ON THE RATIONALITY PROBLEM FOR FORMS OF MODULI SPACES OF STABLE MARKED CURVES OF POSITIVE GENUS

MATHIEU FLORENCE AND ZINOVY REICHSTEIN

Abstract. Let M_{g,n} (respectively, \overline{M}_{g,n}) be the moduli space of smooth (respectively stable) curves of genus g with n marked points. Over the field of complex numbers, it is a classical problem in algebraic geometry to determine whether or not M_{g,n} (or equivalently, \overline{M}_{g,n}) is a rational variety. Theorems of J. Harris, D. Mumford, D. Eisenbud and G. Farkas assert that M_{g,n} is not unirational for any n ⩾ 0 if g ⩾ 22. Moreover, P. Belorousski and A. Logan showed that M_{g,n} is unirational for only finitely many pairs (g, n) with g ⩾ 1. Finding the precise range of pairs (g, n), where M_{g,n} is rational, stably rational or unirational, is a problem of ongoing interest.

In this paper we address the rationality problem for twisted forms of \overline{M}_{g,n} defined over an arbitrary field F of characteristic ≠ 2. We show that all F -forms of \overline{M}_{g,n} are stably rational for g = 1 and 3 ⩽ n ⩽ 4, g = 2 and 2 ⩽ n ⩽ 3, g = 3 and 1 ⩽ n ⩽ 14, g = 4 and 1 ⩽ n ⩽ 9, g = 5 and 1 ⩽ n ⩽ 12.

1. Introduction

Let M_{g,n} (respectively \overline{M}_{g,n}) be the moduli space of smooth (respectively stable) curves of genus g with n marked points. Recall that these moduli spaces are defined over the prime field (ℚ in characteristic zero and 𝔽_{p} in characteristic p). The purpose of this paper is to address the rationality problem for twisted forms of \overline{M}_{g,n}. Recall that a form of a scheme X defined over a field F is another scheme Y, also defined over F, such that X and Y become isomorphic over the separable closure F^{sep}. We will use the terms “form”, “twisted form” and “F -form” interchangeably throughout this paper. Forms of \overline{M}_{g,n} are of interest because they shed light on the arithmetic geometry of \overline{M}_{g,n}, and because they are coarse moduli spaces for natural moduli problems in their own right; see [FR17, Remark 2.4].

This paper is a sequel to [FR17], where we considered twisted forms of \overline{M}_{0,n}. The main results of [FR17] can be summarized as follows.

Theorem 1.1. Let F be a field of characteristic ≠ 2 and n ⩾ 5 be an integer. Then (a) all F -forms of \overline{M}_{0,n} are unirational.

(b) If n is odd, all F -forms of \overline{M}_{0,n} are rational.

2010 Mathematics Subject Classification. 14E08, 14H10, 14G27, 14H45. Key words and phrases. Stable curves, marked curves, moduli spaces, rationality problem, group

actions, twisted varieties, Weil restriction.

Florence was partially supported by the French National Agency (Project GeoLie ANR-15-CE40-0012). Reichstein was partially supported by National Sciences and Engineering Research Council of Canada

grant No. 253424-2017.

(c) If n is even, then there exist fields E/F and E-forms of \overline{M}_{0,n} that are not stably rational (or even retract rational) over E.

In the present paper we will study the rationality problem for forms of \overline{M}_{g,n} in the case, where g ⩾ 1. Here the the rationality problem for the usual (split) moduli space \overline{M}_{g,n} (or equivalently, for M_{g,n}) over the field of complex numbers is already highly non-trivial. Theorems of J. Harris, D. Mumford, D. Eisenbud [HM82, EH87] and G. Farkas [Fa11] assert that if g ⩾ 22, then M_{g,0} is not unirational (and hence, neither is M_{g,n} for any n ⩾ 0). Moreover, work of P. Belorousski [Bel98] (for g = 1) and A. Logan [Lo03] (for g ⩾ 2) tells us that M_{g,n} is unirational for only finitely many pairs (g, n) with g ⩾ 1. Finding the precise range of pairs (g, n), where M_{g,n} is rational, stably rational or unirational, is a problem of ongoing interest. In particular, over ℂ, M_{g,n} is known to be rational for 1 ⩽ n ⩽ r_{g} and not unirational for n ⩾ n_{g}, where

\begin{matrix}g & 1 & 2 & 3 & 4 & 5 \\ r_{g} & 10 & 12 & 14 & 15 & 12 \\ n_{g} & 11 & - & - & 16 & 15 \end{matrix} ,

see [Lo03] and [CF07]. Surprisingly, we have not been able to find specific values for n_{2} and n_{3} in the literature, even though Logan showed that they exist; see [Lo03, Theorem 2.4]. The main result of the present paper is as follows.

Theorem 1.2. Let F be a field of characteristic ≠ 2. Then every F -form of \overline{M}_{g,n} is stably rational over F if

g = 1 and 3 ⩽ n ⩽ 4 (Theorems 4.1 and 5.1),

g = 2 and 2 ⩽ n ⩽ 3 (Theorems 6.1 and 7.1),

g = 3 and 1 ⩽ n ⩽ 14 (Theorem 8.1),

g = 4 and 1 ⩽ n ⩽ 9 (Theorem 9.1),

g = 5 and 1 ⩽ n ⩽ 12 (Theorem 10.1).

Several remarks are in order.

(1) (Stable) rationality of every form of \overline{M}_{g,n} is a priori much stronger than (stable) rationality of \overline{M}_{g,n} itself. For example, \overline{M}_{1,1} ≃ ℙ^{1} is obviously rational, but its forms are conic curves which are not unirational in general.

(2) Theorem 1.2 also holds for (g, n) = (1, 2) (respectively, (2, 1)), provided char(F) = 0 (respectively, char(F) ≠ 2, 3); see Remark 2.11.

(3) By [DR15, Theorem 6.1(b)], every F -form of \overline{M}_{1,n} is unirational for 3 ⩽ n ⩽ 9.

(4) In some cases, with additional work, our proofs can be modified to establish rationality of all forms of \overline{M}_{g,n}, rather than just stable rationality. To keep our arguments as uniform and transparent as possible, we will be satisfied with establishing stable rationality in the present paper.

Roughly speaking, the approach taken in [CF07] and [BCF09] to prove the rationality of M_{g,n} over ℂ for small g and n, is to find an intrinsic map of a general curve of genus g with n marked points to some projective space ℙ(V), thus reducing the rationality problem for \overline{M}_{g,n} to the rationality problem for a quotient variety of the form X/PGL(V). The PGL(V)-variety X is studied separately in each case; the rationality of X/PGL(V) is often established using the so-called “no-name lemma” [Do85]. Our proof of Theorem 1.2 proceeds along similar lines, with an important caveat: in order to establish (stable) rationality of all twisted forms of \overline{M}_{g,n}, our geometric constructions need to be S_{n}-equivariant, i.e., symmetric in the n marked points on the curve. This gives rise to an S_{n} ×PGL(V)-equivariant birational isomorphism between \overline{M}_{g,n} and a rational quotient of the form X/PGL(V), where X is a suitable variety. A theorem of B. Fantechi and A. Massarenti [FM14] (see Theorem 2.9 and Corollary 2.10 below) then allows us to conclude that every F -form of \overline{M}_{g,n} is birationally isomorphic to ^{T} X/PGL(V), for a suitable S_{n}-torsor T → Spec(F). Here ^{T} X denotes the twist of X by T; see Section 2b. To prove stable rationality for quotients of this form we appeal to Proposition 3.6, a convenient variant of the “no-name lemma” developed in Section 3. While our argument follows the same general pattern for all pairs (g, n) covered by Theorem 1.2, our proof relies on case-by-case analysis carried out in Sections 4-10. In each case our argument takes advantage of some numerical coincidence, such as the degree a certain line bundle on a general marked curve being prime to the dimension of its space of global section. Why such numerical coincidences occur in every case and why the situation of Theorem 1.1(c), where some forms of \overline{M}_{0,n} are stably rational and others are not, does not reproduce itself for any of the pairs (g, n) covered by Theorem 1.2, remains a mystery to us.

2. Preliminaries

All algebraic groups in this paper will be assumed to be affine, and all algebraic varieties to be quasi-projective.

2a. Group actions and rational quotients. Consider the action of an algebraic group G on an integral algebraic variety X defined over a field F. A rational quotient for this action is, by definition, and F -variety Y such that F (Y) = F (X)^{G}. Clearly Y is unique up to birational isomorphism. The natural inclusion F (Y) ↪ F (X) induces a rational map π: X −−→ Y which is called the rational quotient map. By a theorem of Rosenlicht [Ro56, Theorem 2], there exists a dense open subvariety Y_{0} ⊂ Y and a G-invariant dense open subvariety X_{0} ⊂ X such that π_{|X_{0}} : X_{0} → Y_{0} is regular, and the preimage of any y ∈ Y_{0} is a single G-orbit in X_{0}; see also [BGR17, Section 7].

We say that the action of G on X is generically free if there exists a dense open subset U ⊂ X such that the scheme-theoretic stabilizer G_{x} = {1} for every x ∈ U. For a generically free action, Rosenlicht’s theorem can be strengthened as follows: the open subvarieties Y_{0} ⊂ Y and X_{0} ⊂ X can be chosen so that π_{|X_{0}} : X_{0} → Y_{0} is, in fact, a G-torsor; see [BF03, Theorem 4.7].

2b. Twisting. Let G be an algebraic group defined over a field F, X be an F -variety endowed with a G-action, and P → Spec(F) be a G-torsor. The twisted variety ^{P} X is defined as ^{P} X := (X × P)/G. Here X × P is, in fact, a G-torsor over (X × P)/G; in particular, X × P → (X × P)/G is a geometric quotient and a rational quotient (as in Section 2a above). A G-equivariant morphism of F -varieties f: Y → X gives rise to a G-equivariant morphism f × id: X × P → Y × P which descends to an F -morphism P f: ^{P} X →^{P} Y. Similarly a G-equivariant rational map f: X −−→ Y of F -varieties induces a rational map ^{P} f: ^{P} X −−→ ^{P} Y. For the basic properties of the twisting operation we refer the reader to [Flo08, Section 2] or [DR15, Section 3]. In particular, we will repeatedly use the following facts in the sequel.

Lemma 2.1. Let G be an algebraic group defined over a field F and f: X → Y be a G-equivariant morphism of F -varieties and P → Spec(F) be a G-torsor.

(a) If X = Z × Y and f is projection to the first factor, then ^{P} X ≃_{F} ^{P} Z × ^{P} Y and P f is also projection to the first factor.

(b) If f is an open (respectively closed) immersion, then so is ^{P}f.

(c) If f is dominant (respectively birational, respectively an isomorphism), then so is Pf.

(d) If f is a vector bundle of rank r, then so is ^{P}f. In particular, in this case ^{P}X is rational over ^{P}Y. Here we are assuming that G acts on X by vector bundle automorphisms. That is, for any g ∈ G and y ∈ Y, g restricts to a linear map between the fibers f^{−1}(y) and f^{−1}(g(y)).

Proof. (a)-(c) See [DR15, Corollary 3.4].

(d) The first assertion is a consequence of Hilbert’s Theorem 90. The second assertion follows from the first, since the vector bundle ^{P} π: ^{P} X → ^{P} Y becomes trivial after passing to some dense Zariski open subset of ^{P} Y. □

The F -forms of a variety X are in a natural bijective correspondence with H^{1}(F, Aut(X)); see [Se97, II.1.3]. (Recall that all varieties in this paper are assumed to be quasiprojective.) Here Aut(X) is a functor which associates to the scheme S/F the abstract group Aut(X_{S}). In general this functor is not representable by an algebraic group defined over F. If it is, one usually says that Aut(X) is an algebraic group. In this case the bijective correspondence between H^{1}(F, Aut(X)) (which may be viewed as a set of Aut(X)-torsors P → Spec(F)) and the set of F -forms of X (up to F -isomorphism) can be described explicitly as follows. An Aut(X)-torsor P → Spec(F) corresponds to the twisted variety ^{P} X, and a twisted form Y of X corresponds to the isomorphism scheme P = Isom_{F} (X, Y), which is naturally an Aut(X)-torsor over Spec(F); see [Se97, Section III.1.3], [Sp98, Section 11.3].

2c. Étale algebras. An étale algebra A/F is a commutative F -algebra of the form F_{1} × ⋯ × F_{r}, where each A_{i} is a finite separable field extension of F. n-dimensional étale algebras over A are F -forms of the split étale algebra A = F × ⋯ × F (n times). The automorphism group of this split algebra is the symmetric group S_{n}, permuting the n factors of F. Thus n-dimensional étale algebras over F are in a natural bijective correspondence with the Galois cohomology set H^{1}(F, S_{n}); see, e.g., [Ser03, Examples 2.1 and 3.2].

2d. Weil restriction. Let A be an étale algebra over F and X → Spec(A) be a variety defined over A. The Weil restriction (or Weil transfer) of X to F is, by definition an F -variety R_{A/F} (X) satisfying

(2.2)  Mor_{F} (Y, R_{A/F} (X)) ≃ Mor_{A}(Y_{A}, X)

where Y_{A} := Y ×_{Spec(F)} Spec(A), Mor_{F} (Y, Z) denotes the set of F-morphisms Y → Z, and ≃ denotes an isomorphism of functors (in Y). For generalities on this notion we refer the reader to [BLR90, Section 7.6]. For a brief summary, see [Ka00, Section 2]. In particular, it is shown in [BLR90, Theorem 4] that if X is quasi-projective over A, then R_{A/F} (X) exists. Note that uniqueness of R_{A/F} (X) follows from (2.2) by Yoneda’s lemma.

The following properties of Weil restriction will be helpful in the sequel.

Lemma 2.3. Let A/F be an étale algebra and X be a (quasi-projective) variety defined over A. Then

(a) If X is an algebraic group over A, then R_{A/F} (X) is naturally an algebraic group over F.

(b) Let V be a free A-module of finite rank, and X = 𝔸_{A}(V) be the associated affine space. Then R_{A/F} (X) = 𝔸_{F} (V), where we view V as an F -vector space.

(c) If X and Y are birationally isomorphic over A, then R_{A/F} (X) and R_{A/F} (Y) are birationally isomorphic over F.

(d) If X is a rational variety over A, then R_{A/F} (X) is rational over F.

Proof. (a) The structure of an algebraic group on X is given by the identity element 1 : Spec(F) → X, the multiplication map μ: X × X → X and the inverse i: X → X, satisfying certain commutative diagrams, such as

\xymatrix{X × X × X \ar[r]^{id×μ} \ar[d]_{μ×id} & X × X \ar[d]^{μ} \\ X × X \ar[r]^{μ} & X}

for associativity. Since the Weil transfer is functorial with respect to morphisms and direct products (see [BLR90, p. 192] or [Ka00, Proposition 1.1]), we see that 1, μ and i induce the identity R_{A/F} (1), multiplication R_{A/F} (μ) and inverse R_{A/F} (i) on R_{A/F} (X). Applying Weil restriction to the appropriate commutative diagrams, we see that R_{A/F} (X) is an algebraic group with respect to R_{A/F} (1), R_{A/F} (μ) and R_{A/F} (i).

(b) follows directly from (2.2). For details, see [Ka00, Lemma 1.2].

(c) Since X and Y are birationally isomorphic, there exists a variety U defined over A and open immersions i: U ↪ X and j: U ↪ Y. In fact, after replacing U by an open subvariety, we may assume that U is quasi-projective (we may even assume that U is affine). Since Weil restriction commutes with open immersions, i and j induce open immersions of R_{A/F} (U) into R_{A/F} (X) and R_{A/F} (Y), respectively, and part (c) follows.

(d) By our assumption, X is birationally isomorphic to Y = 𝔸^{d} over A, where d is the dimension of X. By part (c), R_{A/F} (X) and R_{A/F} (Y) are birationally isomorphic over F, and by part (b), R_{A/F} (Y) is an affine space over F. □

In the special case, where X is defined over F, the Weil transfer R_{A/F} (X_{A}) can be explicitly described as follows. The symmetric group S_{n} acts on the n-fold direct product X^{n} by permuting the factors. If P → Spec(F) is a S_{n}-torsor, and A/F is the étale algebra of degree n representing the class of P in H^{1}(F, S_{n}), then R_{A/F} (X_{A}) = ^{P} (X^{n}); see, e.g., [DR15, Proposition 3.2].

2e. Special groups. An algebraic group G defined over a field F is called special if H^{1}(E, G) = {1} for every field extension E/F. In other words, every G-torsor P → Spec(E) is split. Special groups were introduced by J.-P. Serre [Se58], who showed, among other things, that every special group is affine and connected. A direct product of special groups is again special. The general group GL_{n} is special by Hilbert’s Theorem 90. If A/F is an étale algebra and G is a special group over F then the Weil restriction R_{A/F} (G_{A}) is a special group over F by Shapiro’s Lemma. In particular, R_{A/F} (𝔾_{m}) is special.

Over an algebraically closed field of characteristic 0, special groups were classified by A. Grothendieck [Gro58]. For a partial classification of special reductive groups over an arbitrary field due to M. Huruguen, see [Hu16].

2f. Twisted groups and their actions. Let H and G be algebraic group. Assume G acts on H by group automorphisms; denote this action by

(2.4)  (g, h) ↦ ^{g}h .

Let P → Spec(F) be a G-torsor.

Lemma 2.5. ^{P} H is naturally an algebraic group.

Proof. We argue as in the proof of Lemma 2.3(a). The structure of an algebraic group on H is given by the G-equivariant morphisms,

1: Spec(F) → H, μ: H × H → H and i: H → H,

the identity, multiplication and inverse. To check that

P 1: Spec(F) → ^{P}H, ^{P}μ: ^{P}H × ^{P}H → ^{P}H and ^{P}i: ^{P}H → ^{P}H

satisfy the properties of the identity, the multiplication and the inverse for ^{P}H, encode the group axioms for H into commutative diagrams and twist each diagram by P. □

Proposition 2.6. (a) Suppose G and H both act on an algebraic variety X defined over F, and these actions skew-commute, in the sense that

(2.7)  g · (h · x) = ^{g}h · (g · x)

for any g ∈ G, h ∈ H and x ∈ X. Let P → Spec(F) be a G-torsor. Then (a) the H-action on X naturally induces an ^{P} H-action on ^{P} X.

(b) Furthermore, assume that f: X → Y is a G-equivariant H-torsor. (Here H acts trivially on Y.) Then ^{P} f is an ^{P} H-torsor. In particular, if ^{P} H is special and rational, then ^{P} X is rational over ^{P}Y.

Note that the term “skew-commute” defined in part (a), is somewhat ambiguous, because it depends on the action (2.4) of G on H. In most cases the action (2.4) is clear from the context, and making an explicit reference to it complicates the terminology. We hope this ambiguity will not cause any confusion for the reader.

Proof. (a) Condition (2.7) tell us that the H-action map f: H ×X → X is G-equivariant. Twisting this map by P and remembering that ^{P}(H × X) = ^{P}H × ^{P} X by Lemma 2.1(a), we obtain a morphism

Pf: ^{P}H ×^{P} X →^{P} X .

It remains to show that ^{P}f is a group action. Since f is a group action, the diagram

\xymatrix{H × H × X \ar[r]^{id×f} \ar[d]_{μ×id} & H × X \ar[d]^{f} \\ H × X \ar[r]^{f} & X}

commutes. Here μ: H × H → H is the multiplication map. By (2.7), this entire diagram is G-equivariant. Twisting it by P we obtain the commutative diagram

\xymatrix{^{P}H × ^{P}H × ^{P}X \ar[r]^{id×^{P}f} \ar[d]_{^{P}μ×id} & ^{P}H × ^{P}X \ar[d]^{^{P}f} \\ ^{P}H × ^{P}X \ar[r]^{^{P}f} & ^{P}X}

which tells us that ^{P}f is a group action.

(b) To check that ^{P}f is an H-torsor, we may pass to the separable closure F^{sep} of F. Over F^{sep}, P becomes split. Hence, ^{P}H becomes isomorphic to H, and ^{P}f: ^{P}X → ^{P}Y becomes the same as f: X → Y, which we know is a H-torsor.

If ^{P}H is special, then ^{P}f splits over a dense open subset of ^{P}Y. Thus ^{P}X is birationally isomorphic to ^{P}H ×_{F} ^{P}Y. If ^{P}H is also rational over F, this tells us that ^{P}X is rational over ^{P}Y, as desired. □

2g. Automorphism of marked curves. The following well-known result will be repeatedly used in the sequel. Proofs can be found, e.g., in [Ha77, Corollary 4.4.7] for g = 1 and [Ha77, Exercise V.1.11] for g ⩾ 2.

Proposition 2.8. Suppose 2g +n ⩾ 5. Then Aut(C, p_{1}, …, p_{n}) = {1} for a general point (C, p_{1}, …, p_{n}) of \overline{M}_{g,n} (or equivalently, of M_{g,n}). □

2h. Automorphisms and forms of \overline{M}_{g,n}. The following theorem is the starting point of our investigation.

Theorem 2.9. (A. Massarenti [Mas13], B. Fantechi and A. Massarenti [FM14]) Let F be a field of characteristic ≠ 2. If g, n ⩾ 1, (g, n) ≠ (2, 1) and 2g + n ⩾ 5, then the natural embedding S_{n} → Aut_{F} (\overline{M}_{g,n}) is an isomorphism.

Using the bijective correspondence between F -forms of X and Aut(X)-torsors P → Spec(F) described at the end of Section 2b, we obtain the following.

Corollary 2.10. For F, g, n as in Theorem 2.9 every F -form of \overline{M}_{g,n} is F -isomorphic to P \overline{M}_{g,n} for some S_{n}-torsor P → Spec(F). □

Remark 2.11. Theorem 1.2 also holds in the following cases.

(a) g = 2 and n = 1, and char(F) = 0,

(b) g = 1 and n = 2 and char(F) ≠ 2 or 3. In case (a), \overline{M}_{2,1} has no non-trivial automorphisms by [FM14, Theorem 1] and hence, no non-split forms. On the other hand, the split form of \overline{M}_{2,1} is known to be rational; see [CF07].

In case (b), the automorphism group of \overline{M}_{1,2} is non-trivial; however, it is special see [FM14, Proposition 2.4]. As a consequence, \overline{M}_{1,2} has no non-split forms (see [DR15, Remark 6.4]) and the split form of \overline{M}_{1,2} is rational (see [CF07]).

Remark 2.12. We do not know if \overline{M}_{g,n} can be replaced by M_{g,n} in the statement of Theorem 2.9. If so, then \overline{M}_{g,n} can also be replaced by M_{g,n} in the statements of Theorems 1.2. The proof remains unchanged.

3. Anti-versal group actions

Recall that an action of an algebraic group G on an integral variety X defined over a field F is (weakly) versal if ^{P} X(E) ≠ ∅ for every field extension E/F, and for every G-torsor P → Spec(E); see [DR15]. We will now consider group actions that are far from being versal in the following sense.

Definition 3.1. We will say that the action of G on a geometrically integral F -variety X is anti-versal if ^{P} X(E) = ∅ for any field extension E/F, and any non-split G-torsor P → Spec(E).

Example 3.2. The G-action on X = G by translations is anti-versal. Indeed, by the definition of the twisting operation, ^{P} X ≃ P for any G-torsor P → Spec(E), and P (E) ≠ ∅ if and only if P is split.

Example 3.3. The natural action of PGL(V) on X = ℙ(V) is anti-versal, for any finitedimensional F -vector space V. Indeed, if E/F is a field extension and P → Spec(E) is a PGL_{n}-torsor, then ^{P} X is the Brauer-Severi variety associated to P. This variety has an E-point if and only if P is split over E.

Lemma 3.4. (a) If X ⟶ Y is a G-equivariant morphism between geometrically integral G-varieties, and the G-action on Y is anti-versal, then so is the G-action on X.

(b) If the actions of G on X and H on Y are anti-versal, then so is the product action of G × H on X × Y.

(c) Suppose that A/F is an étale algebra, and that X is an F -variety, equipped with an anti-versal action of the group G := PGL_{n}. If [A : F] is prime to n, then the induced PGL_{n}-action on the Weil transfer R_{A/F} (X) is anti-versal.

Proof. Let E/F be a field extension and P → Spec(E) be a G-torsor.

(a) Twisting the morphism X → Y by P, we obtain a morphism ^{P} X → ^{P} Y. If ^{P} X has an E-point, then so does ^{P} Y. Since Y is anti-versal, P is split, as desired.

(b) Let E/F be a field extension. A G × H-torsor over Spec(E) is of the form P ×_{E} U, where P → Spec(E) is a G-torsor and U → Spec(E) is an H-torsor. By Lemma 2.1(a)

P ×U(X × Y) ≃_{F} ^{P ×U}X × ^{P ×U}Y ≃_{F} ^{P} X × ^{U}Y .

If this twisted variety has an E-point, then ^{P} X(E) ≠ ∅ and ^{U}Y (E) ≠ ∅. Since both X and Y are anti-versal, we conclude that both P and U are split, and hence, so is P ×_{E} U.

(c) First note that ^{P} R_{A/F} (X) is naturally isomorphic to R_{A/F} (^{P} X). Thus ^{P} R_{/F} (X) has an E-point if and only if X has an E ⊗_{F} A-point; see (2.2). Since the PGL_{n}-action on X is anti-versal, this tells us that P is split by E ⊗_{F} A. On the other hand, since [A : F] is prime to n, this implies that P is split over E, as desired. □ Example 3.5. Lemma 3.4(a) implies that if a G-variety Y is anti-versal, then Z × Y (with diagonal G-action) is also anti-versal for any G-variety Z. In particular, taking Y = G, with translation G-action, as in Example 3.2, we see that Z × G is anti-versal for any G-variety Z. Proposition 3.6. Consider a generically free action of an algebraic group G on an integral variety X defined over a field F. Suppose there exists a G-equivariant rational map X −−→ Y such that the G-action on Y is anti-versal. Then

(a) the rational quotient map q : X −−→ X/G has a rational section. In other words, there exists a G-equivariant birational isomorphism X ≃ (X/G) × G.

(b) If the group G is rational over F, then X is rational over X/G. Proof. (b) is an immediate consequence of (a). To prove (a), let X_{0} be the domain of the rational map X −−→ Y. Note that X_{0} is a G-invariant open subvariety of X; in particular, the G-action on X_{0} is generically free. By Lemma 3.4(a), the G-action on X_{0} is anti-versal. After replacing X by X_{0}, we may assume that the G-action on X is anti-versal.

Recall that the function field E = F (X/G) of X/G is, by definition, the field of Ginvariant functions on X. That is, E = F (X)^{G}. Since the action of G on X is generically free, the rational quotient map

q : X −−→ X/G

gives rise to a G-torsor over the generic point of X/G; see Section 2a. Denote this torsor by P → Spec(E). The twist ^{P} X is the generic fiber of the natural map

(X × X)/G ⟶ X/G,

induced by the first projection. This map has a section, induced by the diagonal embedding X ⟶ X × X. Hence, the twist ^{P} X has an E-point. By the definition of anti-versality, this means that P is trivial. Equivalently, q has a rational section, and X ≃ (X/G) × G, as claimed. □

The following lemma supplies us with a family of anti-versal actions for G = PGL_{n}, generalizing Example 3.3. These examples will be used in place of Y in our subsequent applications of Proposition 3.6. Lemma 3.7. Let V be an n-dimensional F -vector space and ρ: GL(V) → GL(W) be a finite-dimensional representation of GL(V) over a field F, such that ρ(t · Id_{V}) = t^{d} · Id_{W}. Suppose 1 ⩽ m < n. If d and m are both coprime to n, then the action of PGL(V) on the Grassmannian Gr(m, W) induced by ρ is anti-versal. In particular, if d and n are coprime, then the PGL(V)-action on PGL(W) is anti-versal. Proof. The second assertion is obtained from the first by setting m = 1. To prove the first assertion, consider the diagram

\xymatrix{1 \ar[r] & 𝔾_{m} \ar[r] \ar[d]^{×d} & GL(V) \ar[r] \ar[d]^{ρ} & PGL(V) \ar[r] \ar[d]^{ρ̄} & 1 \\ 1 \ar[r] & 𝔾_{m} \ar[r] & GL(W) \ar[r] & PGL(W) \ar[r] & 1}

and the associated connecting maps

\xymatrix{[P] \ar[d]^{ρ̄^{∗}} & ∈ & H^{1}(E, PGL(V)) \ar[r]^{∂} \ar[d]^{ρ̄^{∗}} & H^{2}(E, 𝔾_{m}) \ar[d]^{×d} \\ [U] & ∈ & H^{1}(E, PGL(W)) \ar[r]^{∂} & H^{2}(E, 𝔾_{m}).}

Let E/F be a field extension, P → Spec(E) be a PGL(V)-torsor, [P] be the class of P in H^{1}(E, PGL(V)) and U → Spec(E) be the PGL(W)-torsor representing the image of [P] in H^{1}(E, PGL(W)). Since P is a PGL(V)-torsor and dim(V) = n, we have

(3.8)  n · ∂([P]) = 0 in Br(E).

Now observe that the twised variety ^{P} Gr(m, W) = ^{U} Gr(m, W) is the generalized Brauer-Severi variety BS(m, A), where A is the central simple algebra whose Brauer class is ∂([U]) = d · ∂([P]). Our goal is to show that if BS(m, A) has an E-point, then P is split. Indeed, assume that BS(m, A) has an E-point. Then by [Ar81, (3.4)], the index of A divides m. Thus m · ∂([U]) = 0 or equivalently,

(3.9)  md · ∂([P]) = 0 in Br(E).

Since md and n are coprime, (3.8) and (3.9) tell us that ∂([P]) = 0 in Br(E). Equivalently, P is split over E, as desired. □

4. Forms of \overline{M}_{1,3}

In this section we will prove the following.

Theorem 4.1. Let F be a field of characteristic ≠ 2. Then every F -form of \overline{M}_{1,3} is stably rational.

For the rest of this section, V will denote a 3-dimensional F -vector space, and W_{3} := S^{3}(V^{∗}) will denote the 10-dimensional space of cubic forms on V. Consider the closed subvariety X of ℙ(W_{3}) × ℙ(V)^{3} given by

X := {(C, p_{1}, p_{2}, p_{3}) | p_{1}, p_{2}, p_{3} ∈ C, and p_{1}, p_{2}, p_{3} are collinear}.

Here points of ℙ(W_{3}) are viewed as cubic curves in ℙ(V) = ℙ^{2}. The group S_{3} acts on X by permuting p_{1}, p_{2}, p_{3}; this action commutes with the natural action of PGL(V) on X.

Lemma 4.2. (a) The rational quotient X/PGL(V) is S_{3}-equivariantly birationally isomorphic to M_{1,3}.

(b) The PGL(V)-action on X is generically free.

Proof. (a) The natural map f: X −−→ M_{1,3} sending the quadruple (C, p_{1}, p_{2}, p_{3}) (with C smooth) to its isomorphism class is clearly S_{3}-equivariant and factors through the rational quotient X/PGL(V). We claim that the induced S_{3}-equivariant rational map \overline{f}: X/PGL(V) −−→ M_{1,3} is a birational isomorphism.

To prove this claim, we will construct the inverse \overline{h}: M_{1,3} → X/PGL(V) to \overline{f} as follows. Given a point (C, p_{1}, p_{2}, p_{3}) ∈ M_{1,3}, consider the invertible sheaf O_{C}(p_{1} + p_{2} + p_{3}). By the Riemann-Roch theorem, the space of global sections H^{0}(C, O_{C}(p_{1} + p_{2} + p_{3})) is 3dimensional. Identifying this space with V^{∗} we obtain an embedding

h: C ↪ ℙ(V)

of C into ℙ(V) = ℙ^{2} as a curve of degree 3. The element of V^{∗} corresponding to 1 ∈ H^{0}(C, O_{C}(p_{1} + p_{2} + p_{3})) cuts out a projective line L ⊂ ℙ(V) passing through h(p_{1}), h(p_{2}) and h(p_{3}). This shows that (h(C), h(p_{1}), h(p_{2}), h(p_{3})) ∈ X. Note that (h(C), h(p_{1}), h(p_{2}), h(p_{3})) ∈ X is not intrinsically defined by (C, p_{1}, p_{2}, p_{3}), because it depends on the isomorphism we have chosen between V^{∗} with H^{0}(C, O_{C}(p_{1} + p_{2} + p_{3})). On the other hand, the image of (h(C), h(p_{1}), h(p_{2}), h(p_{3})) in X/PGL(V) does not depend on this choice. Using the universal property of M_{1,3} one readily checks that there is a rational map \overline{h}: M_{1,3} −−→ X/PGL(V) which sends (C, p_{1}, p_{2}, p_{3}) to the image of (h(C), h(p_{1})h(p_{2}), h(p_{3})) ∈ X in X/PGL(V).

It remains to show that \overline{f}: X/PGL(V) −−→ M_{1,3} and \overline{h}: M_{1,3} −−→ X/PGL(V) are mutually inverse rational maps. The composition \overline{f} \overline{h}: M_{1,3} −−→ M_{1,3} is clearly the identity map: \overline{h} embeds (C, p_{1}, p_{2}, p_{3}) into ℙ(V) = ℙ^{2}, and \overline{f} “forgets” this embedding and returns (C, p_{1}, p_{2}, p_{3}) as an abstract curve with three marked points. On the other hand, to see that \overline{h} \overline{f}: X/PGL(V) −−→ X/PGL(V) is the identity map, note that if C is a smooth cubic curve in ℙ(V) with three distinct marked collinear points p_{1}, p_{2} and p_{3}, then O_{C}(p_{1} +p_{2} + p_{3}) = O_{C}(1). In other words, we can identify the space of global sections H^{0}(C, O_{C}(p_{1} + p_{2} + p_{3})) with V^{∗} so that h(C, p_{1}, p_{2}, p_{3}) = (C, p_{1}, p_{2}, p_{3}).

(b) Suppose H ⊂ PGL(V) is the stabilizer subgroup of (C, p_{1}, p_{2}, p_{3}) ∈ X in general position. One readily checks that the natural homomorphism H → Aut(C, p_{1}, p_{2}, p_{3}) is injective. Since the map f defined in part (a) is dominant, we may assume that Aut(C, p_{1}, p_{2}, p_{3}) = {1} is the trivial group; see Proposition 2.8. Thus H = {1} as well. □

Proof of Theorem 4.1. By Corollary 2.10, every F -form of \overline{M}_{1,3} is isomorphic to ^{P} \overline{M}_{1,3} for some S_{3}-torsor P → Spec(F). By Lemma 4.2(a), ^{P} \overline{M}_{1,3} is birationally isomorphic to P X/PGL(V). It thus remains to show that ^{P} X/PGL(V) is stably rational over F. We will fix the S_{3}-torsor P for the rest of the proof.

Claim 1: ^{P} X is rational over ^{P} X/PGL(V).

We will deduce Claim 1 from Proposition 3.6(b), with G = PGL(V) and Y = ℙ(V^{∗}). To apply Proposition 3.6, it suffices to check that

(i) PGL(V) is rational,

(ii) PGL(V)-action on ^{P} X is generically free,

(iii) the PGL(V)-action on ℙ(V^{∗}) is anti-versal, and

(iv) there exists a PGL(V)-equivariant rational map ^{P} X −−→ ℙ(V^{∗}).

(i) is obvious. To prove (ii), note that the property of being generically free is geometric, i.e., can be checked after passing to the separable closure of our base field F. When we pass to the separable closure of F, ^{P} X becomes isomorphic to X. Thus (ii) follows from Lemma 4.2(b). (iii) follows from Lemma 3.7 with W = V^{∗}, and d = m = 1. To prove (iv), we begin with the S_{3} ×PGL(V)-equivariant rational map X −−→ ℙ(V^{∗}) sending the quadruple (C, p_{1}, p_{2}, p_{3}) ∈ X to the unique line L ⊂ ℙ(V) passing through p_{1}, p_{2} and p_{3}.

Here S_{3} acts trivially on ℙ(V^{∗}). Twisting by P, we obtain a desired PGL(V)-equivariant rational map

P X −−→ ℙ(V^{∗}) .

This completes the proof of Claim 1.

Claim 2: ^{P} X is stably rational over F.

To prove Claim 2, let T → ℙ(V^{∗}) be the tautological line bundle, whose fiber over the line {l = 0} in ℙ(V^{∗}) consists of vectors v ∈ V such that l(v) = 0. Let (T^{3})_{0} be the dense open subset of T^{3} := T ×_{ℙ(V^{∗})} T ×_{ℙ(V^{∗})} T consisting of triples (v_{1}, v_{2}, v_{3}) which impose independent conditions on cubic polynomials ϕ ∈ W_{3}. Let E be the vector subbundle of the trivial bundle W_{3} × (T^{3})_{0} on (T^{3})_{0} consisting of quadruples (ϕ, v_{1}, v_{2}, v_{3}) such that ϕ(v_{1}) = ϕ(v_{2}) = ϕ(v_{3}) = 0. Finally, let E_{0} be obtained from E by removing the zero section, i.e., by requiring that ϕ ≠ 0 in W_{3}. In summary, we have the following diagram of S_{3}-equivariant morphisms

\xymatrix{ & E_{0} \ar[ddl]^{α} \ar@{^{(}->}[r]^{i} & E \ar[d]^{β} \\  &  & (T^{3})_{0} \ar@{^{(}->}[r]^{j} & T^{3} \ar[d]^{γ} \\ X &  &  & ℙ(V^{∗})}

where α: (ϕ, v_{1}, v_{2}, v_{3}) ↦ ([ϕ], [v_{1}], [v_{2}], [v_{3}]) and β, γ are natural projections. Here the horizontal maps i and j are open immersions, β and γ are vector bundles, and α is a (𝔾_{m})^{4}- torsor, where (s, t_{1}, t_{2}, t_{3}) ∈ (𝔾_{m})^{4} acts on E_{0} by (ϕ, v_{1}, v_{2}, v_{3}) ↦ (sϕ, t_{1}v_{1}, t_{2}v_{2}, t_{3}v_{3}). By Lemma 2.1(d), ^{P}(T^{3})_{0} is rational over ^{P}ℙ(V^{∗}) and ^{P}E is rational over ^{P} (T^{3})_{0}.

On the other hand, the actions of G := S_{3} and H := (𝔾_{m})^{4} on E_{0} skew-commute, relative to the S_{3}-action on H given by permuting the last three 𝔾_{m}-factors (and leaving the first factor unchanged). Applying Proposition 2.6(b) to these skew-commuting actions of G := S_{3} and H := (𝔾_{m})^{4} on E_{0}, we see that ^{P}E_{0} is rational over ^{P}X. Indeed, Proposition 2.6(b) applies in this situation, because ^{P} H = 𝔾_{m} × R_{A/F} (𝔾_{m}^{3}) is both special (see Section 2e) and rational.

Putting all of this together, and remembering that S_{3} acts trivially on ℙ(V^{∗}) and thus P ℙ(V^{∗}) = ℙ(V^{∗}), we obtain the following diagram

\xymatrix{ & ^{P}E_{0} \ar[ddl]^{rational} \ar@{^{(}->}[r] & ^{P}E \ar[d]^{rational} \\  &  & ^{P}(T^{3})_{0} \ar@{^{(}->}[r] & ^{P}T^{3} \ar[d]^{rational} \\ ^{P}X &  &  & ℙ(V^{∗}) \ar[d]^{rational} \\  &  &  & Spec(F).}

By Lemma 2.1(b), the horizontal maps ^{P}i and ^{P}j are open immersions (hence, birational), and Claim 2 follows. This completes the proof of Theorem 4.1. □

5. Forms of \overline{M}_{1,4}

In this section we will prove the following.

Theorem 5.1. Let F be a field of characteristic ≠ 2. Then every F -form of \overline{M}_{1,4} is stably rational.

Let V be a 4-dimensional F -vector space and W_{2} := S^{2}(V^{∗}) be the 10-dimensional space of homogeneous quadratic polynomials on V. Consider the closed subvariety X ⊂ Gr(2, W_{2}) × ℙ(V)^{4} given by

X := {(Q, p_{1}, …, p_{4}) | q(p_{1}) = ⋯ = q(p_{4}) = 0 for any q ∈ Q,
and p_{1}, p_{2}, p_{3}, p_{4} are collinear in ℙ(V) }.

The group S_{4} acts on X by permuting the points p_{1}, p_{2}, p_{3}, p_{4}; this action commutes with the natural action of PGL(V).

Lemma 5.2. (a) The rational quotient X/PGL(V) is S_{4}-equivariantly birationally isomorphic to M_{1,4}.

(b) The PGL(V)-action on X is generically free.

Proof. (a) Define a rational map f: X −−→ M_{1,4} as follows. Given (Q, p_{1}, …, p_{4}) ∈ X, choose a basis q_{1}, q_{2} of the 2-dimensional subspace Q ⊂ W_{2} and set f(Q, p_{1}, …, p_{4}) ∈ \overline{M}_{1,4} to be the isomorphism class of (C, p_{1}, …, p_{4}), where C ⊂ ℙ^{3} is the curve given by q_{1} = q_{2} = 0. Clearly (C, p_{1}, …, p_{4}) does not depend on the choice of the basis q_{1}, q_{2} of Q. For (Q, p_{1}, …, p_{4}) in general position in X, C is a smooth curve of genus 1; see, e.g., [Ha77, Exercise I.7.2]. Hence, f is a well-defined rational map. By construction f is S_{4} ×PGL(V) equivariant, where PGL(V) acts trivially on \overline{M}_{1,4}. Thus f factors through the rational quotient X/PGL(V). We claim that the induced S_{3}-equivariant rational map \overline{f}: X/PGL(V) −−→ M_{1,4} is a birational isomorphism.

To prove this claim, we will construct the inverse \overline{h}: M_{1,4} → X/PGL(V) to \overline{f} as follows. Given (C, p_{1}, …, p_{4}) ∈ M_{1,4}, consider the invertible sheaf O_{C}(p_{1} + ⋯ + p_{4}). By the Riemann-Roch theorem, the space of global sections H^{0}(C, O_{C}(p_{1} + ⋯ + p_{4})) is 4-dimensional. Identifying this space with V^{∗}, we obtain an embedding h: C ↪ ℙ(V). The element of V^{∗} corresponding to 1 ∈ H^{0}(C, O_{C}(p_{1} + ⋯ + p_{4})) cuts out a plane L ⊂ ℙ(V) passing through h(p_{1}), …, h(p_{4}). Moreover, by [Ha77, Exercise 3.6(b)], the space H^{0}(ℙ(V), I_{h(C)}(2)) of global sections of the ideal sheaf I_{h(C)}(2) is 2dimensional. Set Q := H^{0}(ℙ(V), I_{h(C)}(2)). Then (Q, h(p_{1}), …, h(p_{4})) ∈ X. Note that while (Q, h(p_{1}), …, h(p_{4})) ∈ X depends on how we identified H^{0}(C, O_{C}(p_{1}+…+p_{4})) with V^{∗}, the image of (Q, h(p_{1}), …, h(p_{4})) in X/PGL(V) does not. By the universal property of M_{1,4} there is a rational map \overline{h}: M_{1,4} −−→ X/PGL(V) which sends (C, p_{1}, …, p_{4}) to the class of (Q, h(p_{1}), …, h(p_{4})) ∈ X in X/PGL(V).

It remains to show that \overline{f}: X/PGL(V) −−→ M_{1,4} and \overline{h}: M_{1,4} −−→ X/PGL(V) are mutually inverse rational maps. To see that the composition \overline{f} \overline{h}: M_{1,4} −−→ M_{1,4} is the identity map, note that \overline{h} presents an abstract marked curve (C, p_{1}, …, p_{4}) as an intersection of a pencil Q of quadrics in ℙ(V) = ℙ^{3} (up to an element of PGL(V)), and \overline{f} “forgets” this presentation and returns (C, p_{1}, …, p_{4}) as an abstract curve with four marked points.

To verify that \overline{h} \overline{f}: X/PGL(V) −−→ X/PGL(V) is the identity map, recall that for (Q, p_{1}, …, p_{4}) ∈ X in general position, f(Q, p_{1}, …, , p_{4}) is the isomorphism class of (C, p_{1}, …, p_{4}), where C is the curve of genus 1 cut out by a basis q_{1}, q_{2} of Q. Since O_{C}(p_{1} + … + p_{4}) is O_{C}(1), up to an element of PGL(V), (h(C), h(p_{1}), …, h(p_{4})) recovers (C, p_{1}, …, p_{4}), and H^{0}(ℙ(V), I_{h(C)}(2)) recovers Q. This completes the proof of part (a).

(b) follows from Proposition 2.8 by the same argument as in the proof of Lemma 4.2(b). □

Proof of Theorem 5.1. By Corollary 2.10, every F -form of \overline{M}_{1,4} is isomorphic to ^{P} \overline{M}_{1,4} for some S_{4}-torsor P → Spec(F). By Lemma 4.2(a), ^{P} \overline{M}_{1,4} is birationally isomorphic to P X/PGL(V). Thus it remains to show that ^{P} X/PGL(V) is stably rational over F. We will fix the S_{3}-torsor P for the rest of the proof.

Claim 1: ^{P} X is rational over ^{P} X/PGL(V).

The proof, based on Proposition 3.6(b), with G = PGL(V) and Y = ℙ(V^{∗}), follows the same pattern as the proof of Claim 1 in the previous section. Once again, in order to apply Proposition 3.6(b), we need to check items (i) - (iv). The proof of (i), (ii) and (iii) is exactly the same. To define a PGL(V)-equivariant rational map ^{P} X −−→ ℙ(V^{∗}), start with the PGL(V)-equivariant rational map X −−→ ℙ(V^{∗}) sending a quadruple (Q, p_{1}, …, p_{4}) in general position to the unique hyperplane in ℙ^{3} passing through p_{1}, …, p_{4}, then twist this map by P. This completes the proof of Claim 1.

Claim 2: ^{P} X is stably rational over F.

To prove Claim 2, let

T → ℙ(V^{∗}) be the tautological line bundle, whose fiber over the plane {l = 0} in ℙ(V) consists of vectors v ∈ V such that l(v) = 0,

(T^{4})_{0} be the dense open subset of T^{4} := T ×_{ℙ(V^{∗})} ⋯ ×_{ℙ(V^{∗})} T (4 times) consisting of quadruples (v_{1}, …, v_{4}) which impose independent conditions on quadrics in ℙ(V),

E be the vector subbundle of the trivial bundle W_{2} ×(T^{4})_{0} on (T^{4})_{0} consisting of tuples (ϕ_{1}, v_{1}, …, v_{4}) such that ϕ(v_{1}) = ⋯ = ϕ(v_{4}) = 0,

E_{0}^{2} be the dense open subset of E^{2} = E ×_{(T^{4})_{0}} E consisting of tuples (ϕ_{1}, ϕ_{2}, v_{1}, …, v_{4}) such that ϕ_{1} and ϕ_{2} are linearly independent, and

X_{0} be the dense open subset of X consisting of tuples (Q, p_{1}, …, p_{4}) such that p_{1}, …, p_{4} span a 2-dimensional projective plane in ℙ(V).

Then we have the following diagram of S_{4}-equivariant morphisms

\xymatrix{ & (E^{2})_{0} \ar[ddl]^{α} \ar@{^{(}->}[r]^{j} & E^{2} \ar[d]^{β} \\  &  & (T^{4})_{0} \ar@{^{(}->}[r]^{k} & T^{4} \ar[d]^{γ} \\ X_{0} \ar@{^{(}->}[r]^{i} & X &  & ℙ(V^{∗})}

where α: (ϕ_{1}, ϕ_{2}, v_{1}, …, v_{4}) ↦ (Span(ϕ_{1}, ϕ_{2}), [v_{1}], [v_{2}], [v_{3}], [v_{4}]) and β, γ are natural projections. Note that the horizontal maps i, j and k are open immersions, β and γ are vector bundles, and α is a GL_{2} × (𝔾_{m})^{4}-torsor. Twisting by P, we obtain the following diagram of F -morphisms:

\xymatrix{ & ^{P} (E^{2})_{0} \ar[ddl]^{rational} \ar@{^{(}->}[r] & ^{P} E^{2} \ar[d]^{rational} \\  &  & ^{P} (T^{4})_{0} \ar@{^{(}->}[r] & ^{P} (T^{4}) \ar[d]^{rational} \\ ^{P} X_{0} \ar@{^{(}->}[r] & ^{P} X &  & ℙ(V^{∗}) \ar[d]^{rational} \\  &  &  & Spec(F).}

Here ^{P} ℙ(V^{∗}) = ℙ(V^{∗}) because S_{4} acts trivially on ℙ(V^{∗}); ^{P} (E^{2}) is rational over ^{P} (T^{4})_{0} and ^{P} (T^{4}) is rational over ℙ(V^{∗}) by Lemma 2.1(d). Finally, to show that ^{P} (E^{2})_{0} is rational over ^{P} X, we apply Proposition 2.6(b) to the skew-commuting actions of S_{4} and (𝔾_{m})^{4} on (E^{2})_{0}. The twisted group

P(GL_{2} × 𝔾_{m}^{4}) = GL_{2} × R_{A/F} (𝔾_{m})

(where A/F is the étale algebra of degree 4 associated to P) is both special (see Section 2e) and rational, and Proposition 2.6(b) applies.

Finally, the horizontal maps ^{P} i, ^{P} j and ^{P} k are open immersions (and hence, birational) by Lemma 2.1(b). Thus the diagram shows that ^{P} X is stably rational over F. This completes the proof of Claim 2 and thus of Theorem 4.1. □

6. Forms of \overline{M}_{2,2}

In this section we will prove the following.

Theorem 6.1. Let F be a field of characteristic ≠ 2. Then every F -form of \overline{M}_{2,2} is stably rational.

For the rest of this section V will denote a 3-dimensional F -vector space, and W_{4} := S^{4}(V^{∗}) will denote the 15-dimensional space of degree 4 homogeneous polynomials on V. Let (ℙ(V^{∗}) × ℙ(V^{∗}))_{0} be the dense open subset of ℙ(V^{∗}) × ℙ(V^{∗}) consisting of pairs (L_{1}, L_{2}) of distinct lines in ℙ(V). Consider the subbundle E of the trivial vector bundle W_{4} × (ℙ(V^{∗}) × ℙ(V^{∗}))_{0} over (ℙ(V^{∗}) × ℙ(V^{∗}))_{0} consisting of triples (ϕ, L_{1}, L_{2}) such that the restriction of ϕ ∈ W_{4} to L_{i} (i = 1, 2) vanishes to second order at the intersection point p = L_{1} ∩ L_{2}. In other words, the plane curve C ⊂ ℙ(V) given by {ϕ = 0} is singular at p and L_{1}, L_{2} are tangent lines to C at p.

Lemma 6.2. (a) The rational quotient ℙ(E)/PGL(V) is S_{2}-equivariantly birationally isomorphic to M_{2,2}.

(b) The PGL(V)-action on ℙ(E) is generically free.

Proof. (a) We begin by defining a S_{2} ×PGL(V)-equivariant rational map

f: ℙ(E) −−→ M_{2,2}

as follows. Given a triple ([ϕ], L_{1}, L_{2}) ∈ ℙ(E), in general position, let C ⊂ ℙ(V) = ℙ^{2} be the quartic curve cut out by ϕ, and p be the point of intersections of L_{1} and L_{2}. Then C has a node at p and is smooth elsewhere, and thus is a curve of genus 1. Blowing up C at p, we obtain a smooth curve C′, with two marked points, p_{1} and p_{2}, corresponding to the tangent lines L_{1} and L_{2} to C at p. We define f([ϕ], L_{1}, L_{2}) to be the isomorphism class of (C′, p_{1}, p_{2}). Clearly any PGL(V)-translate of ([ϕ], L_{1}, L_{2}) will have the same image under f. Thus f descends to an S_{2}-equivariant rational map

\overline{f}: ℙ(E)/PGL(V) −−→ M_{2,2} .

To prove that \overline{f} is a birational isomorphism, we will construct the inverse \overline{h}: M_{2,2} → ℙ(E)/PGL(V) to \overline{f} as follows. Given a point (C′, p_{1}, p_{2}) ∈ M_{2,2}, consider the invertible sheaf O_{C}(K + p_{1} + p_{2}), where K is the canonical divisor on C′. By the Riemann-Roch theorem, the space of global sections H^{0}(C, O_{C}(K +p_{1} +p_{2})) is 3-dimensional. Identifying this space with V^{∗}, we obtain a map

h: C → ℙ(V)

of C into ℙ(V) = ℙ^{2} as a curve of degree 4. Assuming that p_{1} and p_{2} are in general position in C, the image C = h(C′) of this map is a quartic curve with one node, and h: C′ → C is the normalization map; see [H11, Example 5.15]. Moreover, C has two tangent lines at p, L_{1} and L_{2}, which correspond to p_{1} and p_{2} under h. Thus we send (C′, p_{1}, p_{2}) ∈ M_{2,2} to ([ϕ], L_{1}, L_{2}) ∈ ℙ(E), where ϕ ∈ W_{4} is a defining equation for C. Once again, ([ϕ], L_{1}, L_{2}) depends on the isomorphism we have chosen between V^{∗} with H^{0}(C, O_{C}(K + p_{1} + p_{2})), but the image ([ϕ], L_{1}, L_{2}) in ℙ(E)/PGL(V) does not depend on this choice, giving rise to a rational map

\overline{h}: M_{2,2} −−→ ℙ(E)/PGL(V) .

Both \overline{f} and \overline{h} are S_{2}-equivariant; it remains to show that they are mutually inverse. The composition f̄h̄: M_{2,2} −−→ M_{2,2} is clearly the identity map. To prove that the composition h̄ f̄ is the identity map on ℙ(E)/PGL(V), it suffices to establish the following claim. Suppose C ⊂ ℙ(V) is a plane quartic curve with a single node at p, π: C′ → C is the normalization map, and p_{1}, p_{2} ∈ C′ are the preimages of p. Then π^{∗} O_{C}(1) = O_{C′}(K + p_{1} + p_{2}), where K is the canonical divisor on C′.

To prove this claim, note that if L is a general line through p in ℙ(V), cut out by a linear form l ∈ V^{∗}, and q_{1}, q_{2} are the two other intersection points of L with C (other than p), then clearly H^{0}(C′, π^{−1}(q_{1}) + π^{−1}(q_{2})) contains a non-constant function l′/l, where l and l′ ∈ V^{∗} are linearly independent. By the Riemann-Roch Theorem, this implies that π^{−1}(q_{1}) + π^{−1}(q_{2}) = K is the canonical divisor on C′, and the claim follows.

(b) is a consequence of Proposition 2.8 with g = n = 2. The proof is the same as in Lemma 4.2(a). □

Proof of Theorem 6.1. By Corollary 2.10, every F -form of \overline{M}_{2,2} is isomorphic to ^{P} \overline{M}_{2,2} for some S_{2}-torsor P → Spec(F). By Lemma 6.2(a), ^{P} \overline{M}_{2,2} is birationally isomorphic to P ℙ(E)/PGL(V). It thus remains to show that ^{P} ℙ(E)/PGL(V) is stably rational over F. Let us fix the S_{2}-torsor P → Spec(F) for the rest of the proof.

Claim 1: ^{P} ℙ(E) is rational over ^{P} ℙ(E)/PGL(V).

We will deduce Claim 1 from Proposition 3.6(b), with ℙ(E) = ^{P} ℙ(E), G = PGL(V) and Y = ℙ(W_{4}). The group PGL(V) ≃ PGL_{2} is clearly rational. It follows from Lemma 6.2(b) that the PGL(V)-action on ^{P} ℙ(E) is generically free. The PGL(V)-action on Y is antiversal by Lemma 3.7 with W = W_{4}, m = 1, n = dim(V) = 3 and d = 4. It thus remains to construct a PGL(V)-equivariant rational map ^{P} ℙ(E) −−→ Y = ℙ(W_{4}). We begin with the S_{2} ×PGL(V)-equivariant rational map ℙ(E) → ℙ(W_{4}) sending ([ϕ], L_{1}, L_{2}) ∈ ℙ(E) to [ϕ] ∈ ℙ(W_{4}). Here S_{2} acts trivially on ℙ(W_{4}). Twisting by P, we obtain a PGL(V)- equivariant rational map ^{P} ℙ(E) −−→ ^{P} ℙ(W_{4}) = ℙ(W_{4}), as desired. Claim 1 now follows from Proposition 3.6(b).

Claim 2: ^{P} ℙ(E) is stably rational over F.

To prove Claim 2, recall that at the beginning of this section we defined (ℙ(V^{∗})×ℙ(V^{∗}))_{0} as the dense open subset of ℙ(V^{∗}) × ℙ(V^{∗}) consisting of pairs of distinct lines (L_{1}, L_{2}) and E as a vector bundle on (ℙ(V^{∗}) × ℙ(V^{∗}))_{0}. Let E_{0} be the dense open subset obtained from E by removing the zero section. That is, points of E_{0} are triples (ϕ, L_{1}, L_{2}) ∈ E such that 0 ≠ ϕ ∈ W_{4}. In summary, we have a diagram of S_{2}-equivariant maps

\xymatrix{E_{0} \ar@{^{(}->}[r]^{i} \ar[d]_{α} & E \ar[dr]^{β} \\ ℙ(E) &  & (ℙ(V^{∗}) × ℙ(V^{∗}))_{0} \ar@{^{(}->}[r]^{j} & ℙ(V^{∗}) × ℙ(V^{∗})}

given by α(ϕ, L_{1}, L_{2}) = ([ϕ], L_{1}, L_{2}) and β(ϕ, L_{1}, L_{2}) = (L_{1}, L_{2}). Then α is a 𝔾_{m}-torsor, the horizontal maps i and j are open immersions, and β is a vector bundle. Twisting by P, we obtain the following diagram

\xymatrix{^{P} E_{0} \ar@{^{(}->}[r] \ar[d]_{rational} & ^{P} E \ar[dr]^{rational} \\ ^{P} ℙ(E) &  & ^{P} (ℙ(V^{∗}) × ℙ(V^{∗}))_{0} \ar@{^{(}->}[r] & ^{P} (ℙ(V^{∗}) × ℙ(V^{∗})) \ar[d]_{rational} \\  &  &  & Spec(F).}

Here ^{P} E_{0} is rational over ^{P} (ℙ(V^{∗}) × ℙ(V^{∗}))_{0} by Lemma 2.1(d).

To show that ^{P} E_{0} is rational over ^{P} ℙ(E), we appeal to Proposition 2.6(b). The actions of G = S_{2} and H = 𝔾_{m} on E_{0} commute, i.e., skew-commute relative to the trivial action of S_{2} on 𝔾_{m}. Since S_{2} acts trivially on 𝔾_{m}, ^{P} 𝔾_{m} ≃ 𝔾_{m} is special and rational, and Proposition 2.6(b) tells us that ^{P} E_{0} is rational over ^{P} ℙ(E), as desired.

To see that ^{P} (ℙ(V^{∗}) × ℙ(V^{∗})) is rational over F, note that ^{P} (ℙ(V^{∗}) × ℙ(V^{∗})) = R_{A/F} (ℙ(V_{A}^{∗})), where A/F is the 2-dimensional étale algebra associated to the class of P in H^{1}(F, S_{2}); see the last paragraph of Section 2d. On the other hand, R_{A/F} (ℙ(V_{A}^{∗})) is rational over F by Lemma 2.3(d).

Finally, the horizontal maps ^{P} i and ^{P} j are immersions by Lemma 2.1(b). We conclude that ^{P} ℙ(E) is stably rational over F, thus completing the proof of Claim 2 and of Theorem 6.1. □

7. Forms of \overline{M}_{2,3}

In this section we will prove the following.

Theorem 7.1. Let F be a field of characteristic ≠ 2. Then every F -form of \overline{M}_{2,3} is stably rational.

For the rest of this section V will denote a 2-dimensional F -vector space, and W_{2,3} will denote the 12-dimensional space of degree bihomogeneous polynomials on V × V of bidegree (2, 3). The group

G := PGL(V) × PGL(V) ≃ PGL_{2} × PGL_{2}

acts on ℙ(V)^{3} × ℙ(V) by (g, h): (a_{1}, a_{2}, a_{3}, b) ↦ (ga_{1}, ga_{2}, ga_{3}, hb) and the symmetric group S_{3} by permuting a_{1}, a_{2}, a_{3}. Let (ℙ(V)^{3} × ℙ(V))_{0} be the S_{3} ×G-invariant dense open subset of ℙ(V)^{3} × ℙ(V) consisting of quadruples (a_{1}, a_{2}, a_{3}, b) such that (a_{1}, b), (a_{2}, b) and (a_{3}, b) ∈ ℙ(V)×ℙ(V) impose three independent conditions on bihomogeneous polynomials of bidegree (2, 3). Let E be the rank 9 vector subbundle of the trivial bundle W_{2,3} × (ℙ(V)^{3} × ℙ(V))_{0} over (ℙ(V)^{3} × ℙ(V))_{0} consisting of tuples (ϕ, a_{1}, a_{2}, a_{3}, b) such that

ϕ(a_{1}, b) = ϕ(a_{2}, b) = ϕ(a_{3}, b) = 0 .

Denote the associated projective bundle over (ℙ(V)^{3} × ℙ(V))_{0} by ℙ(E). The symmetric group S_{3} acts on ℙ(E) by permuting a_{1}, a_{2}, a_{3}, and G = PGL(V) × PGL(V) acts on ℙ(E) by

(g, h): ([ϕ], a_{1}, a_{2}, a_{3}, b) ↦ ([ϕ^{g,h}], ga_{1}, ga_{2}, ga_{3}, hb)

for any (g, h) ∈ PGL(V) × PGL(V) and a_{1}, a_{2}, a_{3}, b ∈ ℙ(V). Here ϕ^{g,h}(x, y) := ϕ(gx, hy).

Lemma 7.2. (a) The rational quotient ℙ(E)/G is S_{3}-equivariantly birationally isomorphic to M_{2,3}.

(b) The G-action on ℙ(E) is generically free.

Proof. (a) A bihomogeneous polynomial ϕ ∈ W_{2,3} in general position cuts out a smooth curve C ⊂ ℙ(V) × ℙ(V) = ℙ^{1} × ℙ^{1} of genus 2; see [Ha77, Exercise 5.6(c)]. Sending ([ϕ], a_{1}, a_{2}, a_{3}, b) to the isomorphism class of the marked curve (C, p_{1}, p_{2}, p_{3}), where p_{i} = (a_{i}, b) ∈ ℙ(V) × ℙ(V), gives rise to a G × S_{3}-equivariant rational map

f: ℙ(E) −−→ M_{2,3} .

Here G acts trivially on M_{2,3}. We claim that the induced S_{3}-equivariant rational map

\overline{f}: ℙ(E)/G −−→ M_{2,3}

is a birational isomorphism. To prove this claim we will define the inverse

\overline{h}: M_{2,3} −−→ ℙ(E)/G

to f as follows. Let K be the canonical divisor on C. By the Riemann-Roch theorem, the spaces of global sections H^{0}(C, O_{C}(K)) and H^{0}(C, O_{C}(p_{1} + p_{2} + p_{3})) are both 2dimensional. Identifying them with V^{∗}, we obtain ramified covers

π_{1} and π_{2} : C → ℙ(V)

of degrees 2 and 3 respectively; see [H11, Examples 5.11 and 5.13]. (Note that π_{2} is not well defined for every choice of p_{1}, p_{2} and p_{3} but is well defined for p_{1}, p_{2} and p_{3} in general position.) For p_{1}, p_{2}, p_{3} in general position, C′ := (π_{1} × π_{2})(C) is a curve of bidegree (2, 3) in ℙ(V) × ℙ(V), π_{1} × π_{2} is an isomorphism between C and C′, and π_{2}(p_{1}) = π_{2}(p_{2}) = π_{2}(p_{3}) in ℙ(V). Denote a defining polynomial for C′ in ℙ(V) × ℙ(V) by ϕ ∈ W_{2,3} and set a_{i} = π_{1}(p_{i}) for i = 1, 2, 3 and b := π_{2}(p_{1}) = π_{2}(p_{2}) = π_{2}(p_{3}). The resulting point ([ϕ], a_{1}, a_{2}, a_{3}, b) ∈ ℙ(E) depends on how we identified H^{0}(C, O_{C}(K)) and H^{0}(C, O_{C}(p_{1} + p_{2} + p_{3})) with V^{∗}, but its image in ℙ(E)/G does not. This gives rise to a rational map \overline{h}: M_{2,3} −−→ ℙ(E)/G given by

\overline{h}(C, p_{1}, p_{2}, p_{3}) := ([ϕ], a_{1}, a_{2}, a_{3}, b) .

Now one readily checks that \overline{f} and \overline{h} are mutually inverse S_{3}-equivariant birational isomorphisms between ℙ(E)/G and M_{2,3}.

(b) is, once again, a consequence of Proposition 2.8. The proof is the same as in Lemma 4.2(b). □

Proof of Theorem 7.1. Once again, in view of Corollary 2.10 and Lemma 7.2(a), it suffices to show that ^{P} ℙ(E)/G is stably rational over F for every S_{3}-torsor P → Spec(F).

Claim 1: ^{P} ℙ(E) is rational over ^{P} ℙ(E)/G.

We will prove Claim 1 by applying Proposition 3.6(b) to the G = PGL(V) × PGL(V)- action on ^{P} ℙ(E), with

Y = ^{P} (ℙ(V)^{3} × ℙ(V)) = R_{A/F} (ℙ(V)) × ℙ(V),

where A/F is the étale algebra of degree 3 representing the class of P in H^{1}(F, S_{3}). Here the G = PGL(V) × PGL(V)-action is the product of the natural action of the first factor of PGL(V) on R_{A/F} (ℙ(V)) and the second factor of PGL(V) on ℙ(V). To apply Proposition 3.6, we need to check that

(i) G is rational,

(ii) the G-action on ^{P} ℙ(E) is generically free,

(iii) the G-action on Y is anti-versal, and

(iv) there exists a G-equivariant rational map ^{P} ℙ(E) −−→ Y.

(i) is obvious and (ii) follows from Lemma 7.2(b). To prove (iii), note that the PGL(V)- action on ℙ(V) is anti-versal by Example 3.3 (or as a special case of Lemma 3.7). Since [A : F] = 3 is prime to dim(V) = 2, the PGL(V)-action on R_{A/F} (ℙ(V)) is also antiversal by Lemma 3.4(c). Hence, by Lemma 3.4(b), the action of G = PGL(V) × PGL(V) on Y = R_{A/F} (ℙ(V)) × ℙ(V) is also anti-versal. The proof of (iii) is now complete. To proof (iv), twist the natural S_{3} ×G-equivariant projection ℙ(E) → ℙ(V)^{3} × ℙ(V) taking ([ϕ], a_{1}, a_{2}, a_{3}, b) to (a_{1}, a_{2}, a_{3}, b), by P to obtain a desired G-equivariant morphism P ℙ(E) → R_{A/F} (ℙ(V)) × ℙ(V) = Y. This finishes the proof of Claim 1.

Claim 2: ^{P} ℙ(E) is stably rational over F.

To prove Claim 2, let E_{0} be the dense open subset obtained from E by removing the zero section. That is, E_{0} consists of tuples (ϕ, a_{1}, a_{2}, a_{3}, b) ∈ E such that 0 ≠ ϕ ∈ W_{2,3}. Then we have a diagram of S_{2}-equivariant maps

\xymatrix{E_{0} \ar@{^{(}->}[r]^{i} \ar[d]_{α} & E \ar[dr]^{β} \\ ℙ(E) &  & (ℙ(V^{∗})^{3} × ℙ(V^{∗}))_{0} \ar@{^{(}->}[r]^{j} & ℙ(V^{∗})^{3} × ℙ(V^{∗})}

The horizontal maps i and j are open immersions, α: (ϕ, a_{1}, a_{2}, a_{3}, b) ↦ ([ϕ], a_{1}, a_{2}, a_{3}, b) is a 𝔾_{m}-torsor, and β: (ϕ, a_{1}, a_{2}, a_{3}, b) ↦ (a_{1}, a_{2}, a_{3}, b) is a vector bundle. Twisting by P, we obtain the following diagram

\xymatrix{^{P} E_{0} \ar@{^{(}->}[r] \ar[d]_{rational} & ^{P} E \ar[dr]^{rational} \\ ℙ(E) &  & ^{P} (ℙ(V)^{3} × ℙ(V))_{0} \ar@{^{(}->}[r] & ^{P} (ℙ(V)^{3} × ℙ(V)) \ar[d]_{rational} \\  &  &  & Spec(F).}

We will now justify the rationality assertions in this diagram. First note that since β is an S_{3}-equivariant vector bundle,^{P} E_{0} is rational over^{P} (ℙ(V)^{3}×ℙ(V))_{0} by Lemma 2.1(d). The rationality of ^{P} E_{0} over ℙ(E) follows from Proposition 2.6(b), applied to the commuting actions of S_{3} and 𝔾_{m} on E_{0} in the same way as in the previous section. The rationality of P (ℙ(V)^{3} × ℙ(V)) = R_{A/F} (ℙ(V_{A})) × ℙ(V) over F follows from Lemma 2.3(d). (Here A/F is the degree 3 étale algebra associated to the S_{3}-torsor P → Spec(F), as above.)

Finally, the horizontal maps in the above diagram are open immersions by Lemma 2.1(b). Claim 2 now follows from this diagram. This completes the proof of Theorem 7.1. □

8. Forms of \overline{M}_{3,n}

In this section we will prove the following.

Theorem 8.1. Let F be a field of characteristic ≠ 2 and 1 ⩽ n ⩽ 14 be an integer. Then every F -form of \overline{M}_{3,n} is stably rational.

Recall that every non-hyperelliptic curve of genus 3 can is embedded, via its canonical linear system, as a smooth curve of degree 4 in ℙ^{2}. Conversely, every smooth curve of degree 4 in ℙ^{2} is a canonically embedded curve of genus 3; see [Ha77, Example IV.5.2.1].

Let V be a 3-dimensional F -vector space and W_{4} := S^{4}(V^{∗}) be the 15-dimensional space of homogeneous polynomials of degree 4 on V. For 1 ⩽ n ⩽ 14, let ℙ(V)_{0}^{n} be a S_{n}-invariant dense open subset of ℙ(V)^{n} consisting of n-tuples (p_{1}, …, p_{n}) which impose n independent conditions on quartic polynomials. Now consider the rank 15 − n subbundle E → ℙ(V)_{0}^{n} of the rank 15 trivial bundle W_{4} × ℙ(V)_{0}^{n} → ℙ(V)_{0}^{n} consisting of tuples (ϕ, p_{1}, …, p_{n}) such that ϕ(p_{i}) = 0 for i = 1, …, n. Denote the associated projective bundle over ℙ(V)_{0}^{n} by ℙ(E).

Lemma 8.2. (a) M_{3,n} is S_{n}-equivariantly birationally isomorphic to ℙ(E)/PGL(V). (b) The PGL(V)-action on ℙ(E) is generically free.

Proof. (a) Let f: ℙ(E) −−→ M_{3,n} be the S_{n}-equivariant rational map which associates to a point ([ϕ], p_{1}, …, p_{n}) the quartic curve cut out by ϕ ∈ W_{4} with n marked points p_{1}, …, p_{n}. This map factors through ℙ(E)/PGL(V). Note that the actions of PGL(V) and S_{n} commute. Denote the resulting S_{n}-equivariant rational map by \overline{ϕ}: ℙ(E)/PGL(V) −−→ M_{3,n}.

To show that \overline{ϕ} is a birational isomorphism, we will construct the inverse map as follows. Let C be a smooth non-hyperelliptic curve of genus 3, with n marked points x_{1}, …, x_{n}. The canonical linear system H^{0}(C, O_{C}(K)) and V are both 3-dimensional F -vector spaces. Identifying H^{0}(C, O_{C}(K)) with V^{∗}, we obtain a canonical embedding h: C → ℙ(V) as a plane quartic. Set p_{i} := h(x_{i}). If 0 ≠ ϕ ∈ W_{4} is a quartic polynomial vanishing on h(C), then ([ϕ], p_{1}, …, p_{n}) ∈ ℙ(E). Note that ([ϕ], p_{1}, …, p_{n}) ∈ ℙ(E) depends on the isomorphism of F -vector spaces we chose between H^{0}(C, O_{C}(K)) with V^{∗}. On the other hand, the image of ([ϕ], p_{1}, …, p_{n}) in ℙ(E)/PGL(V) does not. This gives rise to a rational map \overline{h}: M_{3,n} −−→ ℙ(E)/PGL(V) which takes (C, x_{1}, …, x_{n}) to ([ϕ], p_{1}, …, p_{n}) as above.

It remains to check that \overline{f}: ℙ(E)/PGL(V) −−→ M_{3,n} and \overline{h}: M_{3,n} −−→ ℙ(E)/PGL(V) are mutually inverse rational maps. To see that

\overline{f} \overline{h}: M_{3,n} −−→ M_{3,n}

is the identity map, note that after \overline{h} embeds (C, x_{1}, …, x_{n}) into ℙ(V) = ℙ^{2} as a canonical curve and \overline{h} “forgets” this embedding and returns (C, x_{1}, …, x_{n}) as an abstract curve with n marked points. On the other hand, \overline{h} \overline{f}: ℙ(E)/PGL(V) −−→ ℙ(E)/PGL(V) is the identity map, because the canonical linear system on C is H^{0}(C, O_{C}(1)); see [Ha77, Example IV.5.2.1]. This completes the proof of Lemma 8.2.

(b) is, once again, a consequence of Proposition 2.8, as in the proof of Lemma 4.2(a). □

Proof of Theorem 8.1. By Corollary 2.10, every F -form of \overline{M}_{3,n} is isomorphic to ^{P} \overline{M}_{3,n} for some S_{n}-torsor P → Spec(F). By Lemma 8.2(a), ^{P} \overline{M}_{3,n} is birationally isomorphic to P ℙ(E)/PGL(V). Thus it remains to show that ^{P} ℙ(E)/PGL(V) is stably rational over F. We will fix the S_{n}-torsor P → Spec(F) for the rest of the proof.

Claim 1: ^{P} ℙ(E) is rational over ^{P} ℙ(E)/PGL(V).

We will deduce Claim 1 from Proposition 3.6(b), with X = ^{P} ℙ(E), G = PGL(V) and Y = ℙ(W_{4}). To see that Proposition 3.6 applies in this situation, note that (i) the group PGL(V) is clearly rational, (ii) the PGL(V)-action on ^{P} ℙ(E) is generically free by Lemma 8.2(b), (iii) the PGL_{n}-action on Y is anti-versal by Lemma 3.7 with W = W_{4}, m = 1, n = dim(V) = 3 and d = 4. Moreover, (iv) there exists a PGL(V)-equivariant rational map ^{P} ℙ(E) −−→ Y = ℙ(W_{4}). To construct it, twist the natural projection ℙ(E) → ℙ(W_{2}) by P. The proof of Claim 1 is now complete.

Claim 2: ^{P} ℙ(E) is stably rational over F.

To prove Claim 2, let E_{0} be the dense open subset obtained from E by removing the zero section. That is, E_{0} consists of tuples (ϕ, p_{1}, …, p_{n}) ∈ E such that 0 ≠ ϕ ∈ W_{3}.

Then we have a diagram of S_{n}-equivariant maps

\xymatrix{E_{0} \ar@{^{(}->}[r]^{i} \ar[d]_{α} & E \ar[dr]^{β} \\ ℙ(E) &  & ℙ(V)_{0}^{n} \ar@{^{(}->}[r]^{j} & ℙ(V)^{n}}

The horizontal maps i and j are open immersions, α: (ϕ, p_{1}, …, p_{n}) ↦ ([ϕ], p_{1}, …, p_{n}) is a 𝔾_{m}-torsor, and β: (ϕ, p_{1}, …, p_{n}) ↦ (p_{1}, …, p_{n}) is a vector bundle. Twisting by P, we obtain the following diagram

\xymatrix{^{P} E_{0} \ar@{^{(}->}[r] \ar[d]_{rational} & ^{P} E \ar[dr]^{rational} \\ ^{P} ℙ(E) &  & ^{P} (ℙ(V)_{0}^{n}) \ar@{^{(}->}[r] & ^{P} (ℙ(V)^{n}) \ar[d]_{rational} \\  &  &  & Spec(F).}

Once again, the horizontal maps are open immersions by Lemma 2.1(b) and the rationality of ^{P} E over ^{P} ℙ(V)_{0}^{n} follows from Lemma 2.1(d). To show that ^{P} E_{0} is rational over P ℙ(E), we apply Proposition 2.6(b) to the commuting actions of G = S_{n} and 𝔾_{m} on E_{0}, as in the previous section. The rationality of ^{P} (ℙ(V)^{n}) over F by Lemma 2.3(d), since P (ℙ(V)^{n}) = R_{A/F} (ℙ(V)), where A/F is the étale algebra associated to the S_{n}-torsor P → Spec(F).

Claim 2 now follows from the above diagram. This completes the proof of Theorem 7.1.

□
9. Forms of \overline{M}_{4,n}

In this section we will prove the following theorem.

Theorem 9.1. Let F be a field of characteristic ≠ 2 and 1 ⩽ n ⩽ 9 be an integer. Then every F -form of \overline{M}_{4,n} is stably rational.

Our proof will be based on the fact that every non-hyperelliptic curve of genus 4 can is embedded, via its canonical linear system, as a degree 6 curve in ℙ^{3}. This embedded canonical curve is a complete intersection of a unique irreducible quadric surface Q and an irreducible cubic surface S in ℙ^{3}. Moreover, the cubic polynomial s which cuts out S, is uniquely determined up to replacing s by s′ = αs + lq, where α ∈ F^{∗} is a non-zero constant, and l is a linear form. Conversely, any irreducible non-singular curve in ℙ^{3}, which is a complete intersection of a quadric surface and a cubic surface, is a canonically embedded curve of genus 4. For proofs of these assertions, see [Ha77, Example IV.5.2.2].

Let V be a 4-dimensional vector space, W_{i} := Sym^{i}(V^{∗}) be the \binom{3+i}{3}-dimensional vector space of homogeneous polynomials of degree i on V.

Lemma 9.2. Assume n ⩽ 9. There exists a S_{n} ×PGL(V)-invariant dense open subset ℙ(V)_{0}^{n} of ℙ(V)^{n} such that for every (p_{1}, …, p_{n}) ∈ ℙ(V)_{0}^{n}, (i) p_{1}, …, p_{n} impose independent conditions on quadric hypersurfaces in ℙ(V), (ii) p_{1}, …, p_{n} impose independent conditions on cubic hypersurfaces in ℙ(V), and (iii) there is a smooth quadric hypersurface Q ⊂ ℙ(V) and a smooth cubic hypersurface

S ⊂ ℙ(V) such that C := Q∩S is a complete intersection smooth irreducible curve passing through p_{1}, …, p_{n}.

Proof. (i) and (ii) are clearly open conditions in ℙ(V)^{n}, invariant under the natural PGL(V) × S_{n}-action. Since dim(W_{2}) = 10, dim(W_{3}) = 20 and we are assuming that n ⩽ 9, there exists a S_{n} ×PGL(V)-invariant dense open subset E_{n} ⊂ ℙ(V)^{n} such that (i) and (ii) hold for every (p_{1}, …, p_{n}) in U_{n}.

Now consider the closed subvariety I of W_{2}×W_{3}×U_{n} consisting of tuples (q, s, p_{1}, …, p_{n}) such that q(p_{i}) = s(p_{i}) = 0 for every i = 1, …, n, along with the natural projections

\xymatrix{ & I \ar[dl]_{π_{1}} \ar[dr]^{π_{2}} \\ W_{2} × W_{3} &  & U_{n} \ar@{^{(}->}[r] & ℙ(V)^{n}}

given by π_{1}(q, s, p_{1}, …, p_{n}) ↦ (q, s) and π_{1}(q, s, p_{1}, …, p_{n}) ↦ (p_{1}, …, p_{n}). Note that π_{2} is a vector bundle over U_{n}; in particular, I is irreducible. Note also that both π_{1} and π_{2} are invariant under the commuting natural GL(V)- and S_{n}-actions. (Here S_{n} acts trivially on W_{2} and W_{3}.) By Bertini’s theorem, there is a GL(V)-invariant dense open subset (W_{2} × W_{3})_{0} consisting of pairs (q, s) such that the quadric surface Q given by q = 0 and the cubic surface S given by s = 0 intersect transversely, in a smooth irreducible curve in ℙ(V) = ℙ^{3}. The preimage I_{0} := π_{1}^{−1}(W_{2} × W_{3}) is a dense open S_{n} ×GL(V)-invariant subset of I. The image π_{2}(I_{0}) contains a dense subset S_{n} ×PGL(V)-invariant subset of ℙ(V)^{n}. This is our desired open subset (ℙ(V)^{n})_{0}. □

From now on, let us assume that n ⩽ 9, so that we can apply Lemma 9.2. Set

E_{2} := {(ϕ, p_{1}, …, p_{n}) | ϕ(p_{1}) = ⋯ = ϕ(p_{n}) = 0} ⊂ W_{2} × ℙ(V)_{0}^{n} .

We will view E_{2} as a rank 10 − n subbundle of the trivial bundle over ℙ(V)_{0}^{n} and will denote the associated projective bundle over ℙ(V)_{0}^{n} by ℙ(E_{2}). Let Λ_{3} → ℙ(E_{2}) be the rank 20 − n vector bundle whose fiber over ([q], p_{1}, …, p_{n}) consist of cubic forms s ∈ W_{3} such that s(p_{1}) = ⋯ = s(p_{n}) = 0. Set Λ_{1.2} to be the rank 4 vector subbundle of Λ_{3} whose fiber over ([q], p_{1}, …, p_{n}) consist of cubic forms s ∈ W_{3} of the form s = l · q as l ranges over W_{1} = V^{∗}. All of these vector bundles are equivariant with respect to the natural commuting actions of GL(V) (by coordinate changes in V) and S_{n} (by permuting the points p_{1}, …, p_{n}).

Lemma 9.3. Assume 1 ≤ n ⩽ 9. Let Λ_{3}/Λ_{1,2} denote the quotient vector bundle of rank 16 − n over ℙ(E_{2}), and ℙ(Λ_{3}/Λ_{1,2}) denote the associated projective bundle. Then

(a) M_{4,n} is S_{n}-equivariantly birationally isomorphic to ℙ(Λ_{3}/Λ_{1,2})/PGL(V).

(b) The PGL(V)-action on ℙ(Λ_{3}/Λ_{1,2}) is generically free.

Proof. (a) Consider the natural rational S_{n}-equivariant map

Λ_{3} −−→ M_{4,n}

which takes a point (q, s, p_{1}, …, p_{n}) ∈ Λ_{3} to the smooth irreducible curve C ⊂ ℙ(V) of degree 6 obtained by intersecting the quadric surface {q = 0} and the cubic surface {s = 0}. Here (p_{1}, …, p_{n}) denotes a point of (ℙ(V)^{n})_{0}, q is a quadratic polynomial vanishing at p_{1}, …, p_{n}, and s is a cubic polynomial vanishing at p_{1}, …, p_{n}. For a point of Λ_{3} in general position C is a curve of genus 4 in ℙ(V) = ℙ^{3}, and p_{1}, …, p_{n} ∈ C. This gives rise to a rational map Λ_{3} −−→ M_{4,n} taking (q, s, p_{1}, …, p_{n}) to the isomorphism class of (C, p_{1}, …, p_{n}). This map descends to a rational map f: ℙ(Λ_{3}/Λ_{1,2}) −−→ M_{4,n}. Moreover, f is S_{n} ×PGL(V)-equivariant, and factors through the rational quotient ℙ(Λ_{3}/Λ_{1,2})/PGL(V), yielding an S_{n}-equivariant rational map

(9.4)  \overline{f}: ℙ(Λ_{3}/Λ_{1,2})/PGL(V) −−→ M_{4,n}.

In order to complete the proof of the lemma, it remains to show that \overline{f} is a birational isomorphism. We will do so by constructing the inverse

(9.5)  \overline{h}: M_{4,n} −−→ ℙ(Λ_{3}/Λ_{1,2})/PGL(V) .

Let C be a non-hyperelliptic curve of genus 4 with n marked points x_{1}, …, x_{n}. The canonical linear system H^{0}(C, O_{C}(K)) and V are both 3-dimensional F -vector spaces. Identifying H^{0}(C, O_{C}(K)) with V^{∗}, we obtain a canonical embedding h: C → ℙ(V). Let p_{i} := h(x_{i}), q ∈ W_{2} a homogeneous quadratic form on V vanishing on h(C), and 0 ≠ s ∈ W_{3} be a cubic form vanishing on h(C). The discussion at the beginning of this section shows that h uniquely determines the point ([s], q, p_{1}, …, p_{n}) in ℙ(Λ_{3}/Λ_{1,2}). Moreover, the projection of this point to ℙ(Λ_{3}/Λ_{1,2})/PGL(V) depends only on (C, x_{1}, …, x_{n}) and not on the isomorphism of F -vector spaces we chose between H^{0}(C, O_{C}(K)) and V^{∗}. This gives rise to a rational map (9.5) which takes (C, x_{1}, …, x_{n}) to ([s], q, p_{1}, …, p_{n}).

It remains to show that the S_{n}-equivariant rational maps (9.4) and (9.5) are mutually inverse. To see that the composition

ϕ̄ h̄: M_{4,n} −−→ M_{4,n}

is the identity map, note that \overline{h} embeds (C, x_{1}, …, x_{n}) into ℙ(V) = ℙ^{3} as a canonical curve and \overline{ϕ} “forgets” this embedding and returns (C, x_{1}, …, x_{n}) as an abstract curve with n marked points. On the other hand,

\overline{h} \overline{f}: ℙ(Λ_{3}/Λ_{1,2})/PGL(V) −−→ ℙ(Λ_{3}/Λ_{1,2})/PGL(V)

is the identity map, because the canonical linear system on a complete intersection of a smooth quadric surface and a smooth cubic surface in ℙ^{3} is H^{0}(C, O_{C}(1)); see [Ha77, Example IV.5.2.2]. This completes the proof of Lemma 9.3.

(b) We argue as in the proof of Lemma 4.2(b). The stabilizer H of a point of ℙ(Λ_{3}/Λ_{1,2}) represented by (s, q, p_{1}, …, p_{n}) naturally embeds in Aut(C, p_{1}, …, p_{n}), where C is the curve in ℙ(V) = ℙ^{3} given by q = s = 0. For (s, q, p_{1}, …, p_{n}) in general position in Λ_{3}, C is a smooth irreducible curve of genus 4. Moreover, since the rational map f constructed in part (a) is dominant, we may assume that Aut(C, p_{1}, …, p_{n}) = {1}; see Proposition 2.8. Thus H = {1}, as desired. □

Proof of Theorem 9.1. By Corollary 2.10, every F -form of \overline{M}_{4,n} is isomorphic to ^{P} \overline{M}_{4,n} for some S_{n}-torsor P → Spec(F). On the other hand, in view of Lemma 9.3(a), it suffices to show that the quotient variety ^{P} ℙ(Λ_{3}/Λ_{1,2})/PGL(V) is stably rational. We will fix the S_{n}-torsor P → Spec(F) for the rest of the proof.

Claim 1: ^{P} ℙ(Λ_{3}/Λ_{1,2}) is rational over ^{P} ℙ(Λ_{3}/Λ_{1,2})/PGL(V).

We will deduce Claim 1 from Proposition 3.6(b), with X = ^{P} ℙ(Λ_{3}/Λ_{1,2}) and G = PGL(V) and Y = Gr(5, W_{3}). To prove that Proposition 3.6(b) applies in this situation, note that (i) the group PGL(V) ≃ PGL_{4} is rational, (ii) The PGL(V)-action on P ℙ(Λ_{3}/Λ_{1,2}) is generically free by Lemma 9.3(b), and (iii) the PGL(V)-action on Y is anti-versal by Lemma 3.7 with W = W_{3}, n = dim(V) = 4, d = 3 and m = 5. Finally, (iv) there exists PGL(V)-equivariant rational map ^{P} ℙ(Λ_{3}/Λ_{1,2}) → Y = Gr(5, W_{3}). To construct this rational map, start with α: Λ_{3} −−→ Gr(5, W_{3}) taking (q, s, p_{1}, …, p_{n}) to the subspace of W_{3} spanned by the cubic forms s and l · q, as l ranges over V^{∗}. For (q, s, p_{1}, …, p_{n}) ∈ Λ_{3} in general position this subspace of W_{3} will be 5-dimensional. One readily checks that α descends to a PGL(V) × S_{n}-equivariant rational map

\overline{α}: ℙ(Λ_{3}/Λ_{1,2}) → Gr(5, W_{3}) ,

where S_{n} acts trivially on Gr(5, W_{3}). Twisting by P, we obtain a desired PGL(V)- equivariant rational map

P \overline{α}: ^{P} ℙ(Λ_{3}/Λ_{1,2}) −−→ Gr(5, W_{3}) ,

This completes the proof of Claim 1.

Claim 2: ^{P} ℙ(Λ_{3}/Λ_{1,2}) is stably rational over F.

To prove Claim 2, let (Λ_{3}/Λ_{1,2})_{0} be the dense open subset obtained from Λ_{3}/Λ_{1,2} by removing the zero section. Then we have a diagram of S_{n}-equivariant maps

\xymatrix{(Λ_{3}/Λ_{12})_{0} \ar[ddd]^{α} \ar@{^{(}->}[r] & Λ_{3}/Λ_{1,2} \ar[dd]^{β} \\  &  & (E_{2})_{0} \ar@{^{(}->}[r] \ar[dl]^{γ} & E_{2} \ar[dd]^{δ} \\  & ℙ(E_{2}) \\ ℙ(Λ_{3}/Λ_{1,2}) &  &  & ℙ(V)_{0}^{n} \ar@{^{(}->}[r] & ℙ(V)^{n}}

Here the horizontal maps are open immersions, α and γ are 𝔾_{m}-torsors, and β and δ are vector bundles. Twisting by P and applying Lemma 2.1(d) and Proposition 2.6(b), as we did in the previous sections, we obtain the following diagram

\xymatrix{^{P} (Λ_{3}/Λ_{12})_{0} \ar[ddd]_{rational} \ar@{^{(}->}[r] & ^{P} (Λ_{3}/Λ_{1,2}) \ar[dd]_{rational} \\  &  & ^{P} (E_{2})_{0} \ar@{^{(}->}[r] \ar@{-->}[dl]^{rational} & ^{P} E_{2} \ar[dd]^{rational} \\  & ^{P} ℙ(E_{2}) \\ ^{P} ℙ(Λ_{3}/Λ_{1,2}) &  &  & ^{P} (ℙ(V)_{0}^{n}) \ar@{^{(}->}[r] & ^{P} (ℙ(V)^{n}) \ar[d]^{rational} \\  &  &  &  & Spec(F).}

Here the horizontal maps are open immersions by Lemma 2.1(b), and the rationality assertions in this diagram are established in the same way as in the proof of Claim 2 in the previous section. In particular, ^{P} (Λ_{3}/Λ_{1,2}) is rational over ^{P} ℙ(E_{2}) and ^{P} E_{2} is rational over ^{P} (ℙ(V)^{n}) by Lemma 2.1(d). The rationality of ^{P} (Λ_{3}/Λ_{1,2})_{0} over ^{P} ℙ(Λ_{3}/Λ_{1,2}) and of P (E_{2})_{0} over ^{P} ℙ(E_{2}) follows from applying Proposition 2.6(b) to the commuting actions of G = S_{n} and H = 𝔾_{m}. Finally, the rationality of ^{P} (ℙ(V)^{n}) over F follows from Lemma 2.3(d), since ^{P} (ℙ(V)^{n}) = R_{A/F} (ℙ(V)_{A}), where A/F is the étale algebra associated to the S_{n}-torsor P → Spec(F). Claim 2 now follows from the above diagram, and the proof of Theorem 9.1 is complete. □

10. Forms of \overline{M}_{5,n}

In this section we will prove the following.

Theorem 10.1. Let F be a field of characteristic ≠ 2 and 1 ⩽ n ⩽ 12 be an integer. Then every F -form of \overline{M}_{5,n} is stably rational.

Our proof will be based on the fact that a general curve of genus 5 (and more precisely, any smooth curve of genus 5 with no g_{3}^{1}) can is embedded, via its canonical linear system, as a complete intersection of three quadric hypersurfaces in ℙ^{4}. Conversely, any irreducible non-singular curve, which is a complete intersection of three quadrics in ℙ^{4}, is a canonically embedded curve of genus 5. For proofs of these assertions, see [Ha77, Example IV.5.5.3].

Let V be a 5-dimensional F -vector space and W_{2} := S^{2}(V^{∗}) be the 15-dimensional space of homogeneous polynomials of degree 2 on V. For 1 ⩽ n ⩽ 14, let (ℙ(V)^{n})_{0} be a S_{n}-invariant dense open subset of ℙ(V)^{n} consisting of n-tuples (p_{1}, …, p_{n}) which impose n independent conditions on qudratic polynomials. Now consider the vector bundle E → (ℙ^{4})_{0}^{n} of rank 15 − n whose fiber of E over (p_{1}, …, p_{n}) consists of homogeneous quadratic polynomials vanishing at p_{1}, …, p_{n}. Denote the Grassmannian bundle of 3-dimensional subspaces associated to E by Gr(3, E). The natural action of PGL(V) on ℙ(V) induces PGL(V)-actions on ℙ(V)^{n}, ℙ(V)_{0}^{n}, E and Gr(3, E).

Lemma 10.2. (a) M_{5,n} is S_{n}-equivariantly birationally isomorphic to Gr(3, E)/PGL(V). (b) The PGL(V)-action on Gr(3, E) is generically free.

Proof. (a) Let f: Gr(3, E) −−→ M_{5,n} be the S_{n}-equivariant rational map which associates to a point (L, p_{1}, …, p_{n}) the curve {ϕ = 0 | ϕ ∈ L} with n marked points p_{1}, …, p_{n}. Here L denotes a 3-dimensional subspace of E_{(p_{1},...,p_{n})}. Since f(L, p_{1}, …, p_{n}) remains unchanged when we translate (L, p_{1}, …, p_{n}) by an element of PGL(V), f factors through Gr(3, E)/PGL(V). Denote the resulting rational map by \overline{f}: Gr(3, E)/PGL(V) −−→ M_{3,n}. Note that the actions of PGL(V) and S_{n} on Gr(3, E) commute. Hence, the rational map \overline{ϕ} is S_{n}-equivariant.

To show that \overline{f} is a birational isomorphism, we will construct the inverse map as follows. Let C be a smooth curve of genus 5 with no g_{3}^{1}, equipped with n marked points x_{1}, …, x_{n}. Identifying V^{∗} with the canonical linear system on C (both are 5-dimensional F -vector spaces) we obtain a canonical embedding h: C → ℙ(V). Set p_{i} := h(x_{i}). The space L := H^{0}(ℙ(V), I_{h(C)}(2)) of global sections of the ideal sheaf I_{h(C)}(2) is 3-dimensional. The resulting point (L, p_{1}, …, p_{n}) of Gr(3, E) depends on the choice of isomorphism between V^{∗} and H^{0}(C, O_{C}(K)) but the projection of (L, p_{1}, …, p_{n}) to Gr(3, E)/PGL(V) does not. This gives rise to a rational map \overline{h}: M_{5,n} −−→ Gr(3, E)/PGL(V) which takes (C, x_{1}, …, x_{n}) to (L, p_{1}, …, p_{n}), as above.

Now observe that the composition \overline{f} \overline{h}: M_{5,n} −−→ M_{5,n} is the identity map. Indeed, \overline{h} embeds (X, x_{1}, …, x_{n}) into ℙ(V) as a canonical curve, and \overline{h} “forgets” this embedding and returns (X, x_{1}, …, x_{3}) as an abstract curve with n marked points. On the other hand, \overline{h} \overline{f}: Gr(3, E)/PGL(V) −−→ Gr(3, E)/PGL(V) is the identity map, because the canonical linear system on a complete intersection X of three smooth quadric hypersurfaces in ℙ^{4} is H^{0}(X, O_{X}(1)); see [Ha77, Example IV.5.5.3]. This completes the proof of Lemma 10.2.

(b) is deduced from Proposition 2.8 (with g = 5) in the same way as in the proof of Lemma 4.2(b). □

Proof of Theorem 10.1. By Corollary 2.10, every F -form of \overline{M}_{5,n} is isomorphic to ^{P} \overline{M}_{5,n} for some S_{n}-torsor P over Spec(F). By Lemma 10.2(b), ^{P} \overline{M}_{5,n} is birationally isomorphic to ^{P} Gr(3, E)/PGL(V). Thus it remains to show that ^{P} Gr(3, E)/PGL(V) is stably rational over F. We fix the PGL(V)-torsor P → Spec(F) for the rest of the proof.

Claim 1: ^{P} Gr(3, E) is rational over ^{P} Gr(3, E)/PGL(V).

We will deduce Claim 1 from Proposition 3.6(b), with X = ^{P} Gr(3, E), G = PGL(V) and Y = Gr(3, W_{2}). To check that Proposition 3.6 applies in this situation, note that (i) the group PGL(V) ≃ PGL_{5} is rational, (ii) the PGL(V)-action on Gr(3, E) is generically free by Lemma 10.2(a), (iii) the PGL(V)-action on Y is anti-versal by Lemma 3.7 with W = W_{2}, n = dim(V) = 5, d = 2 and m = 3, and (iv) there exists a PGL(V)-equivariant map ^{P} Gr(3, E) → Y = ℙ(3, W_{2}). To construct this map, recall that E is, by definition, a subbundle of the trivial bundle over ℙ(V)_{0}^{n} with fiber W_{2}. Twisting the tautological PGL(V) × S_{n}-equivariant morphism

Gr(3, E) → Gr(3, W_{2})

by P, we obtain a desired PGL(V)-equivariant morphism

P Gr(3, E) → ^{P} Gr(3, W_{2}) = Gr(3, W_{2}) .

This completes the proof of Claim 1.

Claim 2: ^{P} Gr(3, E) is stably rational over F.

To prove Claim 2, let (E^{3})_{0} be a dense open subset of E^{3} := E ×_{ℙ(V)_{0}^{n}} E ×_{ℙ(V)_{0}^{n}} E whose elements are tuples (q_{1}, q_{2}, q_{3}, p_{1}, …, p_{n}) such that q_{1}, q_{2} and q_{3} ∈ W_{2}(V) are linearly independent over F. Consider the following diagram of S_{n}-equivariant morphisms

\xymatrix{(E^{3})_{0} \ar@{^{(}->}[r]^{i} \ar[d]_{α} & E^{3} \ar[dr]^{β} \\ Gr(3, E) &  & ℙ(V)_{0}^{n} \ar@{^{(}->}[r]^{j} & ℙ(V)^{n}}

The horizontal maps i and j are open immersions,

α: (q_{1}, q_{2}, q_{3}, p_{1}, …, p_{n}) ↦ (Span(q_{1}, q_{2}, q_{3}), p_{1}, …, p_{n})

is a GL_{3}-torsor, and β: (q_{1}, q_{2}, q_{3}, p_{1}, …, p_{n}) ↦ (p_{1}, …, p_{n}) is a vector bundle. Twisting by P, and applying Lemma 2.1, Lemma 2.3 and Proposition 2.6(b), as in the previous sections, we obtain the following diagram

\xymatrix{^{P} (E^{3})_{0} \ar@{^{(}->}[r] \ar[d]_{rational} & ^{P} E^{3} \ar[dr]^{rational} \\ Gr(3, E) &  & ^{P} (ℙ(V)_{0}^{n}) \ar@{^{(}->}[r] & ^{P} (ℙ(V)^{n}) \ar[d]_{rational} \\  &  &  & Spec(F).}

This completes the proof of Claim 2 and thus of Theorem 10.1. □

Acknowledgements

We are grateful to A. Massarenti, M. Mella, M. Talpo and A. Vistoli for helpful comments.

References

[Ar81] M. Artin, Brauer-Severi varieties, in Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), 194–210, Lecture Notes in Math., 917, Springer, Berlin. MR0657430

[BCF09] E. Ballico, G. Casnati, C. Fontanari, On the birational geometry of moduli spaces of pointed curves, Forum Math. 21, no. 5, 935–950, 2009.

[BGR17] J. Bell, D. Ghioca, Z. Reichstein, On a dynamical version of a theorem of Rosenlicht, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XVII (2017), 187-204.

[BF03] G. Berhuy and G. Favi, Essential dimension: a functorial point of view (after A. Merkurjev), Doc. Math. 8 (2003), 279–330. MR2029168

[Bel98] P. Belorousski, Chow rings of moduli spaces of pointed elliptic curves, PhD Thesis, University of Chicago, 1998.

[BLR90] S. Bosch, W. Lütkebohmert and M. Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 21, Springer-Verlag, Berlin, 1990. MR1045822

[BM13] A. Bruno and M. Mella, The automorphism group of \overline{M}_{0,n}, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 949–968. MR3085097 [CF07] G. Casnati, C. Fontanari, On the rationality of moduli spaces of pointed curves, J. Lond. Math.

Soc. (2) 75, no. 3: 582–596, 2007.

[Do85] I. V. Dolgachev, Rationality of fields of invariants, in Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), 3–16, Proc. Sympos. Pure Math., 46, Part 2, Amer. Math. Soc., Providence, RI. MR0927970

[DR15] A. Duncan, Z. Reichstein, Versality of algebraic group actions and rational points on twisted varieties, J. Algebraic Geom. 24 (2015), no. 3, 499–530. MR3344763

[EH87] D. Eisenbud, J. Harris, The Kodaira dimension of the moduli space of curves of genus ≥ 23, Invent. Math. 90 (1987), no. 2, 359–387. MR0910206 (88g:14027)

[FM14] B. Fantechi, A. Massarenti, On the rigidity of moduli of curves in arbitrary characteristic, Int. Math. Res. Not. IMRN 2017, no. 8, 2431–2463. MR3658203

[Fa11] G. Farkas, Birational aspects of the geometry of \overline{M}_{g}, in Surveys in differential geometry. Vol. XIV. Geometry of Riemann surfaces and their moduli spaces, 57–110, Surv. Differ. Geom., 14, Int. Press, Somerville, MA. MR2655323 (2011g:14065).

[FR17] M. Florence, Z. Reichstein, The rationality problem for forms of \overline{M}_{0,n}, http://www.math.ubc.

ca/~reichst/moduli3-10-17.pdf

[Flo08] M. Florence, On the essential dimension of cyclic p-groups, Invent. Math. 171 (2008), no. 1, 175–189.

[Gro58] A. Grothendieck, La torsion homologique et les sections rationnelles, Exposé 5, Séminaire C. Chevalley, Anneaux de Chow et applications, IHP, (1958).

[HM82] J. Harris and D. Mumford, On the Kodaira dimension of the moduli space of curves, Invent. Math. 67 (1982), no. 1, 23–88. MR0664324 (83i:14018)

[H11] J. Harris Geometry of algebraic curves, notes by A. Mathew, 2011 http://www.math.harvard.

edu/~amathew/287y.pdf

[Ha77] R. Hartshorne, Algebraic geometry, Springer, New York, 1977. MR0463157 (57 #3116)

[Hu16] M. Huruguen, Special reductive groups over an arbitrary field, Transform. Groups 21 (2016), no. 4, 1079–1104. MR3569568

[Ka00] N. A. Karpenko, Weil transfer of algebraic cycles, Indag. Math. (N.S.) 11 (2000), no. 1, 73–86. MR1809664

[Lo03] A. Logan, The Kodaira dimension of moduli spaces of curves with marked points, Amer. J. Math. 125, no. 1, 105–138, 2003. th. Soc., Providence, RI, 1969. MR0252145

[Mas13] A. Massarenti, The automorphism group of \overline{M}_{g},_{n}, J. Lond. Math. Soc. (2) 89 (2014), no. 1, 131–150. MR3174737

[Ro56] M. Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401–443. MR0082183

[Se58] J.-P. Serre, Espaces fibrés algébriques, in: Anneaux de Chow et Applications, Séminaire C. Chevalley, 1958, exposé 1. Reprinted in J.-P. Serre, Exposés de séminaires 1950–1999, deuxième édition, augmentée, Documents mathématiques 1, Société mathématique de France 2008, 107–140.

[Se97] J.-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 1997.

[Ser03] J.-P. Serre, Cohomological invariants, Witt invariants, and trace forms. In Cohomological invariants in Galois cohomology, volume 28 of Univ. Lecture Ser., pp. 1–100. Amer. Math. Soc., Providence, RI, 2003. Notes by Skip Garibaldi.

[Sp98] T. A. Springer, Linear algebraic groups, second edition, Progress in Mathematics, 9, Birkhäuser Boston, Inc., Boston, MA, 1998. MR1642713

Institut de Mathématiques de Jussieu, Université Paris 6, place Jussieu, 75005 Paris, France

E-mail address: mathieu.florence@imj-prg.fr

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

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