follows from Lemma 6.1, applied to the projection map α: X × Z ⟶ X. To prove the opposite inequality, let c = cd(X × Z, G) and choose a canonical form map F: X × Z −−→ X × Z such that dim F (G · (x, z)) = c. Suppose F is induced by f: X × Z −−→ G, i.e., F (x, z) = f(x, z) · (x, z), as in Definition 3.1. It is now easy to see that for z_{0} ∈ Z in general position, the map f_{z_{0}} : X −−→ G given by f_{z_{0}}(x) = f(x, z_{0}) gives rise to a canonical form map F_{z_{0}} : X −−→ X such that dim F_{0}(G · x) = c. In other words, cd(X, G) ≤ c, as claimed. □

7.2. Proposition. Let V be a generically free linear representation of G.

(a) If X is an irreducible generically free G-variety then cd(X, G) ≤ cd(V, G).

(b) If W is another generically free G-representation, then cd(V, G) = cd(W, G).

Proof. (a) By [R, Corollary 2.20], there is a dominant rational map α: X × 𝔸^{d} −−→ V of G-varieties, where d = dim(V), and G acts trivially on 𝔸^{d}. Now

cd(X, G) =^{by Lemma 7.1} cd(X × 𝔸^{d}, G) ≤^{by Lemma 6.1} cd(V, G) ,

as claimed.

(b) cd(W, G) ≤ cd(V, G) by part (a). To prove the opposite inequality, interchange the roles of V and W. □

7.3. Definition. We define the canonical dimension cd(G) of an algebraic group G to be cd(V, G), where V is a generically free linear representation of G. By Proposition 7.2(a) this number is independent of the choice of V. Moreover, by Proposition 7.2(b) cd(G) = max{cd(X, G)}, as X ranges over all irreducible generically free G-varieties.

7.4. Corollary. Suppose W is a linear representation of G such that Stab_{G}(w) is finite for w ∈ W in general position. Then cd(G) ≤ cd(W, G).

Proof. Let V be a generically free linear representation of G. Then so is X = V × W. The desired inequality is now a consequence of Lemma 6.1, applied to the projection map α: V × W ⟶ W. □

7.5. Lemma. (a) cd(G) ≤ cd(H) + dim(G) − dim(H), for any closed subgroup H ⊂ G.

(b) cd(G) = cd(G^{0}).

(c) cd(G) = 0 if and only if G^{0} is special.

(d) cd(G_{1} × G_{2}) ≤ cd(G_{1}) + cd(G_{2}).

Proof. (a) Follows from Lemma 4.2, with X = V = generically free linear representation of G.

(b) Immediate from Lemma 4.4.

(c) By part (b), we may assume G = G^{0} is connected. The desired conclusion now follows from Lemma 4.8.

(d) Follows from Lemma 4.6, by taking X_{i} = V_{i} to be a generically free representation of G_{i} for i = 1, 2. □

7.6. Example. Consider the subgroup

H = \{(\begin{matrix}A & b_{1} ; ⋮ ; b_{n−1} \\ 0 … 0 & 1 \end{matrix}) | A ∈ GL_{n−1} , b_{1}, …, b_{n−1} ∈ k\} ,

of G = PGL_{n}. The Levi subgroup of H is special (it isomorphic to GL_{n−1}); hence, H itself is special. (This follows from the theorem of Grothendieck stated in Section 2.8 or alternatively, from [San, Lemma 1.13].) Since dim(H) = n^{2} − n, Lemma 7.5(a) yields cd(PGL_{n}) ≤ n − 1. In particular, cd(PGL_{2}) = 1. (Note that cd(PGL_{2}) ≥ 1 by Lemma 7.5(c)).

Alternatively, we can deduce the inequality cd(PGL_{n}) ≤ n−1 by applying Lemma 6.1 to the projection map M_{n} × M_{n} ⟶ M_{n} to the first factor, where PGL_{n} acts on M_{n} by conjugation. The PGL_{n}-action on M_{n} × M_{n} is generically free; hence, cd(PGL_{n}) = cd(M_{n} × M_{n}, PGL_{n}). On the other hand, cd(M_{n}, PGL_{n}) = 0; see Example 3.7. Now Lemma 6.1 tells us that

cd(PGL_{n}) = cd(M_{n} × M_{n}, PGL_{n}) ≤ cd(M_{n}, PGL_{n}) + n − 1 − 0 = n − 1 .

For a third proof of this inequality, see Example 9.9.

8. Splitting fields

Throughout this section we will assume that G/k is a connected linear algebraic group. Unless otherwise specified, the fields E, K, L, etc., are assumed to be finitely generated extensions of the base field k.

Let X be a generically free irreducible G-variety, E = k(X)^{G} = k(X/G), π: X −−→ X/G be the rational quotient map and F: X −−→ X be a canonical form map. Recall that F commutes with π, so that F (X) may be viewed as an algebraic variety over E.

8.1. Lemma. Let X be a generically free G-variety such that G-orbits in X are separated by regular invariants and let F: X −−→ X be a canonical form map. Suppose α ∈ H^{1}(E, G) is the class represented by X. Then for any field extension K/E the following conditions are equivalent:

(a) α_{K} = 1,

(b) X is rational over K,

(c) F (X) is unirational over K,

(d) K-points are dense in F (X),

(e) F (X) has a K-point.

Proof. We begin by proving the lemma in the case where K = E.

(a) ⇒ (b): If α = 1 then X is birationally isomorphic to X/G × G (over X/G). Now recall that the underlying variety of a connected algebraic group G is rational over k. Hence, X/G×G is rational over X/G, i.e., X is rational over E.

(b) ⇒ (c): The rational map F: X −−→ F (X) is, by definition, dominant. If X is rational, this makes F (X) unirational.

(c) ⇒ (d) and (d) ⇒ (e) are obvious.

(e) ⇒ (a): An E-point in F (X) is a rational section s: X/G −−→ F (X) ⊂ X for π. The existence of such a section implies that X is split, and hence, so is α; see Section 2.4.

To prove the general case, note that since the G-orbits in X are separated by regular invariants, we can choose a regular model of the rational quotient variety X/G, so that the rational quotient map π: X ⟶ X/G is regular and its fibers are exactly the G-orbits in X. After making X/G smaller if necessary, we may also assume that our field extension K/E is represented by a surjective morphism Y ⟶ X/G of algebraic varieties. Then α_{K} is represented by the G-variety X_{K} = X ×_{X/G} Y.

We claim that the morphism π_{K}: X_{K} ⟶ Y (projection to the second component) separates the G-orbits in X_{K}. Indeed, if for some x_{1}, x_{2} ∈ X, z_{1} = (x_{1}, y) and z_{2} = (x_{2}, y) ∈ X_{K} have the same second component then π(x_{1}) = π(x_{2}). We conclude that x_{1} and x_{2} are in the same G-orbit in X and consequently, z_{1} and z_{2} are in the same G-orbit in X_{K}, as claimed. This shows that π_{K} is a rational quotient map for the G-action on X_{K}; cf. [PV, Lemma 2.1].

We now define a rational map F_{K}: X_{K} −−→ X_{K} by F_{K}(x, y) = (F (x), y). Since π_{K} ∘F_{K} = π_{K}, F_{K} is a canonical form map; see Remark 3.2. Moreover, F_{K}(X_{K}) = F (X) ×_{X/G} Y. Replacing X by X_{K} and F by F_{K}, we reduce the lemma to the case we settled at the beginning of the proof (where K = E). □

Let α ∈ H^{1}(E, G). As usual, we will call a field extension K/E a splitting field for α if the image α_{K} of α under the natural map H^{1}(E, G) ⟶ H^{1}(K, G) is split. If α is represented by a generically free G-variety X, with k(X)^{G} = E then we will also sometimes say that K is a splitting field for X.

8.2. Proposition. Suppose α ∈ H^{1}(E, G).

(a) If cd(α) = 1 then there exist 0 ≠ a, b ∈ E such that a field extension K/E splits α if and only if the quadratic form q(x, y, z) = x^{2} + ay^{2} + bz^{2} is isotropic over K. In particular, α has a splitting field K/E of degree 2.

(b) If cd(α) = 2 then α has a splitting field K/E of degree 2, 3, 4, or 6.

Note that if K/E is a splitting field for α then [K : E] = 1 is impossible in either part. Indeed, otherwise α itself is split, and cd(α) = 0 by see Lemma 4.8(a).

Proof. Choose a canonical form map F: X −−→ X, such that dim F (X) − dim(X/G) = cd(α). By Lemma 8.1, F (X) is unirational over every splitting field K of α; in particular, it is unirational over the algebraic closure \overline{E} of E.

(a) Here F (X) is a curve over E, and Lüroth’s theorem tells us that F (X) is rational over \overline{E}. It is well known that any such curve is birationally isomorphic to a conic Z in ℙ_{E}^{2} (see, e.g., [MT, Proposition 1.1.1]) and that K-points are dense in Z if and only if Z(K) ≠ ∅ (see, e.g., [MT, Theorem 1.2.1]). Writing the equation of Z ⊂ ℙ_{E}^{2} in the form x^{2} + ay^{2} + bz^{2} = 0, we deduce the first assertion of part (a). The second assertion is an immediate consequence of the first; for example, K = E(\sqrt{−a}) is a splitting field for α.

(b) Here F (X) is a surface over E, which becomes unirational over the algebraic closure \overline{E}. By a theorem of Castelnuovo, F (X) is, in fact, rational over \overline{E}. Let Z be a complete smooth minimal surface, defined over E, which is birationally isomorphic to F (X) via ϕ: F (X) −−→^{∼} Z and let U ⊂ Z be an open subset such that ϕ is an isomorphism over U. Part (b) now follows from Lemma 8.1 and Lemma 8.3 below. □

8.3. Lemma. Let E be a field of characteristic zero, Z be a complete minimal surface defined over E and rational over \overline{E}, and let U be a dense open subset of Z (defined over E). Then U contains a K-point for some field extension K/E of degree 1, 2, 3, 4 or 6.

Proof. By the Enriques-Manin-Iskovskih classification Z is a conic bundle or a del Pezzo surface; see [I, Theorem 1] or [MT, Theorem 3.1.1]. Note that Z = ℙ^{2}, listed as a separate case in [I, Theorem 1], is, in fact, a del Pezzo surface. (We remark however, that the lemma is obvious in this case, since E-points are dense in ℙ_{E}^{2}.)

If f: Z ⟶ C is a conic bundle over a rational curve C, then after replacing E by a quadratic extension E′, we may assume that C_{E′} ≃ ℙ_{E′}^{1}. For every E′-point z ∈ C, f^{−1}(z) is a rational curve over E′. Taking z ∈ C_{E′} so that f^{−1}(z) ∩ U ≠ ∅, we can choose an extension K/E′ of degree 1 or 2 so that f^{−1}(z)_{K} ≃ ℙ_{K}^{1}. Now [K : E] = 1, 2 or 4, and K-points are dense in f^{−1}(z), so that one of them will lie in U.

From now on we may assume that Z is a del Pezzo surface. Recall that the anticanonical divisor −Ω_{Z} on a del Pezzo surface is ample, and the degree d = Ω_{Z} · Ω_{Z} can range from 1 to 9.

If d = 1 the linear system | − 2Ω_{Z}| defines a (ramified) double cover f: Z ⟶ Q, where Q is a quadric cone in ℙ_{E}^{3}; see [I, p. 30]. Then Q_{E′} ≃ ℙ_{E′}^{2} for some extension E′/E of degree 1 or 2. Now choose an E′-point x ∈ f(U) ⊂ Q and split f^{−1}(x) over a field extension K/E′ of degree 1 or 2. Then [K : E] = 1, 2 or 4 and U contains a K-point.

If d = 2 then the linear system | − Ω_{Z}| defines a (ramified) double cover Z ⟶ ℙ_{E}^{2} (see [I, p. 30]), and points of degree 2 are dense in Z.

If 3 ≤ d ≤ 9 then it is enough to show that Z(K) ≠ ∅ for some field extension K/E of degree 1, 2, 3, 4 or 6. Indeed, if Z(K) ≠ ∅ then Z_{K} is unirational over K (see [MT, Theorem 3.5.1]) and thus K-points are dense in Z. Note also that for 3 ≤ d ≤ 9, Z is isomorphic to a surface in ℙ^{d} of degree d. Intersecting this surface by two hyperplanes in general position, we see that Z has a point of degree dividing d. This proves the lemma for d = 3, 4, and 6.

For d = 5 and 7, Z always has an E-point (see [MT, Theorem 7.1.1]), so the lemma holds trivially in these cases. For d = 8, Z has a point of degree dividing 4 and for d = 9, Z has a point of degree dividing 3 (see [MT, p. 80]). The proof of the lemma is now complete. □

8.4. Example. Suppose α ∈ H^{1}(K, PGL_{n}) is represented by a central simple algebra of index d. Then the degree of every splitting field for α is divisible by

d (cf. e.g, [Row, Theorem 7.2.3]); hence, cd(α) ≥ \begin{cases}2, if d ≥ 3, \\ 3, if d ≠ 1, 2, 3, 4 or 6. \end{cases} .

In particular, cd(PGL_{n}) ≥ \begin{cases}2, if n ≥ 3, \\ 3, if n ≠ 1, 2, 3, 4 or 6. \end{cases} For sharper results on

cd(PGL_{n}), see Section 11.

8.5. Example. Let V be a generically free linear representation of G = F_{4}, E_{6} or E_{7} (adjoint or simply connected), K = k(V)^{G} and α ∈ H^{1}(K, G) be the class represented by the G-variety V. Then the degree of any splitting field L/K for α ∈ H^{1}(k(V)^{G}, G) is divisible by 6; [RY_{2}, p. 223]. We conclude that cd(G) ≥ 2 for these groups.

8.6. Example. cd(G) ≥ 3, if G = E_{8} or adjoint E_{7}; see [RY_{2}, Corollaries 5.5 and 5.8].

8.7. Remark. Let G be a connected linear algebraic group defined over k and let H be a finite abelian p-subgroup of G, where p is a prime integer. Recall that the depth of H is the smallest value of i such that [H : H∩T] = p^{i}, as T ranges over all maximal tori of G; see [RY_{2}, Definition 4.5]. Note that H has depth 0 if and only if it lies in a torus of G. A prime p is called a torsion prime for G if and only if G has a finite abelian p-subgroup of depth ≥ 1. (This is one of many equivalent definitions of torsion primes; see [St, Theorem 2.28].) The inequalities of Examples 8.4 - 8.6 may be viewed as special cases of the following assertion:

Suppose a connected linear algebraic group G has a p-subgroup H of depth d.

(a) If cd(G) ≤ 1 then p^{d} = 1 or 2.

(b) If cd(G) ≤ 2 then p^{d} = 1, 2, 3 or 4. The proof is immediate from Proposition 8.2 and [RY_{2}, Theorem 4.7] (where we take X to be a generically free linear representation of G).

9. Generic splitting fields

9.1. Definition. Let K/E be a (finitely generated) field extension. A (finitely generated) field extension L/E is said to be

(a) a specialization of K/E if there is a place ϕ: K ⟶ L ∪ {∞}, defined over E,

(b) a rational specialization of K/E if there is an embedding K ↪ L(t_{1}, …, t_{r}), over E, for some r ≥ 0.

In geometric language, (a) and (b) can be restated as follows. Suppose the field extensions K/E and L/E are induced by dominant rational maps V −−→ Z and W −−→ Z of irreducible algebraic k-varieties, respectively. Then

(a′) W is a specialization of V if there is a rational map W −−→ V, such that the diagram

\xymatrix{W \ar@{-->}[dr] \ar@{-->}[r] & V \ar@{-->}[d] \\  & Z}

commutes.

(b′) W is a rational specialization of V if for some r ≥ 0 there is a dominant map W × 𝔸^{r} −−→ V such that the diagram

\xymatrix{W × 𝔸^{r} \ar@{-->}[dr] \ar@{->>}[r] & V \ar@{-->}[d] \\  & Z}

commutes.

9.2. Remark. In the definition of rational specialization we may assume without loss of generality that r = max {0, trdeg_{E}(L)−trdeg_{E}(K)}; see [Roq_{2}, Lemma 1].

9.3. Definition. Let α ∈ H^{1}(E, G). A splitting field K/E for α is called generic (respectively, very generic) if every splitting field L/E for α is a specialization (respectively, a rational specialization) of K/E.

9.4. Remark. It is easy to see that a rational specialization is a specialization (cf. [Sal_{2}, Lemma 11.1]). Consequently, a very generic splitting field is generic.

9.5. Remark. The generic splitting field of α in Definition 9.3 is the same as the generic splitting field for the twisted group _{α}G defined by Kersten and Rehmann [KR].

9.6. Lemma. Let G be a connected algebraic group, X be an irreducible generically free G-variety, E = k(X)^{G} = k(X/G) and F: X −−→ X be a canonical form map. Then k(F (X))/E is a very generic splitting field for the class α ∈ H^{1}(E, G) represented by X.

Proof. After replacing X by a G-invariant open subset, we may assume that G-orbits in X are separated by regular invariants. The generic point of F (X) is a k(F (X))-point; hence, by Lemma 8.1, F (X)/E is a splitting field for α.

It remains to show that every splitting field L/E for α is a rational specialization of k(F (X))/E. After replacing X by a smaller G-invariant dense open subset, we may assume that L/E is induced by a surjective morphism Y ⟶ X/G of algebraic varieties. Then α_{L} = 1_{L} is represented by the generically free G-variety X_{L} = X ×_{X/G} Y. Since X_{L} is split, it is rational over L; see Lemma 8.1. The morphisms

X_{L} ⟶^{pr_{1}} X ⟶^{F} F (X) ⟶^{π} X/G

now tell us that k(F (X)) ↪ k(X_{L}) = L(t_{1}, …, t_{r}), over E, where t_{1}, …, t_{r} are independent variables and r = dim(G). (Here pr_{1} is the projection X_{L} = X ×_{X/G} Y ⟶ X to the first factor.) This shows that L/E is a rational specialization of k(F (X))/E, as claimed. □

9.7. Proposition. Let E/k be a finitely generated field extension. Then for every α ∈ H^{1}(E, G),

cd(α) = min {trdeg_{E}(K) | K/E is a generic splitting field for α } = min {trdeg_{E}(L) | L/E is a very generic splitting field for α }.

Proof. Let X be a generically free G-variety representing α; in particular, E = k(X)^{G} = k(X/G). Since cd(X, G) is, by definition, the minimal value of dim F (X)−dim(X/G) = trdeg_{E} k(F (X)), as F ranges over all canonical form maps X −−→ X, Lemma 9.6 tells us that

cd(α) = cd(X, G) ≥ min{trdeg_{E}(L) | L/E is a very splitting field for α }.

Now let K/E be a generic splitting field for α. It remains to show that

(9.8)  cd(X, G) ≤ trdeg_{E}(K)

Choose a variety Y whose function field k(Y) is K; the inclusion E ⊂ K then gives rise to a rational map Y −−→ X/G. By Lemma 9.6 (with F = id), k(X)/E is a very generic (and hence, a generic; cf. Remark 9.4) splitting field of α. Since k(X)/E and K/E are both generic splitting fields, each is a specialization of the other. Geometrically, this means that there exist rational maps f_{1} : X −−→ Y and f_{2} : Y −−→ X such that the diagram

\xymatrix{X \ar@{-->}[r]^{f_{1}} \ar@{-->}[dr] & Y \ar@{-->}[r]^{f_{2}} \ar@{-->}[d] & X \ar@{-->}[dl] \\  & X/G}

commutes. After replacing Y by the closure of the graph of f_{2} in Y × X, and f_{2} by the projection from this graph to X, we may assume that f_{2} is a morphism. Now F = f_{2} ∘ f_{1} is a well defined rational map X −−→ X which commutes with the rational quotient map π: X −−→ X/G. By Remark 3.2, F is a canonical form map. Thus

cd(X, G) ≤ dim F (X) − dim X/G ≤ dim f_{2}(Y) − dim X/G ≤
dim(Y) − dim(X/G) = trdeg_{k}(K) − trdeg_{k}(E) = trdeg_{E}(K)

Thus completes the proof of (9.8) and thus of Proposition 9.7. □

9.9. Example. Let α ∈ H^{1}(E, PGL_{n}) be represented by a central simple E-algebra A and let K be the function field of the Brauer-Severi variety of A. Then K/E is a very generic splitting field for α (see, e.g., [Sal_{2}, Corollary 13.9]) and trdeg_{E}(K) = n − 1. By Proposition 9.7, cd(α) ≤ n − 1. This gives yet another proof of the inequality cd(PGL_{n}) ≤ n − 1 of Example 7.6.

10. The canonical dimension of a functor

The results of the previous section naturally lead to the following definitions. Let F be a functor from the category Fields_{k} of finitely generated extensions of the base field k to the category Sets^{∗} of pointed sets. We will denote the marked element in F(E) by 1_{E} (and sometimes simply by 1, if the reference to the field E is clear from the context). Given a field extension L/E, we will denote the image of α ∈ F(E) in F(L) by α_{L}.

The notions of splitting field and generic splitting field naturally extend to this setting. That is, given α ∈ F(E), we will say that L/E is a splitting field for α if α_{L} = 1_{L}. We will call a splitting field K/E for α generic (respectively, very generic) if every splitting field L/E for α is a specialization (respectively, a rational specialization) of K/E. Moreover, we can now define cd(α) by

cd(α) = min {trdeg_{E}(K) | K/E is a generic splitting field for α }

and cd(F) by

cd(F) = max {cd(α) | E/k is a finitely generated extension, α ∈ F(E) }.

Proposition 9.7 says that if G is a connected linear algebraic group and F = H^{1}(_{−}, G) then the above definition of cd(α) agrees with Definition 3.5. Moreover, Definition 7.3 tells us that for this F, cd(F) = cd(G).

Note that none of the above definitions require k to be algebraically closed. In particular, it now makes sense to talk about the canonical dimension of an algebraic group defined over a non-algebraically closed field. This opens up interesting directions for future research, but we shall not pursue them in this paper. Instead we will continue to assume that k is algebraically closed, and our main focus will remain on the functors H^{1}(_{−}, G). However, even in this (more limited but already very rich) context, we will take advantage of the notion of canonical dimension for a functor by considering certain subfunctors of H^{1}(_{−}, G).

We also remark that it is a priori possible that for some functors F, some fields E/k and some α ∈ F(E) there will not exist a generic splitting field; if this happens, then, according to our definition, cd(α) = cd(F) = ∞. However, Proposition 9.7 tells us that this does not occur for any functor of the form H^{1}(_{−}, G), where G is a linear algebraic group, and consequently, for any of its subfunctors.

10.1. Example. Isomorphic functors clearly have the same canonical dimension. In particular, suppose G is a linear algebraic group and U is a normal unipotent subgroup of G. Then the natural map H^{1}(_{−}, G) ⟶ H^{1}(_{−}, G/U) is an isomorphism (see, e.g., [San, Lemma 1.13]) and hence, cd(G) = cd(G/U). Taking U to be the unipotent radical of G, we see that that cd(G) = cd(G_{red}), where G_{red} is the Levi subgroup of G.

The following simple lemma slightly extends the observation that isomorphic functors have the same canonical dimension. This lemma will turn out to be surprisingly useful in the sequel.

10.2. Lemma. Suppose τ: F_{1} ⟶ F_{2} is a morphism of functors with trivial kernel. Then for every finitely generated field extension E/k,

(a) cd(α) = cd(τ(α)) for any α ∈ F_{1}(E).

(b) cd(F_{1}) ≤ cd(F_{2}).

(c) Moreover, if τ is surjective then cd(F_{1}) = cd(F_{2}).

Proof. Since τ has trivial kernel, α and τ(α) have the same splitting fields and hence, the same generic splitting fields. This proves part (a). Parts (b) and (c) follow from part (a) and the definition of cd(F). □

10.3. Example. Recall that the cohomology set H^{1}(_{−}, PSO_{2n}) classifies pairs (A, σ), where A is a central simple algebras of degree 2n with an orthogonal involution σ of determinant 1; see [KMRT, p. 405]. (Note that [KMRT] uses the symbol PGO^{+} instead of PSO.) Consider the morphism of functors f: H^{1}(_{−}, SO_{2n}) ⟶ H^{1}(_{−}, PSO_{2n}) sending a quadratic form q of dimension 2n to the pair (M_{2n}(K), σ_{q}), where σ_{q} is the involution of M_{2n}(K) associated to q. We claim that f has trivial kernel. Indeed, q ∈ Ker(K) ⟺ q gives rise to the standard (transposition) involution on M_{2n}(K) ⟺ q is the 2ndimensional form ⟨a, a, …, a, a⟩ for some a ∈ K^{∗}; cf. e.g., [KMRT, p. 14]. On the other hand, since we are assuming that K contains an algebraically closed base field k of characteristic zero (and in particular, K contains a primitive 4th root of unity), the form ⟨a, a⟩ is hyperbolic (cf., e.g., [Lam, Theorem I.3.2]), and hence, so is q. This shows that f has trivial kernel, as claimed. Lemma 10.2(b) now tells us that cd(PSO_{2n}) ≥ cd(SO_{2n}).

10.4. Example. The exact sequence

1 ⟶ μ_{2} ⟶ Spin_{n} ⟶^{π} SO_{n} ⟶ 1

of algebraic groups gives rise to the exact sequence

\xymatrix{SO_{n}(_{−}) \ar[r]^{δ} & H^{1}(_{−}, μ_{2}) \ar[r] & H^{1}(_{−}, Spin_{n}) \ar[r]^{π_{∗}} & H^{1}(_{−}, SO_{n})}

of cohomology sets, where δ is the spinor norm; see, e.g., [Gar, p. 688]. Since −1 is a square in k, the unit form is hyperbolic, hence δ is surjective and thus π_{∗} has trivial kernel. On the other hand, the image of π_{∗} consists of quadratic forms q of discriminant 1 such that

q′ = \begin{cases}q, if n is even, \\ q ⊕ ⟨1⟩, if n is odd \end{cases}

has trivial Hasse-Witt invariant. Thus cd(Spin_{n}) = cd(HW_{n}), where HW_{n} is the set of n-dimensional quadratic forms q such that q′ has trivial discriminant and trivial Hasse-Witt invariant.

10.5. Example. Define the functors Pf_{r} and GPf_{r} by Pf_{r}(E) = r-fold Pfister forms defined over E and GPf_{r}(E) = scaled r-fold Pfister forms defined over E. In other words,

GPf_{r}(E) = {⟨c⟩ ⊗ q | c ∈ E^{∗}, q ∈ Pf_{r}(E)} .

Taking c = 1 above, we see that Pf_{r} is a subfunctor of GPf_{r}; hence, cd(Pf_{r}) ≤ cd(GPf_{r}). On the other hand, since q and ⟨c⟩ ⊗ q have the same splitting fields for every q ∈ Pf_{r}(E) and every c ∈ E^{∗}, we actually have equality cd(Pf_{r}) = cd(GPf_{r}).

Now suppose q ∈ Pf_{r}(E). Let q′ be a subform of q of dimension 2^{r−1} + 1. The argument in [KS, p. 29] shows that K = E(q′) is a generic splitting field for α. Recall that E(q′) is defined as the function field of the quadric hypersurface q′ = 0 in ℙ_{E}^{2^{r−1}}; in particular, trdeg_{E} E(q) = 2^{r−1} − 1. Proposition 9.7 now tells us that cd(q) ≤ 2^{r−1} − 1. On the other hand, if q is anisotropic, a theorem of Karpenko and Merkurjev [KM, Theorem 4.3] tells us that, in fact cd(q) = 2^{r−1} − 1. We conclude that

(10.6)  cd(GPf_{r}) = cd(Pf_{r}) = 2^{r−1} − 1 .

We remark that the setting considered by Karpenko and Merkurjev in [KM] is a bit different from ours in that they call a field K/E splitting for a quadratic form q/E if q_{K} is isotropic, where as we use this term to indicate that q_{K} is hyperbolic. However, if q is a Pfister form then the two definitions coincide; cf., e.g., [Lam, Corollary 10.1.6].

10.7. Example. Consider the exceptional group G_{2}. The functors H^{1}(_{−}, G_{2}) and Pf_{3} are isomorphic; see, e.g., [KMRT, Corollary 33.20]. Hence, cd(G_{2}) = cd(Pf_{3}) = 3; see (10.6).

11. Groups of type A

In this section we will study canonical dimensions of the groups GL_{n} /μ_{d} and SL_{n} /μ_{e}, where e divides n. We define the functor

(11.1)  C_{n,e}: Fields_{k} ⟶ Sets^{∗}

by C_{n,e}(E/k) = {isomorphism classes of central simple E-algebras of degree n and exponent dividing e}. The marked element in C_{n,e}(E/k) is the split algebra M_{n}(E). Clearly, C_{n,e} is a subfunctor of H^{1}(_{−}, PGL_{n}).

11.2. Lemma. Let n and d be positive integers and e be their greatest common divisor. Then cd(GL_{n} /μ_{d}) = cd(GL_{n} /μ_{e}) = cd(SL_{n} /μ_{e}) = cd(C_{n,d}) = cd(C_{n,e}).

Proof. By Lemma 2.6, there are surjective morphisms of functors H^{1}(_{−}, GL_{n} /μ_{d}) ⟶ C_{n,d} ,

H^{1}(_{−}, GL_{n} /μ_{e}) ⟶ C_{n,e} and
H^{1}(_{−}, SL_{n} /μ_{e}) ⟶ C_{n,e}

with trivial kernels. Basic properties of the index and the exponent of a central simple algebra tell us that C_{n,d} = C_{n,e}; the rest follows from Lemma 10.2(c). □

11.3. Lemma. Let n and e be positive integers such that e divides n, (a) If e′ | e and n′ | n then cd(SL_{n} /μ_{e}) ≥ cd(SL_{n′} /μ_{e′}).

(b) Suppose n = n_{1}n_{2} and e = e_{1}e_{2}, where e_{i} | n_{i} and n_{1}, n_{2} are relatively prime. Then

cd(SL_{n} /μ_{e}) = cd(SL_{n_{1}} /μ_{e_{1}} × SL_{n_{2}} /μ_{e_{2}}) ≤ cd(SL_{n_{1}} /μ_{e_{1}}) + cd(SL_{n_{2}} /μ_{e_{2}})

(c) Let n = ∏ p^{a_{p}} be the prime factorization of n (here the product is taken over all primes p and a_{p} = 0 for all but finitely many primes) and m = ∏_{p|e} p^{a_{p}}. Then cd(SL_{n} /μ_{e}) = cd(SL_{m} /μ_{e}),

Proof. (a) The morphism of functors C_{n′,e′} ⟶ C_{n,e} given by A ↦ M_{\frac{n}{n′}} (A) has trivial kernel. By Lemma 10.2(b), cd(C_{n,e}) ≥ cd(C_{n′,e′}). The desired inequality now follows from Lemma 11.2.

(b) First note that the functors

H^{1}(_{−}, SL_{n_{1}} /μ_{e_{1}} × SL_{n_{2}} /μ_{e_{2}}) and H^{1}(_{−}, SL_{n_{1}} /μ_{e_{1}}) × H^{1}(_{−}, SL_{n_{2}} /μ_{e_{2}})

are isomorphic. Thus by Lemma 2.6, there is a surjective morphism of functors

H^{1}(_{−}, SL_{n_{1}} /μ_{e_{1}} × SL_{n_{2}} /μ_{e_{2}}) ⟶ C_{n_{1},e_{1}} × C_{n_{2},e_{2}}

with trivial kernel. By Lemma 10.2(c),

cd(SL_{n_{1}} /μ_{e_{1}} × SL_{n_{2}} /μ_{e_{2}}) = cd(C_{n_{1},e_{1}} × C_{n_{2},e_{2}})

and by Lemma 11.2, cd(SL_{n} /μ_{e}) = cd(C_{n,e}). The equality in part (b) now follows from the fact that for relatively prime n_{1} and n_{2} the functors C_{n_{1}, e_{1}} × C_{n_{2}, e_{2}} and C_{n,e} are isomorphic via (A_{1}, A_{2}) ↦ A_{1} ⊗ A_{2}. The inequality in part (b) is a special case of Lemma 7.5(d).

(c) By Lemma 11.2, cd(SL_{n} /μ_{e}) = cd(C_{n,e}) and cd(SL_{m} /μ_{e}) = cd(C_{m,e}). On the other hand, basic properties of the index and the exponent of a central simple algebra tell us that the functors C_{m,e} and C_{n,e} are isomorphic via A ↦ M_{n/m}(A). □

11.4. Theorem. Let α ∈ H^{1}(E, PGL_{n}) be the class of a division algebra A of degree n = p^{i}, where p is a prime. Then cd(α) = n − 1.

Let X be the Brauer-Severi variety of A. By a theorem of Karpenko [K_{1}, Theorem 2.1] every rational map X −−→ X defined over E is necessarily dominant. (For a related stronger result, see Merkurjev [M_{2}, Section 7.2].) Theorem 11.4 is an easy consequence of this fact; we outline the argument below.

Proof. The function field K = E(X) is a generic splitting field for A; in particular, as we pointed out in Example 9.9, cd(α) ≤ n − 1. To prove the opposite inequality, assume the contrary: A has a generic splitting field L/E of transcendence degree < n − 1. Let Y be a variety (defined over E) with function field L. Since k(X)/E and K/E are both generic splitting fields for α, each is a specialization of the other. Arguing as in the proof of Proposition 9.7, we see that there exist rational maps f_{1} : X −−→ Y and f_{2} : Y −−→ X such that the diagram

\xymatrix{X \ar@{-->}[r]^{f_{1}} \ar@{-->}[dr] & Y \ar@{-->}[r]^{f_{2}} \ar@{-->}[d] & X \ar@{-->}[dl] \\  & X/G}

commutes. Moreover, after replacing Y by the closure of the graph of f_{2} in Y × X, we may assume that f_{2} is regular. Now f_{2} ∘ f_{1} is a well defined rational map X −−→ X, and

dim f_{2} ∘ f_{1}(X) ≤ dim(Y) < n − 1 = dim(X) ,

contradicting Karpenko’s theorem. This concludes the proof of Theorem 11.4. □

11.5. Corollary. Suppose n = p^{i}n_{0} and e = p^{j}, where gcd(p, n_{0}) = 1 and

i ≥ j. Then cd(SL_{n} /μ_{e}) = \begin{cases}0, if j = 0, \\ p^{i} − 1, if j ≥ 1. \end{cases}

Proof. If j = 0 then SL_{n} /μ_{e} = SL_{n} is special and hence, has canonical dimension 0. Thus we only need to consider the case where j ≥ 1. By Lemma 11.3(c), cd(SL_{n} /μ_{e}) = cd(SL_{p^{i}} /μ_{p^{j}}). Thus we may also assume that n_{0} = 1, i.e., n = p^{i}.

By Lemma 11.2, cd(SL_{n} /μ_{e}) = cd(C_{n,e}). Since C_{n,e} is, by definition, a subfunctor of H^{1}(_{−}, PGL_{n}), Example 7.6 tells us that cd(SL_{n} /μ_{e}) ≤ n − 1 = p^{i} − 1. To prove the opposite inequality, let A be a division algebra of degree p^{i} and exponent p^{j} (such algebras are known to exist; see, e.g., [Row, Appendix 7C]) and let α be the class of A in H^{1}(E, PGL_{n}). Then by Theorem 11.4, cd(SL_{n} /μ_{e}) ≥ cd(α) = n − 1, as desired. □

11.6. Corollary. Let n = 2^{i}n_{0}, where i ≥ 1 and n_{0} is odd. Then cd(PSp_{n}) = 2^{i} − 1.

Here PSp stands for the projective symplectic group. Note that these groups are sometimes denoted by the symbol PGSp; see, e.g., [KMRT, p. 347].

Proof. Recall that every α ∈ H^{1}(_{−}, PSp_{n}) is represented by a pair (A, σ), where A is a central simple algebra of degree 2n and exponent ≤ 2, and σ is a symplectic involution on A. A central simple algebra has a symplectic involution if and only if its exponent is 1 or 2; moreover, a symplectic involution of a split algebra is necessarily hyperbolic. In other words, the morphism of functors

H^{1}(_{−}, PSp_{n}) ⟶ C_{n,2} ,

given by α ↦ A is surjective and has trivial kernel. Here C_{n,2} is the functor of central simple algebras of degree n and exponent dividing 2, as in (11.1). Thus

cd(PSp_{n}) = cd(C_{n,2}) =^{by Lemma 11.2} cd(SL_{n} /μ_{2}) =^{by Corollary 11.5} 2^{i} − 1 ,

as claimed. □

12. Orthogonal and Spin groups

12.1. Lemma. (a) cd(SO_{n−1}) ≤ cd(SO_{n}) for every n ≥ 2. Moreover, equality holds if n is even.

(b) cd(Spin_{n−1}) ≤ cd(Spin_{n}) for every n ≥ 2. Moreover, equality holds if n is even.

(c) cd(SO_{n}) ≥ cd(Spin_{n}) for every n ≥ 2. Moreover, if n ≥ 2^{r}, where r ≥ 3 is an integer, then cd(Spin_{n}) ≥ 2^{r−1} − 1.

Proof. (a) The morphism τ: H^{1}(_{−}, SO_{n−1}) → H^{1}(_{−}, SO_{n}), sending a quadratic form q to ⟨1⟩ ⊕ q has trivial kernel. Lemma 10.2(b) now tells us that cd(SO_{n−1}) ≤ cd(SO_{n}).

To prove the opposite inequality for n is even, let q = ⟨a_{1}, …, a_{n}⟩ ∈ SO_{n}. Then q = ⟨a_{1}⟩ ⊗ q̃, where q̃ = ⟨1, a_{1}a_{2}, …, a_{1}a_{n}⟩ lies in the image of τ. (Note that here we use the assumption that n is even to conclude that q̃ has discriminant 1.) Since q and q̃ have the same splitting fields, cd(q) = cd(q̃). On the other hand, since q̃ lies in the image of τ, Lemma 10.2(a) tells us that cd(q̃) ≤ cd(SO_{n−1}). Thus cd(q) ≤ cd(SO_{n−1}) and consequently, cd(SO_{n}) ≤ cd(SO_{n−1}), as desired.

(b) is proved by the same argument as (a), using the identity cd(Spin_{n}) = cd(HW_{n}) of Example 10.4. The first assertion follows from the fact that τ restricts to a morphism HW_{n−1} ⟶ HW_{n}. In the proof of the second assertion, the key point is that if a quadratic form q = ⟨a_{1}⟩ ⊗ q̃, of even dimension n, has trivial discriminant and trivial Hasse-Witt invariant then so does q̃; the rest of the argument goes through unchanged.

(c) The first inequality follows from the fact that HW_{n} is a subfunctor of H^{1}(_{−}, SO_{n}). To prove the second inequality, note that by part (b) we may assume n = 2^{r}. Since the discriminant and the Hasse-Witt invariant of an r-fold Pfister form are both trivial for any r ≥ 3, we see that Pf_{r} is a subfunctor of HW_{n} and thus

cd(Spin_{n}) = cd(HW_{n}) ≥ cd(Pf_{r}) = 2^{r−1} − 1 ;

see Example 10.5. □

12.2. Example. (a) cd(SO_{n}) = \begin{cases}0, if n = 1 or 2, \\ 1, if n = 3 or 4, \\ 3, if n = 5 or 6. \end{cases}

(b) cd(Spin_{n}) = 0 for n = 3, 4, 5 or 6.

(c) cd(Spin_{n}) = 3 for n = 7, 8, 9 or 10.

Proof. (a) Note that SO_{1} = {1}, SO_{3} ≃ PGL_{2}, and SO_{6} ≃ SL_{4} /μ_{2}. Hence, cd(SO_{1}) = 0, and (by Corollary 11.5) cd(SO_{3}) = 1 and cd(SO_{6}) = 3. The remaining cases follow from Lemma 12.1(a).

(b) In view of Lemma 12.1(b), it is enough to show that cd(Spin_{6}) = 0. By the Arason-Pfister theorem [Lam, Theorem 10.3.1], the only 6-dimensional form with trivial discriminant and trivial Hasse-Witt invariant is the split form. In other words, HW_{6} is the trivial functor and thus

cd(Spin_{6}) = cd(HW_{6}) = 0 .

Alternative proof of (b): Exceptional isomorphisms of simply connected simple groups tell us that Spin_{3} ≃ SL_{2}, Spin_{5} ≃ Sp_{4} and Spin_{6} ≃ SL_{4} are all special and hence, have canonical dimension 0; cf. Lemma 7.5(c).

(c) Using the Arason-Pfister theorem once again, we see that every 8dimensional quadratic form with trivial discriminant and trivial Hasse-Witt invariant, is a scaled Pfister form; see [Lam, Corollary 10.3.3]. Thus

cd(Spin_{7}) = cd(Spin_{8}) = cd(GPf_{3}) = 3 ;

see Example 10.5. On the other hand, by a theorem of Pfister every q ∈ HW_{10} is isotropic, i.e., has the form ⟨1, −1⟩ ⊕ q′, where q′ ∈ HW_{8}; see [Pf, Proof of Satz 14] (cf. also [KM, Theorem 4.4]).

Applying Lemma 10.2(a) to the morphism HW_{8} ⟶ HW_{10} given by q′ ⟶ ⟨1, −1⟩ ⊕ q′, we see that

cd(q) = cd(q′) ≤ cd(HW_{8}) = cd(Spin_{8}) .

This shows that cd(Spin_{9}) = cd(Spin_{10}) ≤ cd(Spin_{8}) = 3. The opposite inequality is given by Lemma 12.1(b). □

12.3. Proposition. cd(SO_{2m}) ≤ \frac{m(m−1)}{2} for every m ≥ 1.

Proof. Write SO_{2m} = SO(q), where q is a non-degenerate quadratic form on k^{2m}. Let X = Gr_{iso}(m, 2m) be the Grassmannian of maximal (i.e., mdimensional) q-isotropic subspaces of k^{2m}, i.e., of m-dimensional subspaces contained in the quadric Q ⊂ k^{2m} given by q = 0. It is well known that X is a projective variety with two irreducible components X_{1} and X_{2}, each of dimension \frac{m(m−1)}{2}; see e.g., [GH, Section 6.1]. Using the Witt Extension Theorem (see, e.g., [Lam, p. 26]), it is easy to see that the full orthogonal group O(q) acts transitively on X and SO(q) acts transitively on each component X_{i} (i = 1, 2). Fix an isotropic subspace L ∈ X_{1} and let P = Stab_{SO(q)}(L). By Lemma 7.5(a), with G = SO(q) and H = P, we have

cd(SO_{2m}) ≤ cd(P) + dim(SO_{2m}) − dim(P) =

cd(P) + dim(X_{1}) = cd(P) + \frac{m(m − 1)}{2} .

It remains to show that cd(P) = 0. We claim that the Levi subgroup of P is naturally isomorphic to GL(L) ≃ GL_{m} via f: P ⟶ GL(L), where f(g) = g_{| L}. Once this claim is established, Example 10.1 tells us that cd(P) = cd(GL_{m}) = 0. (The last equality follows from the fact that GL_{m} is special.)

To prove the claim, note that by the Witt Extension Theorem, f is a surjective homomorphism. It remains to show that Ker(f) is unipotent. Indeed, choose a basis e_{1}, …, e_{2m} of k^{2m} so that

q(x_{1}e_{1} + ⋯ + x_{2m}e_{2m}) = x_{1}x_{m+1} + ⋯ + x_{m}x_{2m}

and L is the span of e_{1}, …, e_{m}. Then every g ∈ Ker(f) has the form

(12.4)  g = (\begin{matrix}I_{m} & A \\ O_{m} & B \end{matrix}) ,

for some m × m-matrices A and B. (Here O_{m} and I_{m} are, respectively, the zero and the identity m × m-matrices.) The condition that g ∈ O(q) translates into

(12.5)  g (\begin{matrix}O_{m} & I_{m} \\ I_{m} & O_{m} \end{matrix}) g^{Transpose} = (\begin{matrix}O_{m} & I_{m} \\ I_{m} & O_{m} \end{matrix}) .

Substituting (12.4) into (12.5), we see that B = I_{m}. Formula (12.4) now shows that g is unipotent; consequently, Ker(f) is a unipotent group, as claimed. □

12.6. Conjecture. cd(SO_{2m−1}) = cd(SO_{2m}) = \frac{m(m−1)}{2} for every m ≥ 1.

12.7. Remark. Conjecture 12.6 was recently proved by Karpenko [K_{2}]; for an alternative proof due to Vishik, see [V].

13. Groups of low canonical dimension

13.1. Theorem. Assume that G is simple. Then cd(G) = 1 if and only if G ≃ SL_{2m} /μ_{2} or PSp_{2m}, where m is an odd integer.

Proof. First of all, observe that if G ≃ SL_{2m} /μ_{2} or PSp_{2m}, with m odd, then indeed, cd(G) = 1; see Corollary 11.5 and Corollary 11.6. Thus we only need to show that no other simple group has this property. Our proof relies on the classification of simple algebraic groups; cf., e.g., [KMRT, §24 and 25]. (Note that [KMRT] uses the symbols O^{+} and PGSp instead of SO and PSp. Recall also that we are working over an algebraically closed base field k of characteristic zero.) We begin by observing that cd(G) ≥ 2 for every simple group of exceptional type; see Examples 8.5, 8.6 and 10.7.

Now suppose cd(G) = 1 and G is of type A. Then G ≃ SL_{n} /μ_{e}, where e divides n. Let p be a prime dividing e. Then we can write e = p^{j}e_{0} and n = p^{i}n_{0}, where i ≥ j ≥ 1, and gcd(p, e_{0}) = gcd(p, n_{0}) = 1. Then

1 = cd(G) ≥^{by Lemma 11.3(a)} SL_{p^{i}} /μ_{p^{j}} ≥^{by Corollary 11.5} p^{i} − 1 ,

which is only possible if p = 2 and i = 1. This implies that e cannot be divisible by 4 or by any prime p ≥ 3; in other words, e = 1 or 2. If e = 1 then G = SL_{n} is special and thus cd(G) = 0. If e = 2 then i = 1 implies that n ≡ 2 (mod 4), as claimed.

Next suppose G is of type C. Then G is isomorphic to Sp_{2m} or PSp_{2m}. The groups Sp_{2m} are special and thus have canonical dimension 0. By Corollary 11.6, cd(PSp_{2m}) = 1 if and only if m is odd. This completes the proof of Theorem 13.1 for groups of type A or C.

Now suppose G is of type B or D. We have already considered some of these groups. In particular,

• cd(SO_{n}) ≥ cd(SO_{5}) = 3 for any n ≥ 5 (see Lemma 12.1(a) and Example 12.2(a)),

• cd(PSO_{2n}) ≥ cd(SO_{2n}) ≥ 3 for any n ≥ 3 (see Example 10.3),

• cd(Spin_{n}) = 0 for n = 3, 4, 5, 6 (see Example 12.2(b)), and

• cd(Spin_{n}) ≥ cd(Spin_{7}) = 3 for any n ≥ 7 (see Lemma 12.1 and Example 12.2(c)).

We also remark that SO_{2} ≃ 𝔾_{m} and SO_{4} ≃ (SL_{2} × SL_{2})/μ_{2} are not simple, and SO_{3} ≃ PGL_{2} = SL_{2} /μ_{2} was considered above.

This covers every simple group of type B; the only simple groups of type D we have not yet considered are G = Spin_{4n}^{±} (n ≥ 2); cf., e.g., [KMRT, Theorems 25.10 and 25.12]. The natural projection π: Spin_{4n} ⟶ SO_{4n} factors through Spin_{4n}^{±}:

π: Spin_{4n} ⟶^{f} Spin_{4n}^{±} ⟶ SO_{4n} .

Since π_{∗}: H^{1}(_{−}, Spin_{4n}) ⟶ H^{1}(_{−}, SO_{4n}) has trivial kernel (see Example 10.4), so does f_{∗}: H^{1}(_{−}, Spin_{4n}) ⟶ H^{1}(_{−}, Spin_{4n}^{±}). Now for any n ≥ 2,

cd(Spin_{4n}^{±}) ≥^{by Lemma 10.2(b)} cd(Spin_{4n}) ≥^{by Lemma 12.1(b)} cd(Spin_{8}) =^{by Example 12.2(c)} 3 .

This completes the proof of Theorem 13.1. □

13.2. Remark. The above argument also shows that if a simple classical group G has canonical dimension 2 then either (i) G ≃ SL_{3m} /μ_{3}, where m is prime to 3 or possibly (ii) G ≃ SL_{6m} /μ_{6}, where m is prime to 6. In case (i), we know that cd(G) = cd(PGL_{3}) = 2; see Corollary 11.5. In case (ii), cd(G) = cd(PGL_{6}) (see Corollary 11.3(c)); we do not know whether this number is 2 or 3.

14. The functor of orbits and homogeneous forms

We now briefly recall the definition of essential dimension of a functor, due to Merkurjev [M_{1}].

Let F be a functor from the category of all field extensions K of k to the category of sets. (For our purposes, it is sufficient to consider only finitely generated extensions K/k.) Given α ∈ F(K), we define ed(α) as the minimal value of trdeg_{k}(K_{0}), where k ⊂ K_{0} ⊂ K and α lies in the image of the natural map F(K_{0}) ⟶ F(K). The essential dimension ed(F) of the functor F is then defined as the maximal value of ed(α), as α ranges over F(K) and K ranges over all field extensions of k. In the special case, where G is an algebraic group and F = H^{1}(_{−}, G) we recover the numbers defined in Section 2.1: ed(α) = ed(X, G), where α ∈ H^{1}(K, G) and X is a generically free G-variety representing α. Moreover, ed(H^{1}(_{−}, G)) = ed(G). For details, see [BF_{2}].

Now to each G-variety X we will associate the functor Orb_{X,G} given by Orb_{X,G}(L) = X(L)/ ∼, where a ∼ b for a, b ∈ X(L), if a = g · b for some g ∈ G(L). Given an L-point a ∈ X(L), we shall denote a (mod ∼) by [a] ∈ Orb_{X,G}(L). Using this terminology, Definition 3.5 can be rewritten as follows.

14.1. Proposition. ed [η] = cd(X, G) + dim X/G, where η ∈ X(k(X)) is the generic point of X.

Proof. Let Y be a variety with function field k(Y) = L. Then z ∈ X(L) may be viewed as a rational map ϕ_{z}: Y −−→ X and g ∈ G(L) as a rational map f_{g}: Y −−→ G. The point g · z of X(L) corresponds to the map F_{z,g}: Y −−→ X given by F_{z,g}(y) = f_{g}(y) · ϕ_{z}(y). Consequently, the definition of ed [z] can be rewritten as

(14.2)  ed [z] = min_{g∈G(L)} { trdeg_{k} k(F_{z,g}(Y)) } .

Now set z = η, L = k(X), Y = X, and ϕ = id_{X}. The element g ∈ G(L) is then a rational map f = f_{g}: X −−→ G, F = F_{z,g}: X −−→ X is, by definition, a canonical form map (see Definition 3.1) and the proposition follows from (14.2) and Definition 3.5. □

14.3. Remark. By the definition of ed(Orb_{X,G}), we have

ed(Orb_{X,G}) ≥ ed [η] .

We do not know whether or not equality holds in general.

For the rest of this paper we will focus on the following example. Let

N = (\begin{matrix}n + d − 1 \\ d \end{matrix}) and let X = 𝔸^{N} be the space of degree d forms in n

variables x = (x_{1}, …, x_{n}). That is, elements of 𝔸^{N} are forms p(x_{1}, …, x_{n}) of degree d and elements of ℙ^{N−1} are hypersurfaces p(x_{1}, …, x_{n}) = 0. The generic point of 𝔸^{N} is the “general” degree d form in n variables as

ϕ_{n,d}(x) = ∑_{i_{1}+···+i_{n}=d} a_{i_{1},...,i_{n}}x_{1}^{i_{1}} … x_{n}^{i_{n}} ∈ K[x_{1}, …, x_{n}] ,

where a_{i_{1},...,i_{n}} are independent variables, K = k(𝔸^{N}) is the field these variables generate over k. Then

(14.4)  ed [ϕ_{n,d}] = min_{g∈GL_{n}(K)} trdeg_{k} k(b_{i_{1},...,i_{d}} |i_{1} + ⋯ + i_{n} = d) ,

where

ϕ_{n,d}(g · x) = ∑_{i_{1}+···+i_{n}=d} b_{i_{1},...,i_{n}}x_{1}^{i_{1}} … x_{n}^{i_{n}} .

The generic point of ℙ^{N−1} is the “general” degree d hypersurface ϕ_{n,d}(x) = 0 in ℙ^{N−1}(K), which we denote by H_{n,d}.

14.5. Lemma. Let N = (\begin{matrix}n + d − 1 \\ d \end{matrix}) and

D = dim(𝔸^{N}/ GL_{n}) = dim(ℙ^{N−1}/ PGL_{n}) = dim(𝔸^{N}/(𝔾_{m} × GL_{n})) .

(Here 𝔾_{m} acts on 𝔸^{N} by scalar multiplication.) Then (a) ed [ϕ_{n,d}] = D + cd(𝔸^{N}, GL_{n}).

(b) ed [H_{n,d}] = D + cd(ℙ^{N−1}, GL_{n}) = D + cd(ℙ^{N−1}, PGL_{n}) = = D + cd(𝔸^{N}, 𝔾_{m} × GL_{n}).

Proof. Part (a) and the first equality in part (b) are immediate consequences of Proposition 14.1. To complete the proof of (b), note that

cd(ℙ^{N−1}, GL_{n}) = cd(ℙ^{N−1}, PGL_{n})

by Lemma 4.10(b), and

cd(ℙ^{N−1}, GL_{n}) = cd(𝔸^{N}, 𝔾_{m} × GL_{n})

by Proposition 4.11 (with H = 𝔾_{m} × {1}). □

14.6. Corollary. Let K = k(𝔸^{N}) (as above) and K_{0} = k(ℙ^{N}). Given g ∈ GL(K), let F_{g} be the field extension of k generated by elements of the form

\frac{b_{i_{1},...,i_{n}}}{b_{j_{1},...,j_{n}}}, where ϕ_{n,d}(g · x) = ∑_{i_{1}+···+i_{n}=d} b_{i_{1},...,i_{n}}x_{1}^{i_{1}} … x_{n}^{i_{n}},

and i_{1}, …, i_{n}, j_{1}, …, j_{n} ≥ 0 satisfy i_{1} + ⋯ + i_{n} = j_{1} + ⋯ + j_{n} = d and b_{j_{1},...,j_{n}} ≠ 0. Then

ed [H_{n,d}] = min_{g∈GL_{n}(K_{0})} trdeg_{k}(F_{g}) = min_{g∈GL_{n}(K)} trdeg_{k}(F_{g}) .

Proof. The first equality is an immediate consequence of the definition of ed [H_{n,d}]. To prove the second equality, we use the identity ed [H_{n,d}] = D + cd(𝔸^{N}, 𝔾_{m} × GL_{n}) of Lemma 14.5. Proposition 14.1 now tells us that

ed [H_{n,d}] = min_{g∈GL_{n}(K), c∈K^{∗}} trdeg_{k}(cb_{i_{1},...,i_{n}} | i_{1} + ⋯ + i_{n} = d) .

The minimum is clearly attained if c = b_{j_{1},...,j_{n}}^{−1}, for some (and thus any) j_{1}, …, j_{n} such that b_{j_{1},...,j_{n}} ≠ 0, and the corollary follows. □

In view of (14.4), it is natural to think of ed [ϕ_{n,d}] as the minimal number of independent parameters required to define the general form of degree d in n variables. Corollary 14.6 says that ed [H_{n,d}] can be similarly interpreted as the minimal number of independent parameters required to define the general degree d hypersurface in ℙ^{n−1}. Comparing the expressions for these numbers given by (14.4) and Corollary 14.6, we see that they are closely related.

14.7. Corollary. ed [H_{n,d}] ≤ ed [ϕ_{n,d}] ≤ ed [H_{n,d}] + 1. □

14.8. Remark. Lemma 14.5(b) shows that the number ed [H_{n,d}] is the essential dimension of the generic form ϕ_{n,d} in the sense of [BF_{1}], i.e., the essential dimension of the 𝔾_{m} × GL_{n}-orbit of the generic point of 𝔸^{N}. Note that the emphasis in [BF_{1}] is on the essential dimension of the functor Hypersurfaces_{n,d} = Orb_{𝔸^{N} ,𝔾_{m}×GL_{n}} (which is denoted there by F_{d,n}), and, more specifically, on the functor Hypersurfaces_{3,3} (which is denoted there by Cub_{3}). As we pointed out in Remark 14.3, ed(Hypersurfaces_{n,d}) ≥ ed [H_{n,d}] but we do not know whether or not equality holds.

15. Essential dimensions of homogeneous forms I

15.1. Theorem. Let n and d be positive integers such that d ≥ 3 and (n, d) ≠ (2, 3), (2, 4) or (3, 3). Then

ed [H_{n,d}] = N − n^{2} + cd(GL_{n} /μ_{d}) = N − n^{2} + cd(SL_{n} /μ_{gcd(n,d)}) ,

where N = (\begin{matrix}n + d − 1 \\ d \end{matrix}).

Proof. First observe that by Lemma 11.2, cd(GL_{n} /μ_{d}) = cd(SL_{n} /μ_{gcd(n,d)}), so only the first equality needs to be proved.

Secondly, under our assumption on n and d, the PGL_{n}-action on ℙ^{N−1} is generically free. For n = 2 this is classically known (cf., e.g., [PV, p. 231]), for n = 3, this is proved in [B] and for n ≥ 4 in [MM]. Substituting

D = dim(ℙ^{N−1}/ PGL_{n}) = dim(ℙ^{N−1}) − dim(PGL_{n}) = N − n^{2}

into Lemma 14.5(b), we reduce the theorem to the identity

(15.2)  cd(𝔸^{N}, 𝔾_{m} × GL_{n}) = cd(GL_{n} /μ_{d}) .

To prove (15.2), observe that the normal subgroup

S = {(t^{−d}, t) | t ∈ 𝔾_{m}} ⊂ 𝔾_{m} × 𝔾_{m} ⊂ 𝔾_{m} × GL_{n}

acts trivially on 𝔸^{N}. Since S is special, we have

(15.3)  cd(𝔸^{N}, 𝔾_{m} × GL_{n}) =^{by Lemma 4.10(b)}
cd(𝔸^{N}, (𝔾_{m} × GL_{n})/S) =^{by Definition 7.3} cd((𝔾_{m} × GL_{n})/S) ,

where the last equality is a consequence of the fact that the PGL_{n}-action on ℙ^{N−1} (and hence, the (𝔾_{m} × GL_{n})/S-action on 𝔸^{N}) is generically free.

Finally, consider the homomorphism GL_{n} ⟶ (𝔾_{m} × GL_{n})/S given by g ↦ (1, g), modulo S. Since we are working over an algebraically closed field k of characteristic zero, this homomorphism is surjective, and its kernel is exactly μ_{d}. Thus (𝔾_{m} × GL_{n})/S ≃ GL_{n} /μ_{d}. Combining this with (15.3), we obtain (15.2). □

The results of Section 11 can now be used to determine ed [H_{n,d}] for many values of n and d (and produce estimates for others). In particular, combining Theorem 15.1 with Corollary 11.5, we deduce Theorem 1.1 stated in the Introduction.

The number ed [ϕ_{n,d}] appears to be harder to compute than ed [H_{n,d}]. By Corollary 14.7, ed [ϕ_{n,d}] = ed [H_{n,d}] or ed [ϕ_{n,d}] = ed [H_{n,d}] + 1, but for general n and d, we do not know which of these cases occurs. One notable exception is the case where n and d are relatively prime.

15.4. Corollary. Suppose d ≥ 3, gcd(n, d) = 1 and (n, d) ≠ (2, 3). Then

(a) ed [H_{n,d}] = (\begin{matrix}n + d − 1 \\ d \end{matrix}) − n^{2} and

(b) ed [ϕ_{n,d}] = (\begin{matrix}n + d − 1 \\ d \end{matrix}) − n^{2} + 1.

Proof. Part (a) is a special case of Theorem 1.1 (with j = 0). We can also deduce it directly from Theorem 15.1 by noting that SL_{n} is a special group and thus cd(SL_{n}) = 0.

(b) In view of Corollary 14.7, we only need to prove that ed [ϕ_{n,d}] ≥ ed [H_{n,d}] + 1 or equivalently, cd(𝔸^{N}, GL_{n}) ≥ 1; see Lemma 14.5(a). Recall that the central subgroup μ_{d} of GL_{n} acts trivially on 𝔸^{N}, and (under our assumptions on n and d) the induced GL_{n} /μ_{d}-action is generically free. Thus the stabilizer in general position for the GL_{n}-action on 𝔸^{N} is μ_{d}, and

cd(𝔸^{N}, GL_{n}) ≥^{by Proposition 5.5(c)} ed(μ_{d}) =^{by [BR, Theorem 6.2]} 1 ,

as claimed. □

15.5. Remark. We have proved that if n and d are relatively prime then cd(ℙ^{N−1}, GL_{n}) = 0 but cd(𝔸^{N}, GL_{n}) = 1. In particular, this shows that the equality cd(X, G) = cd(X/H, G) of Proposition 4.11 fails for X = 𝔸^{N}, G = GL_{n} and H = 𝔾_{m}. Note that Proposition 4.11 does not apply in this situation because the H-action on X is not generically free (the subgroup μ_{d} acts trivially).

16. Essential dimensions of homogeneous forms II

In this section we will study ed [ϕ_{n,d}] and ed [H_{n,d}] for the pairs (n, d) not covered by Theorem 15.1. We begin with a simple lemma.

16.1. Lemma. ed [H_{2,d}] ≤ d − 2 for any d ≥ 3.

In the sequel we will only need this lemma for d = 3 and 4. For n ≥ 5, Theorem 1.1 (with n = 2) gives a stronger result, namely,

ed [H_{2,d}] = \begin{cases}d − 2 if d is even, \\ d − 3 if d is odd. \end{cases}

However, the proof of the lemma below is valid for any d ≥ 3.

Proof. The linear transformation g ∈ GL_{2}(K) given by

x_{1} ↦ x_{1} − \frac{a_{d−1,1}}{n a_{d,0}} , x_{2} ↦ x_{2}

reduces the generic binary form

ϕ_{2,d}(x_{1}, x_{2}) = a_{d,0}x_{1}^{d} + a_{d−1,1}x_{1}^{d−1}x_{2} + ⋯ + a_{0,d}x_{2}^{d}

to

ϕ_{2,d}(g · (x_{1}, x_{2})) = b_{d,0}x_{1}^{d} + b_{d−2,2}x_{1}^{d−2}x_{2}^{2} + ⋯ + b_{1,d−1}x_{1}x_{2}^{d−1} + b_{0,d}x_{2}^{d}

for some b_{i,d−i} ∈ K = k(a_{0,d}, …, a_{d,0}). Composing this linear transformation with

x_{1} ↦ \frac{b_{0,d}}{b_{1,d−1}} x_{1} , x_{2} ↦ x_{2}

(and, by abuse of notation, denoting the composition by g once again), we may further assume b_{1,d−1} = b_{0,d}. The field F_{g} = k(b_{i,d−i}/b_{0,d} | i = 1, …, d), defined in the statement of Corollary 14.6, now has transcendence degree ≤ d − 2. By Corollary 14.6 we conclude that ed [H_{2,d}] ≤ n − 2. □

We are now ready to proceed with the main result of this section.

16.2. Proposition. (a) ed [ϕ_{n,1}] = ed [H_{n,1}] = 0.

(b) ed [ϕ_{n,2}] = n and ed [H_{n,2}] = n − 1.

(c) ed [ϕ_{2,3}] = 2 and ed [H_{2,3}] = 1.

(d) ed [ϕ_{2,4}] = 3 and ed [H_{2,4}] = 2.

(e) ed [H_{3,3}] = 3.

We do not know whether ed [ϕ_{3,3}] is 3 or 4.

Proof. (a) A linear form l(x_{1}, …, x_{n}) over K can be reduced to just x_{1} by applying a linear transformation g ∈ GL_{n}(K). Thus ed [ϕ_{n,1}] = ed [H_{n,1}] = 0.

(b) Here d = 2, N = n(n + 1)/2, and elements of 𝔸^{N} are quadratic forms in n variables. Diagonalizing the generic quadratic form ϕ_{n,2} over K, we see that ed [ϕ_{n,2}] ≤ n and ed [H_{n,2}] ≤ n − 1. In view of Corollary (14.7) it suffices to show that ed [ϕ_{n,2}] = n.

The GL_{n}-action on 𝔸^{N} has a dense orbit, consisting of non-singular forms. In particular, D = dim(𝔸^{N}/ GL_{n}) = 0, so that by Lemma 14.5(a)

ed [ϕ_{n,2}] = cd(𝔸^{N}, GL_{n}) .

Since the stabilizer of a non-singular form is the orthogonal group O_{n}, 𝔸^{N} is birationally G-equivariantly isomorphic to GL_{n} / O_{n}. Thus

ed [ϕ_{n,2}] = cd(𝔸^{N}, GL_{n}) = cd(GL_{n} / O_{n}, GL_{n}) =^{by Corollary 5.7(b)}
ed(O_{n}) =^{by [R, Theorem 10.3]} n .

This completes the proof of part (b).

(c) By Lemma 16.1, ed [H_{2,3}] ≤ 1. Thus in view of Corollary 14.7, we only need to show that ed [ϕ_{2,3}] ≥ 2.

Here N = 4, and the GL_{2}-action on 𝔸^{4} has a dense orbit consisting of binary cubic forms with three distinct roots. Applying Lemma 14.5(a), with D = dim(𝔸^{4}/ GL_{2}) = 0, as in part (b), we obtain

ed [ϕ_{2,3}] = cd(𝔸^{4}, GL_{2}) = cd(GL_{2} /S, GL_{2}) =^{by Corollary 5.7(b)} ed(S) ,

where S ⊂ GL_{2} is the stabilizer of a binary cubic form with three roots, say of x^{3} + y^{3}. Note that S is a finite group and that matrices that multiply x and y by third roots of unity form a subgroup of S isomorphic to (ℤ/3ℤ)^{2}. Thus ed(S) ≥ ed (ℤ/3ℤ)^{2} = 2 (cf. [BR, Lemma 4.1(a) and Theorem 6.1]), as desired.

(d) By Lemma 16.1, ed [H_{2,4}] ≤ 2. In view of Corollary 14.7, it remains to prove the inequality ed [ϕ_{2,4}] ≥ 3. Note that since the invariant field k(𝔸^{5})^{GL^{2}} is generated by one element (namely, the cross-ratio of the four roots of the quartic binary form), we have D = dim(𝔸^{5}/ GL_{2}) = 1. Thus we only need to show that

(16.3)  cd(𝔸^{5}, GL_{2}) ≥ 2 .

Let S be the stabilizer of f ∈ 𝔸^{5} (i.e., of a degree 4 binary form) in general position. By Proposition 5.5(c),

cd(𝔸^{5}, GL_{2}) ≥ ed(S) .

To compute ed(S), recall that the stabilizer of f in PGL_{2} is isomorphic to (ℤ/2ℤ)^{2}; cf. e.g., [PV, p. 231]. It is now easy to see that S fits into the sequence

{1} ⟶ ℤ/4ℤ ⟶ S ⟶ (ℤ/2ℤ)^{2} ⟶ {1} .

In particular, S is a finite group which admits a surjective homomorphism onto (ℤ/2ℤ)^{2}. Thus S is neither cyclic nor odd dihedral and consequently, ed(S) ≥ 2; see [BR, Theorem 6.2(a)]. This concludes the proof of (16.3) and thus of part (d). For the sake of completeness, we remark that since S is a finite subgroup of GL_{2}, we also have ed(S) ≤ 2 and thus ed(S) = 2.

(e) Here N = 10, and the rational quotient ℙ^{9}/ GL_{3} is the j-line, so that D = dim ℙ^{9}/ GL_{3} = 1. Thus we only need to show

(16.4)  cd(ℙ^{9}, GL_{3}) = 2 .

An element of ℙ^{9} (i.e., a plane cubic curve) in general position can be written as F_{λ} = x^{3}+y^{3}+z^{3}+3λxyz. Denote the stabilizer of F_{λ} by S ⊂ GL_{3}. We will deduce (16.4) from Corollary 6.2(b). Indeed, let N be the normalizer of S in GL_{3}. Since GL_{3} is a special group, e(GL_{3}, S) = ed(S), e(GL_{3}, N) = ed(N) (see Lemma 5.4(c)), and Corollary 6.2(b) assumes the following form:

ed(S) ≤ cd(ℙ^{9}, GL_{3}) ≤ ed(N) − dim(S) + dim(N) .

Let \overline{S} and \overline{N} be the images of S and N in PGL_{3}, under the natural projection GL_{3} ⟶ PGL_{3}. Note that \overline{S} is a finite group (this follows from the fact that D = dim ℙ^{9}/ PGL_{3} = 1). In particular, dim(S) = 1. It thus suffices to show:

(e_{1}) ed(S) ≥ 2,

(e_{2}) \overline{N} is a finite subgroup of PGL_{3} (and consequently, dim(N) = 1). (e_{3}) ed(N) ≤ 2. The inequality (e_{1}) is a consequence of [RY_{1}, Corollary 7.3], with G = S

and

(16.5)  H = <diag(1, ζ, ζ^{2}), σ> ≃ (ℤ/3ℤ)^{2} ,

where σ is a cyclic permutation of the variables x, y, z, and ζ is a primitive third root of unity. (Note that [RY_{1}, Corollary 7.3] applies because S has no non-trivial unipotent elements, and the centralizer of H in S is finite.)

To prove (e_{2}), note that \overline{N} is the normalizer of \overline{S} in PGL_{n}. The natural 3-dimensional representation of S ⊂ GL_{3} is irreducible (to see this, restrict to the subgroup H of S defined in (16.5)). Hence, by Schur’s lemma, the centralizer C_{PGL_{n}}(\overline{S}) = \{1\}, so that \overline{N} = N_{PGL_{n}}(\overline{S})/C_{PGL_{n}}(\overline{S}). The last group is naturally isomorphic to a subgroup of Aut(\overline{S}), which is a finite group. This proves (e_{2}).

To prove (e_{3}), consider the natural representation of N ⊂ GL_{3} on 𝔸^{3}. (e_{2}) implies that this representation is generically free; cf. [BF_{1}, Section 1]. Consequently, ed(N) ≤ 3 − dim(N) = 2.

This completes the proof of part (e). □

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Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

   E-mail address: berhuy@math.ubc.ca

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

   E-mail address: reichst@math.ubc.ca