On the rigidity of moduli of curves in arbitrary characteristic

The stack $\overline{\mathcal{M}}_{g,n}$ of stable curves and its coarse moduli space $\overline{M}_{g,n}$ are defined over $\mathbb{Z}$, and therefore over any field. Over an algebraically closed field of characteristic zero, Hacking showed that $\overline{\mathcal{M}}_{g,n}$ is rigid (a conjecture of Kapranov). Bruno and Mella for $g=0$, and the second author for $g\geq 1$ showed that its automorphism group is the symmetric group $S_n$, permuting marked points unless $(g,n)\in\{(0,4),(1,1),(1,2)\}$. The methods used in the papers above do not extend to positive characteristic. We show that in characteristic $p>0$, the rigidity of $\overline{\mathcal{M}}_{g,n}$, with the same exceptions as over $\mathbb{C}$, implies that its automorphism group is $S_n$. We prove that, over any perfect field, $\overline{M}_{0,n}$ is rigid and deduce that, over any field, $Aut(\overline{M}_{0,n})\cong S_{n}$ for $n\geq 5$. Going back to characteristic zero, we prove that for $g+n>4$, the coarse moduli space $\overline M_{g,n}$ is rigid, extending a result of Hacking who had proven it has no locally trivial deformations. Finally, we show that $\overline{M}_{1,2}$ is not rigid, although it does not admit locally trivial deformations, by explicitly computing his Kuranishi family.


Introduction
A remarkable property of the moduli stack M g,n of n-pointed genus g Deligne-Mumford stable curves is that it is defined over Spec(Z) and thus over any commutative ring R via base change M R g,n M g,n Spec(R) Spec(Z) Furthermore, the same is true for the coarse moduli space M R g,n of M R g,n . In this paper we study the rigidity of M K g,n and M K g,n , where K is an arbitrary field, both in the sense of the absence of infinitesimal deformations and of automorphisms not coming form the permutations of the marked points. Along the paper, when no confusion arises, we will write simply M g,n and M g,n for M R g,n and M R g,n .
In [Ka] M. Kapranov proposed the following conjecture about the deformations of a smooth scheme.
Conjecture. Let X be a smooth scheme, and let X 1 be the moduli space of deformations of X, X 2 the moduli space of deformations of X 1 , and so on. Then X dim(X) is rigid, that is it does not have infinitesimal deformations.
Let us recall that, if X is an algebraic stack over a field K, the K-vector space of locally trivial first order infinitesimal deformations of X is H 1 (X , T X ), where T X = Hom(Ω X , O X ). While the K-vector space of first order infinitesimal deformations of X is Ext 1 (Ω X , O X ). In general these two spaces are linked by the following exact sequence The sheaf Ext 1 (Ω X , O X ) is supported on the singular locus of X . In particular, if X is smooth we have Ext 1 (Ω X , O X ) = 0 and Ext 1 (Ω X , O X ) ∼ = H 1 (X , T X ). If X is a curve the conjecture states that M g,n is rigid. Over a field of characteristic zero the rigidity of the stack M g,n has been proven by P. Hacking in [Hac,Theorem 2.1]. Furthermore, by [Hac,Theorems 2.3], the coarse moduli space M g,n does not have locally trivial deformations if (g, n) / ∈ {(1, 2), (2, 0), (2, 1), (3, 0)}. Finally in [Hac,Section 6] Hacking exhibits a moduli space of surfaces that is not rigid. Hacking's argument consists in reducing the vanishing of H 1 (M g,n , T Mg,n ) to the vanishing of the cohomology of certain ψ-classes. Since such ψ-classes are nef and big, he deduces the desired vanishing by the Kodaira vanishing theorem [Hac,Theorems A.1]. However, Kodaira vanishing works only in characteristic zero. At the best of our knowledge, the closest result to Kodaira vanishing in positive characteristic is the Deligne-Illusie vanishing theorem [DI]. However, such theorem works for ample line bundle while in positive characteristic the involved ψ-classes are just semi-ample [Ke]. Nevertheless, in Section 2, for g = 0 we manage to overcome this problem by using Kapranov's morphism f n : M 0,n → P n−3 of Remark 2.5 induced by the psi-class ψ n . Since P n−3 is smooth, such a morphism allows us to translate the desired vanishings to vanishings of line bundles on P n−3 . However, our argument do not really need the smoothness of P n−3 . It would be enough to know that ψ n induces a birational morphism to a scheme with rational quotient singularities. In Section 3 we consider more general compactifications of M g,n , namely the stacks M g,A [n] introduced by B. Hassett [Has] by assigning rational weights A[n] = (a 1 , ..., a n ), 0 < a i ≤ 1 to the markings. In particular, the classical Deligne-Mumford compactification arises for a 1 = ... = a n = 1. In genus zero, some of these spaces appear as intermediate steps of the blow-up construction of M 0,n developed by Kapranov [Ka2], while in higher genus their coarse moduli spaces may be related to the LMMP on M g,n [Moo]. Since any Hassett's stack M g,A[n] receives a birational reduction morphism from M g,n , defined by lowering the weights in Theorem 3.5 we manage to derive the rigidity of M g,A[n] from the rigidity of M g,n .
The results on the rigidity of these stacks in Theorems 2.6, 2.8, 3.5 and Remark 3.6 can be summarized in the following statement.
Theorem. Let g, n be two non-negative integers such that 2g − 2 + n > 0. Over any perfect field if g = 0, and over any field of characteristic zero if g ≥ 1, the stack M g, A[n] is rigid and the pair (M g, A[n] , ∂M g,A[n] ) does not have locally trivial deformations for any vector of weights A[n].

However, the following issue remains open.
Question. If g ≥ 1, is the stack M g,A[n] rigid over a field of positive characteristic?
In Section 4 we study the infinitesimal deformations of the coarse moduli space M g, A[n] . This coarse space is a scheme with finite quotient singularities. Therefore we expect it to be more flexible, from the point of the deformations, than the smooth stack M g,A [n] . Indeed, using the description of the singularities of M 1,2 provided in [Ma,Proposition 2.1], in Theorem 4.1 we prove that M 1,2 is not rigid and that it does not have locally trivial deformations. Finally, in Theorem 4.3 we prove that if g + n ≥ 4 then M g,A [n] does not admit locally trivial deformations for any vector of weights A[n]. In Section 5, by means of the rigidity results of the previous sections, we extend the main results of [BM], [Ma], [MM1] and [MM2] on the automorphisms of moduli of curves over fields different from the complex numbers. As it is shown in Appendix A these results can be easily generalized over suitable algebraically closed fields. The first step consists in considering non algebraically closed fields of characteristic zero. This is not hard. Indeed, in Corollary 5.2, we construct an injective morphism where K is the algebraically closure of K. Then we use the results of Appendix A on the automorphism groups of M K g,A [n] . However, with this approach we do not manage to compute Aut(M K 0,n ) when char(K) = p > 0.
To attack this problem we consider the moduli space M W (K) 0,n and we translate the problem over the field K(ξ) of characteristic zero. Finally, we use the result of Theorem 5.3 about Aut(M K(ξ) 0,n ) in characteristic zero to deduce that Aut(M K 0,n ) ∼ = S n for any n ≥ 5. In Remark 5.7 we observe that these results on M K 0,n hold for most Hassett's spaces appearing in Constructions 1.3, 1.5 and 1.6 as well. The results on the automorphism groups in Theorems 5. 3,5.4,5.5,A.2,A.5,A.7 and in Remarks 5.7 and A.4 can be summarized in the following statement.
A priori, a group scheme over a field of positive characteristic is not necessarily reduced. However, in Proposition 5.6 we show that Aut(M 0,n ) is always reduced.

Notation and Preliminaries
We begin by recalling some basic facts about moduli spaces of weighted pointed curves introduced by B. Hassett in [Has]. Let S be a Noetherian scheme and g, n two non-negative integers. A family of nodal curves of genus g with n marked points over S consists of a flat proper morphism π : C → S whose geometric fibers are nodal connected curves of arithmetic genus g, and sections s 1 , ..., s n of π. A collection of input data (g, A) := (g, a 1 , ..., a n ) consists of an integer g ≥ 0 and the weight data: an element (a 1 , ..., a n ) ∈ Q n such that 0 < a i ≤ 1 for i = 1, ..., n, and Definition 1.1. A family of nodal curves with marked points π : (C, s 1 , ..., s n ) → S is stable of type (g, A) if -the sections s 1 , ..., s n lie in the smooth locus of π, and for any subset {s i 1 , ..., s ir } with non-empty intersection we have a i 1 + ... + a ir ≤ 1, [Has,Theorems 2.1 and 3.8] proved that given a collection (g, A) of input data, there exists a connected and smooth Deligne-Mumford stack M g,A[n] of dimension 3g − 3 + n, smooth and proper over Z, representing the moduli problem of pointed stable curves of type (g, A). The corresponding coarse moduli scheme M g,A[n] is projective over Z. Fixed g, n, consider two collections of weight data A[n], B[n] such that a i ≥ b i for any i = 1, ..., n. Then there exists a birational reduction morphism associating to a curve [C, s 1 , ..., s n ] ∈ M g,A[n] the curve ρ B[n],A[n] ([C, s 1 , ..., s n ]) obtained by collapsing components of C along which K C + b 1 s 1 + ... + b n s n fails to be ample.
Let us focus on the case g = 0. In [Ka2] Kapranov proved that, over an algebraically closed field of characteristic zero, any psi-class ψ i on M 0,n is big and globally generated, and that it induces a birational morphism f i : M 0,n → P n−3 which is an iterated blow-up of linear sub spaces of P n−3 in order of increasing dimension. For instance, this construction has been used by A. Bruno and M. Mella in [BM] to study the biregular fibrations of M 0,n , and by M. Bolognesi in [Bo] for his study of GIT compactifications of M 0,n . Construction 1.3. [Has,Section 6.1] More precisely, fixed (n − 1)-points p 1 , ..., p n−1 ∈ P n−3 in linear general position: (1) Blow-up the points p 1 , ..., p n−2 , then the lines p i , p j for i, j = 1, ..., n−2,..., the (n−5)planes spanned by n − 4 of these points.
(2) Blow-up p n−1 , the lines spanned by pairs of points including p n−1 but not p n−2 ,..., the (n − 5)-planes spanned by n − 4 of these points including p n−1 but not p n−2 . . . .
(r) Blow-up the linear spaces spanned by subsets {p n−1 , p n−2 , ..., p n−r+1 } so that the order of the blow-ups in compatible by the partial order on the subsets given by inclusion, the (r − 1)-planes spanned by r of these points including p n−1 , p n−2 , ..., p n−r+1 but not p n−r ,..., the (n−5)-planes spanned by n−4 of these points including p n−1 , p n−2 , ..., p n−r+1 but not p n−r . . . .
(n − 3) Blow-up the linear spaces spanned by subsets {p n−1 , p n−2 , ..., p 4 }. The composition of these blow-ups is the morphism f n : M 0,n → P n−3 induced by the psi-class ψ n . We denote by W r,s [n] the variety obtained at the r-th step once we finish blowing-up the subspaces spanned by subsets S with |S| ≤ s + r − 2, and by W r [n] the variety produced at the r-th step. In particular W 1,1 [n] = P n−3 and W n−3 [n] = M 0,n .
There are other blow-up constructions of M 0,n , we consider other two of them. The first is due to Kapranov.
Hassett's spaces appearing in [Has,Section 6.3] are strictly related to the construction of M 0,n provided by Keel in [Ke2].
Construction 1.6. [Has,Section 6.3] We start with the variety Y 0 [n] := (P 1 ) n−3 which can be realized as Hassett's space M 0,A[n] where A[n] = (a 1 , ..., a n ) satisfy the following conditions: a i + a j > 1 where {i, j} ⊂ {1, 2, 3}, a i + a j 1 + ... + a jr ≤ 1 for i = 1, 2, 3, {j 1 , ..., j r } ⊆ {4, ..., n}, with r ≥ 2. Let ∆ d be the locus in (P 1 ) n−3 where at least n − 2 − d of the points coincide, that is the d-dimensional diagonal. Let π i : (P 1 ) n−3 → P 1 for i = 1, ..., n − 3 be the projections, and let . We define F 1 and F ∞ similarly and use the same notation for proper transforms. Consider the following sequence of blow-ups ( The variety Y h [n] obtained at the step h can be realized as Hassett's space M 0,A[n] where the weights satisfy the following conditions: Now, we consider another sequence of blow-ups starting from Y n−4 [n].
The variety Y h [n] obtained at the step h can be realized as Hassett's space M 0,A[n] where the weights satisfy the following conditions: Witt vectors. Witt vectors were introduced by E. Witt in [Wi]. In this section we recall some basic properties of the ring of the Witt vectors of a field. For a modern survey on this subject see [Haz]. Let K be field. A Witt vector over K is a sequence (a 0 , ..., a n , ...) of elements of K. We denote by W (K) the set of Witt vectors on K. If A = (a 0 , ..., a n , ... so that W (K) is a discrete valuation ring with a unique maximal ideal generated by (0, 1, 0, ...). Furthermore S n and P n are functions of the first n terms of A and B. Therefore we can truncate the vectors at the n-th entry and define the ring of truncated Witt vectors as W n (K) := {(a 0 , ..., a n−1 ) | a i ∈ K}.
Example 1.7. The first terms of addition and multiplication are given by .., a n ) −→ (a 0 , ..., a n−1 ) induce an inverse system of rings and taking the inverse limit we have Note that the inverse system is induced by the filtration determined by the maximal ideal. Therefore W (K) is a complete discrete valuation ring with residue field K and function field of characteristic zero. We have also an additive shift map which is not a ring homomorphism. Note that W n (K) = W (K)/V n W (K).
Example 1.8. If K = F p then W (K) = Z p the ring of p-adic integers. This ring has a unique maximal ideal m = pZ p and Z p /m ∼ = Z/pZ ∼ = F p . Furthermore W n (K) = Z/p n Z, any ideal of Z p is of the form p i Z p for some i ≥ 0, and Lifting schemes and automorphisms form positive to zero characteristic. Let X 1 be a scheme over a field K with char(K) = p > 0. We would like to construct a scheme X flat over W (K) whose fiber over the unique closed point of Spec(W (K)) is isomorphic to X 1 . Suppose that for any n ≥ 1 there exists a lifting X n → Spec(W n (K)) of X 1 such that X n+1 × Spec(W n+1 (K)) Spec(W n (K)) = X n . Then the limit is a Noetherian formal scheme. The next issue is the algebraization of X, that is the existence of an actual scheme X flat over W (K) induced by the formal scheme X.
Remark 1.9. For instance any smooth projective curve over a perfect field of characteristic p > 0 admits a lifting to characteristic zero. However, by [Har2,Theorem 25.8] over an algebraically closed field of characteristic p ≥ 5, there is a smooth projective 3-fold that does not admit a lifting in characteristic zero.
The moduli space M g,n is projective over Z, so it is defined and flat over any commutative ring. In particular over the ring W (K). Therefore for M g,n the issue of the lifting in positive characteristic does not arise. However, in Section 5 we will attack the problem of lifting automorphisms of M g,n from positive to zero characteristic. In what follows we recall some basic fact about the automorphism group of a scheme. Let X, Y be two schemes flat and projective over a base scheme S. Let us consider the functor By associating to an S-morphism X → Y its graph we get an open S-subscheme representing the functor Hom S (X, Y ). Now, considering morphisms which define relative isomorphisms we get a subfunctor Then if X is a scheme flat and projective over a base scheme S, as a consequence of the existence of the Hilbert scheme Hilb S (X × S X) [FAG,Chapter 5], the group of relative automorphisms Aut S (X) carries a scheme structure itself.
Remark 1.10. For a smooth projective scheme X with ample canonical bundle ω X the fact that Aut(X) has a scheme structure can be seen in a very direct way. Let k be an integer such that L := ω ⊗k X is very ample and let be the corresponding embedding. Let φ ∈ Aut(X) be an automorphism. Then φ * L ∼ = L and φ acts on the space of sections H 0 (X, L). Therefore φ extend to an automorphism of P N and we can identify Aut(X) with a subscheme of P GL(N + 1).

Now, let
A be an Artinian local K-algebra with maximal ideal m, and let X 1 be a smooth scheme over K. Let us assume that X 1 admits a family {X n } n≥1 of infinitesimal deformations. That is, for any n ≥ 1 there exists a scheme X n+1 → Spec(A/m n+1 ) such that X n+1 × Spec(A/m n+1 ) Spec(A/m n ) ∼ = X n . We want to lift an automorphism φ n of X n to an automorphism φ n+1 of X n+1 such that φ n+1|Xn = φ n .
Proposition 1.11. Let A be an Artinian local K-algebra with maximal ideal m, and let X 1 be a smooth scheme over K admitting a family of infinitesimal deformations {X n } n≥1 . For any n ≥ 1 let us consider the exact sequence 0 → m n /m n+1 → A/m n+1 → A/m n → 0. Then there exists an exact sequence Proof. Let us assume X 1 to be affine. By the infinitesimal lifting property [Har2,Proposition 4.4] given an automorphism φ n ∈ Aut(X n ) the set is non-empty and it is a principal homogeneous space under the action of H 0 (X 1 , T X 1 ) ⊗ m n /m n+1 . Now, if X 1 is not affine we can take an affine covering {X i 1 } of X 1 . By [De,Theorem 2.6] the obstruction to construct a lifting φ n+1 of φ n from the lifting φ i n+1 : 12. Proposition 1.11 states that given an n-order infinitesimal deformation φ n ∈ Aut(X n ) of an automorphism φ ∈ Aut(X 1 ) the obstruction to lift φ n to an (n + 1)-order deformation lies in H 1 (X 1 , T X 1 ) ⊗ m n /m n+1 . Furthermore if the obstruction vanishes then the lifting is unique modulo H 0 (X 1 , T X 1 ) ⊗ m n /m n+1 . In particular, if H 0 (X 1 , T X 1 ) = H 1 (X 1 , T X 1 ) = 0 then any φ n ∈ Aut(X n ) induces an unique φ n+1 ∈ Aut(X n+1 ).

Rigidity of M 0,n in positive characteristic
In this section we work over a perfect field K of any characteristic. In the first part we mainly follow [Hac,Section 3]. Let B be the boundary of M g,n and let Ω Mg,n (log(B)) be the sheaf of 1-forms on M g,n with logarithmic poles along B. The dual sheaf is the subsheaf of T Mg,n of vector fields on M g,n tangent to the boundary, that is the sheaf of first order infinitesimal automorphisms of the pair (M g,n , B). The first order locally trivial deformations of the pair (M g,n , B) are parametrized by H 1 (M g,n , T Mg,n (− log(B))).
Let π : U g,n → M g,n be the universal curve over M g,n . So U g,n is the stack of n-pointed genus g curves endowed with an extra section with no smoothness condition. Let Σ be the union of the n sections of π. We define the boundary of U g,n as Let ν : B ν → B be the normalization of the boundary, and let N be the normal bundle of the map B ν → M g,n . Then there is an exact sequence Furthermore, by [Hac,Lemma 3.1] there is a natural isomorphism where ω π is the relative dualizing sheaf of the morphism π.
By the definition of Deligne-Mumford stable curve the line bundle ω π (Σ) is ample on the fibers of π, so π * (ω π (Σ) ∨ ) = 0. Furthermore R i π * (ω π (Σ) ∨ ) = 0 for i > 1 because the fibers of π are curves. By the Leray spectral sequence we have for any i ≥ 0. Then the isomorphism (2.2) induces an isomorphism for any i ≥ 0. Now, the exact sequence (2.1) yields a long exact sequence in cohomology which by the isomorphism 2.3 translates to the long exact sequence Now, let p : U g,n → U g,n be the coarse moduli space of U g,n , and let p * ω π (Σ) be the Q-line bundle on U g,n induced by the line bundle ω π (Σ).
By [Ke,Theorem 0.4] the Q-line bundle p * ω π (Σ) is nef and big. Furthermore, if the base field has positive characteristic. Then p * ω π (Σ) is semi-ample, but this fails in characteristic zero.
In [Hac,Theorem 3.2] Hacking proves that, over a field K of characteristic zero . Furthermore, by [Hac,Corollary 4.4] the Q-line bundle on the coarse moduli space of B ν defined by N ∨ is nef and big on each component. Then Finally combining the vanishings (2.4) and (2.5) Hacking proves that Both the vanishings (2.4) and (2.5) are consequences of [Hac,Theorem A.1] which is a version of Kodaira vanishing for proper and smooth Deligne-Mumford stacks. This theorem derives from the Kodaira vanishing theorem for proper normal varieties. Therefore it holds only in characteristic zero. In what follows our aim is to treat the same problem in positive characteristic. By [Kn,Corollary 3.9] the irreducible components of the normalization B ν of B ⊂ M g,n are finite images of the stacks: where M h,S is the moduli stack of stable curves of genus h labeled by a finite set S. Let π : U h,S → M h,S be the universal family of M h,S , and let σ i : M h,S → U h,S be the sections of π. The ψ-classes over M h,S are defined as Recall that by [Kn,Section 1,2] there exists an isomorphism of stacks c : M g,n+1 → U g,n identifying the last forgetful morphism p n+1 : M g,n+1 → M g,n with the projection π : U g,n → M g,n+1 .
The following lemma links the pullback of N ∨ to the components of the boundary B ν and the line bundle ω π (Σ) with suitable ψ-classes.
Note that Lemma 2.1 allows us to interpret both the vanishings (2.4) and (2.5) as vanishings of the cohomology of suitable ψ-classes. By [Ke,Theorem 0.4] if the base field K has positive characteristic then the Q-line bundle p * ω π (Σ) is nef, big and semi-ample on the coarse moduli space M g,n . Therefore there exists an integer m > 0 such that the Q-line bundle L := p * ω π (Σ) ⊗m defines a birational morphism In what follows we analyze the morphism φ L in the case g = 0. Note that when g = 0 any n-pointed stable curve is automorphisms free and therefore the coarse moduli space M 0,n is smooth and coincides with the stack. Let [C, x 1 , ..., x n ] ∈ M 0,n be the class of a stable curve. By [Kn, Corollaries 1.10 and 1.11], over any field, the line bundle L := ω C (x 1 + ... + x n ) is very ample and h 0 (C, L) = n − 1. Therefore it defines an embedding φ L : C → P n−2 Lemma 2.2. Let (C, x 1 , ..., x n ) be a genus zero n-pointed stable curve. Then the points p i := φ L (x i ) are in linear general position in P n−2 .
Proof. Let us suppose that there are n − 1 of the p i 's, let us say p 2 , ..., p n contained in a hyperplane H ⊂ P n−2 . Then there is a section ξ of ω C (x 1 + ... + x n ) vanishing at x 2 , ..., x n .
Recall that a section of ω C (x 1 + ... + x n ) is a differential form ξ with at most simple poles at x i and if x ∈ C is a node x = C 1 ∩ C 2 then ξ |C 1 and ξ |C 2 have at most simple poles at x and Res x (ξ |C 1 ) = Res x (ξ |C 2 ). Therefore the section ξ is regular at x 2 , ..., x n and has a pole in x 1 . Now, the graph of C is a tree, that is a connected graph without simple cycles. Then there exist at least two components of C intersecting only one other component. Let C 1 be one of these components not containing x 1 , and let C 2 be the unique component intersecting C 1 . The restriction ξ |C 1 has at most one simple pole at C 1 ∩ C 2 and since C 1 is rational we get ξ |C 1 = 0. Furthermore, by the residue condition we have that ξ does not have a pole at C 1 ∩ C 2 . So we can remove C 1 and apply the same argument to C 2 to get ξ |C 2 = 0. Proceeding recursively on the components of C we get ξ = 0.
Proposition 2.3. The line bundle L := ω π (Σ) over the universal curve U 0,n defines an embedding φ L : U 0,n → P(π * L) over M 0,n , and we have the following commutative diagram Proof. By [Kn, Corollaries 1.10 and 1.11] L is π-very ample. So R 1 π * L = 0, and π * L is a rank n − 1 vector bundle on M 0,n . Finally, by Lemma 2.2 there exists a unique isomorphism φ L ∼ = P n−2 × M 0,n over M 0,n .
Proposition 2.4. The line bundle ψ n over M 0,n defines a birational morphism f n : M 0,n → P n−3 .
Proof. By Proposition 2.3 we have an embedding U 0,n P n−2 × M 0,n M 0,n π g φ L Now, take a point [C, x 1 , ..., x n ] ∈ M 0,n , consider the curve φ L (π −1 ([C, x 1 , ..., x n ])) ⊂ P n−2 and its tangent line Now, by construction we have f * n O P n−3 (1) = ψ n , and by the second part of Lemma 2.1 we can identify ψ n with ω π (Σ) which is nef and big by [Ke,Theorem 0.4]. Therefore f n is a birational morphism.
Remark 2.5. The birational morphism f n is nothing but the analogous over an arbitrary field of the morphism constructed by M. Kapranov in [Ka2] over an algebraically closed field of characteristic zero. Moreover Kapranov proved that such morphism factorizes as a sequence of blow-ups along linear subspaces of P n−3 . Now, we are ready to prove the main rigidity result of this section.
Theorem 2.6. Let K be a perfect field. Then that is M 0,n is rigid for any n ≥ 4.
Proof. Recall that we have the exact sequence (2.1) inducing the long exact sequence Let us consider the line bundle ω π (Σ) over U 0,n ∼ = M 0,n . By Lemma 2.1 ω π (Σ) identifies with ψ n+1 . Therefore, by Proposition 2.4 we have a birational morphism [Har,Theorem 5.1] we have that By the projection formula we get Since f is a birational morphism between smooth varieties we have f . Furthermore, since the base field K is perfect by [CR,Theorem 1] we have R i f * ψ ∨ n+1 = 0 for i > 0, and by the Leray spectral sequence we get Summing up we proved the following vanishings: Now, let us consider the dual of the normal bundle N ∨ . By Lemma 2.1 the pullback of N ∨ to M 0,S 1 ∪{n 1 +1} × M 0,S 2 ∪{n 2 +1} is identified with pr * 1 ψ n 1 +1 ⊗ pr * 2 ψ n 2 +1 . Note that since g = 0 we do not have the boundary divisor parametrizing irreducible nodal curves. By Proposition 2.4 the pullback of N ∨ to M 0,S 1 ∪{n 1 +1} × M 0,S 2 ∪{n 2 +1} defines a birational morphism

By Künneth formula we have
Now, by [Har,Theorem 5.1] we get H i (P s 1 −2 × P s 2 −2 , O(−1, −1)) = 0 for i < n − 4. Finally, applying to N ∨ the argument we used in the first part of the proof for ω π (Σ) ∨ we get By the long exact sequence 2.6 we conclude that The last assertion derives easily from the first if n ≥ 6. If n = 4 then M 0,4 ∼ = P 1 and H 1 (P 1 , T P 1 ) = 0 being T P 1 = O P 1 (2). Finally, let us consider the case n = 5. Then boundary B ⊂ M 0,5 is the union of ten irreducible components isomorphic to P 1 . The surface M 0,5 is a Del Pezzo surface of degree five isomorphic to the blow-up of P 2 in four points p 1 , ..., p 4 in linear general position. From this point of view we can interpret the ten irreducible components of B as the four exceptional divisors and the strict transforms of the six lines p i , p j for i, j = 1, ..., We conclude that H 1 (M 0,5 , T M 0,5 ) = 0 that is M 0,5 is rigid.
Remark 2.7. For the benefit of the reader we give another proof of the rigidity of M 0,5 using an argument of deformation theory. Let S be a surface and Z = {p 1 , ..., p n } ⊂ S be a reduced subscheme of dimension zero. Let : S → S be the blow-up of S at Z. Consider the exact sequence 0 → * Ω S → Ω S → i * Ω E/Z → 0 where i : E → S is the exceptional divisor. Note that Hom(i * Ω E/Z , O S ) = 0 and by Grothendieck duality Ext 1 (i * Ω E/Z , O S ) ∼ = i * T E/Z (E). So dualizing the above exact sequence we get for any i ≥ 0 and we get the following exact sequence in cohomology Since the map between the tangent spaces H 1 ( S, T S ) → H 1 (S, T S ) is surjective and the map between the obstruction spaces H 2 (S, * T S ) → H 2 (S, T S ) is injective the map Def S → Def S is smooth of relative dimension 2n − dim H 0 (S, T S ) + dim H 0 ( S, T S ). This means that the obstructions to deforming S are exactly the obstructions to deforming S. The vector space K 2n parametrizes the deformations of Z inside S and the spaces H 0 ( S, T S ), H 0 (S, T S ) parametrize the infinitesimal automorphisms of S and S respectively. If the map H 0 (S, T S ) → K 2n is surjective then H 1 ( S, T S ) ∼ = H 1 (S, T S ) and the deformations of S are induced by deformations of S. Otherwise the deformations of Z inside S induce non-trivial deformations of S. Now, take S = P 2 . Then H 0 (S, T S ) ∼ = T Id Aut(P 2 ) has dimension 8. If n ≤ 4 the map T Id Aut(P 2 ) → K 2n is surjective and H 1 ( S, T S ) ∼ = H 1 (P 2 , T P 2 ). Indeed if n ≤ 4 there is an automorphism mapping Z to any other set of n points in general position and moving Z inside P 2 just induces trivial deformations of S. Furthermore, since P 2 itself is rigid we have H 1 (P 2 , T P 2 ). In particular for n = 4 we get H 1 (M 0,5 , T M 0,5 ) ∼ = H 1 (P 2 , T P 2 ) = 0.
Recall that given a pair (X, Y ) where X is a smooth scheme and Y is a smooth subscheme of X a locally trivial deformation of the pair (X, Y ) is an infinitesimal deformation of Y in X which is locally trivial. By [Hac,Theorem 2.2] over a field of characteristic zero the pair (M g,n , ∂M g,n ) does not have locally trivial deformations. By the proof of Theorem 2.6 we get the same result for g = 0 over any perfect field.
Furthermore the pair (M 0,n , ∂M 0,n ) does not have locally trivial deformations for any n ≥ 4.

On the rigidity of M g,A[n]
Let ρ : M g,A[n] → M g,B[n] be a reduction morphism between Hassett's moduli stacks. By [Has,Proposition 4.5] ρ contract the boundary divisors D I,J = M 0,A I × M g,A J with A I = (a i 1 , ..., a ir , 1), A J = (a j 1 , ..., a j n−r , 1) and c = b i 1 + ... + b ir ≤ 1 for 2 < r ≤ n. By [Has,Remark 4.6] ρ can be factored as a composition of reduction morphisms ρ = ρ 1 • ...  We will need the following commutative algebra lemma.
Lemma 3.1. Let R be a commutative ring. Given the following commutative diagram of Rmodules 0 0 there exists an isomorphism δ : C 1 → C 2 .
Proof. Let c 1 ∈ C 1 be an element. Then there exists a 1 ∈ A 1 such that c 1 = π 1 (a 1 ). We define δ(c 1 ) = π 2 (γ(a 1 )). It is straightforward to check that δ is well defined and that it is an isomorphism. Now, we are ready to explicit the normal bundle of M g,C[s+1] ⊂ M g,B[n] in terms of the first psi-class.
Therefore applying the functor Hom(−, O C ) and taking stalks at the point x 1 ∈ C we get the following exact sequence On the other hand we have the same exact sequence on [C, x 1 , ..., x 1 , ..., x s+1 ] seen as a point in M g,C[s+1] . Therefore we may consider the following diagram where the vertical maps are defined as

By
Proof. Let us write the exceptional divisor as is the second projection. We proceed by induction on the dimension of Hassett's stacks. If n = 3 then any Hassett's space is just a point and H 1 (M 0,3 , ψ ∨ i ) = 0. Furthermore we have H 1 (D I,J , ρ * ψ ∨ i ) = H 1 (D I,J , pr * 2 ψ ∨ 1 ). Now, let us consider the exact sequence B[n] . Since R 1 ρ * ρ * ψ ∨ i = 0 taking the long exact sequence in cohomology we conclude that We will need the following lemma relating the first order infinitesimal deformations of a stack to the deformations of its blow-up along a smooth substack.
Lemma 3.4. Let X be a smooth stack and Z ⊆ X be a smooth substack. Then Furthermore in the following diagram Proof. Let X = Bl Z X be the blow-up and : X → X be the blow-up morphism with exceptional and H i ( −1 (z), L | −1 (z) ) = 0 for any i. This implies that R i * L = 0 for any i. Therefore R i * * N Z/X |E ∼ = R i * Q for any i.
Then the Leray spectral sequence for Q degenerates and we get H i (E, Q) = H i (Z, N Z/X ) for any i ≥ 1.
Now, we are ready to prove the main result of this section. By the formula 2.7 in the proof of Theorem 2.6 we get the statement in the case g = 0 over any perfect field. Let us prove the same for g ≥ 1 over any field of characteristic zero. By Lemma 2.1 the line bundle ψ n on M g,n is identified with the pullback of the line bundle ω π (Σ) via the isomorphism c : M g,n → U g,n−1 . Furthermore, by [Ke,Theorem 0.4] the Q-line bundle p * ω π (Σ) is nef and big, where p : U g,n−1 → U g,n−1 is the map on the coarse moduli space. Since we are over a field of characteristic zero we can apply [Hac,Theorem A.1] to the line bundle p * ω π (Σ). In particular we get H 1 (M g,n , ψ ∨ n ) = H 1 (U g,n−1 , ω π (Σ)) = 0. The situation for the other psi-classes is clearly completely symmetric and we conclude that H 1 (M g,n , ψ ∨ i ) = 0 for any i = 1, ..., n. Now, let us consider the second statement. Since by Theorem 2.6 we have H 1 (M 0,n , T M 0,n ) = 0 over any perfect field and by [Hac,Theorem 2.1] we have H 1 (M g,n , T Mg,n ) = 0 for g ≥ 1 on any field of characteristic zero we may proceed by induction on k and prove the second statement for a single morphism ρ i : M g,A [n] → M g,B [n] . Let E be the exceptional locus of the morphism ρ, and let Z = ρ(E). We denote by T 1 Def (M g,B [n] ,Z) the space of first order infinitesimal deformation of the couple (M g,B [n] , Z). Then we have the following exact sequence ).
By Proposition 3.2 we have N Z/M g,B [n] ∼ = (ψ ∨ 1 ) ⊕(r−1) and by Proposition 3.3 we have Finally, by induction we conclude that Remark 3.6. The same argument used in Theorem 2.8 implies that the pair (M g,A[n] , ∂M g,A[n] ) does not have locally trivial deformations for any n ≥ 4, over any perfect field if g = 0 and over any field of characteristic zero if g ≥ 1.
Remark 3.7. For Hassett's spaces M 0, Ar,s[n] in Construction 1.3, X k [n] in Construction 1.5, and Y h [n] in Construction 1.6 for n − 4 ≤ h ≤ 2n − 9 we can say more. Indeed they admit a birational morphism on P n−3 factorizing Kapranov's morphism of Proposition 2.4. Therefore, the argument used in the proof of Theorem 2.6 yields Furthermore if n = 4, 5 this result holds for i = 0, 1.

On the rigidity of the coarse moduli space M g,A[n]
In this section we consider the coarse moduli space M g,A [n] . If 2g − 2 + n > 0 the M g,A[n] is a smooth Deligne-Mumford stack and therefore M g,A[n] is a normal scheme with finite quotient singularities. If g = 0 we have M 0,A[n] ∼ = M 0,A[n] because any stable n-pointed genus zero weighted curve is automorphism free. By Theorem 3.5 we know that M 0,A[n] is rigid. Therefore we consider the case g ≥ 1. Let us recall that, in characteristic zero, by [Hac,Theorem 2.3] the scheme M g,n does not have locally trivial deformations if (g, n) / ∈ {(1, 2), (2, 0), (2, 1), (3, 0)}. Let us begin with (g, n) = (1, 2). We recall the notion of cyclic quotient singularity. Any such singularity is of the form A n /µ r , where µ r is the group of r-roots of unit. The action µ r A n can be diagonalized, and then written in the form for some a 1 , ..., a r ∈ Z/Z r . The singularity is thus determined by the numbers r, a 1 , ..., a n . Following the notation of [Re], we denote by 1 r (a 1 , ..., a n ) this type of singularity. By [Ma,Proposition 2.1], over an algebraically closed field K with char(K) = 2, 3, the moduli space M 1,2 is a rational surface with four singular points. Two singular points lie in M 1,2 , and are: -a singularity of type 1 4 (2, 3) representing an elliptic curve with automorphism group Z 4 and marked points [0 : 1 : 0] and [0 : 0 : 1]; -a singularity of type 1 3 (2, 4) representing an elliptic curve with automorphism group Z 6 and with marked points [0 : 1 : 0] and [0 : 1 : 1].
The remaining two singular points lie on the boundary divisor parametrizing reducible curves, and are: -a singularity of type 1 6 (2, 4) representing a reducible curve whose irreducible components are an elliptic curve with automorphism group Z 6 and and a smooth rational curve connected by a node; -a singularity of type 1 4 (2, 6) representing a reducible curve whose irreducible components are an elliptic curve with automorphism group Z 4 and a smooth rational curve connected by a node.
Theorem 4.1. The coarse moduli space M 1,2 does not have locally trivial deformations and it is not rigid.
Remark 4.2. The expression of the versal deformation of S is with (µ 1 , µ 2 ) ∈ K 2 . In particular these deformations are not locally trivial.
Now, let us consider Hassett's moduli space M g,A[n] such that g + n ≥ 4. Let ρ : M g,n → M g,A[n] be a reduction morphism. Therefore ρ contracts the boundary divisors D I,J = M 0,A I × M g,A J with A I = (1, ..., 1, 1), A J = (1, ..., 1, 1) and c = a i 1 + ... + a ir ≤ 1 for 2 < r ≤ n. The morphism ρ can be factored as a composition of reduction morphisms ρ = ρ 1 • ... Let us consider the exact sequence in cohomology Since R i ρ * ρ * T M g,A[n] = 0 for i > 0 we conclude On the other hand, if g + n ≥ 4 then H 1 (M g,n , ρ * T M g,A[n] ) ∼ = H 1 (M g,n , T M g,n ) = 0 In what follows our aim is to understand how automorphisms behave with respect to field extensions.
Lemma 5.1. Let K ⊆ L be a field extension, X K a scheme over K and X L := X K × Spec(K) Spec(L). Any morphism f K : X K → X K over K induces a unique morphism f L : X L → X L over L.
Proof. Since X L is the fiber product of X K and Spec(L) over Spec(K) we have the following commutative diagram By the universal property of the fiber product there exists a unique morphism f L : X L → X L making the diagram commutative.
Corollary 5.2. Let K ⊆ L be a field extension, X K a scheme over K and X L := X K × Spec(K) Spec(L). Then there exists an injective morphism of groups χ : Aut(X K ) → Aut(X L ).
Proof. Let φ K ∈ Aut(X K ) be an automorphism. By Lemma 5.1 φ K induces a unique morphism φ L : X L → X L of schemes over L. Clearly, by Lemma 5.1, φ −1 K ∈ Aut(X K ) induces the inverse of φ L in Aut(X L ). Then φ L ∈ Aut(X L ) and we get a morphism of groups Then, a fortiori, φ K = Id X K and χ is injective.
In the following we apply these simple results to moduli spaces of curves.
Proof. Let K be the algebraic closure of K. By Corollary 5.2 on the field extension K ⊆ K we have an injective morphism By Theorem A.1 Aut(M K 0,n ) ∼ = S n , so the morphism χ is an isomorphism. The second part of the statement follows by the same argument and by Theorem A.2. Finally, the same argument gives the statement on the stack M g,n .
Furthermore by Corollary 5.2, and Theorems A.5, A.7, building on the proof of Theorem 5.3 we get the analogous statement for Hassett's moduli spaces. We denote by A A[n] ⊆ S n the group of admissible permutations, see Definition A.6.
Theorem 5.4. Let K be a field of characteristic zero. The automorphism groups of Hassett's spaces appearing in Construction 1.3 for r ≥ 2, in Construction 1.5 for 1 ≤ k ≤ n − 4, and in Construction 1.6 for n − 4 ≤ h ≤ 2n − 9, are given by for any n ≥ 5. Furthermore, if K is any field with char(K) = 2 then for any (g, n) = (2, 1) such that 2g − 2 + n ≥ 3.
Finally, we want to compute the automorphism group of M 0,n over a field of positive characteristic.
Theorem 5.5. Let K be a field of characteristic p > 0 and let M 0,n be the moduli space of pointed rational curves over K. Then Aut(M 0,n ) ∼ = S n for any n ≥ 5.
Proof. Since the moduli space of pointed rational curves is defined over Z we have a family of infinitesimal deformations {(M 0,n ) h } h≥1 , where (M 0,n ) h is a scheme over W h (K), and a scheme M W (K) 0,n → Spec(W (K)). By Proposition 1.11 we have the exact sequence Now, let us assume K to be perfect. By Theorem 2.6 we have Now, H 1 (M 0,n , T M 0,n ) = 0 implies that any automorphism of (M 0,n ) h lifts to an automorphism of (M 0,n ) h+1 . Furthermore H 0 (M 0,n , T M 0,n ) = 0 implies that this lifting is unique. So, starting from an automorphism φ ∈ Aut(M 0,n ) we construct a family of automorphisms φ • = {φ h } h≥1 where φ h is an automorphism of (M 0,n ) h . The direct limit is a formal scheme over the formal spectrum Spf(W (K)) of W (K). By [FAG,Section 8.1.5] φ • induces a unique automorphism φ : M 0,n → M 0,n and by [FAG,Corollary 8.4.7] φ induces an unique automorphism φ of M W (K) 0,n → Spec(W (K)). Now, recall that W (K) is a discrete valuation ring with a closed point x ∈ Spec(W (K)) with residue field K and a generic point ξ ∈ Spec(W (K)) with residue field of characteristic zero. Let (M W (K) 0,n ) ξ be the general fiber of M W (K) 0,n → Spec(W (K)). By taking the restriction of φ to (M W (K) 0,n ) ξ we get a morphism of groups Clearly if φ and ψ are two automorphisms of M 0,n inducing the same automorphism of (M W (K) 0,n ) ξ then φ = ψ because ξ is the general point of Spec(W (K)) and M W (K) 0,n is separated. In particular φ and ψ coincide on the special fiber (M So the morphism χ is injective. Furthermore, by Theorem 5.3 Aut((M W (K) 0,n ) ξ ) ∼ = S n for any n ≥ 5, hence χ is an isomorphism. If K is any field of characteristic p > 0 we can consider its algebraic closure K which is a perfect field of characteristic p. By Corollary 5.2 we have an injective morphism By the first part of the proof we have Aut(M K 0,n ) ∼ = S n for any n ≥ 5. This implies Aut(M 0,n ) ∼ = S n for any n ≥ 5.
is an isomorphism. So C ∼ = C, [C ] := φ([C]) = [C] and φ = Id M g . For any [C] ∈ M g the restriction of φ to the fiber of π 1 defines an automorphism of the fiber. Since g > 2 we conclude that φ is the identity on the general fiber of π 1 so it has to be the identity on M g,1 .
-Assume g = 1, 2. Note that φ restricts to an automorphism of the fibers of π 1 and π 2 . So φ defines an automorphism of the fiber of π 1 with at least two fixed points in the case g = 1, n ≥ 3 and one fixed point in the case g = 2, n ≥ 2. Since the general 2-pointed genus 1 curve and the general 1-pointed genus 2 curves have no non-trivial automorphisms we conclude as before that φ restricts to the identity on the general fiber of π 1 , so φ = Id M g,n .
By [Ma,Theorem 3.10], if K is algebraically closed of characteristic zero we have that Aut(M 2,1 ) is trivial. The proof is based on the fact that any automorphism of M 2,1 preserves the boundary and on Royden's theorem [Moc,Theorem 6.1]. The first statement is true over any algebraically closed field K with char(K) = 2. Lemma A.3. Let K be an algebraically closed field with char(K) = 2. Then any automorphism of M g and of M 2,1 preserves the boundary.
Proof. Let λ be the Hodge class on M g . It is known that λ induces a birational morphism f : M g → X on a projective variety whose exceptional locus is the boundary ∂M g , see [Ru]. Assume that there exists an automorphism φ : M g → M g which does not preserve the boundary. Then there is a point [C] ∈ ∂M g such that φ([C]) = [C ] ∈ M g . Now f •φ is a birational morphism whose exceptional locus is φ −1 (∂M g ), and by the assumption on φ we have φ −1 (∂M g ) ∩ M g = ∅. So we construct a big line bundle on M g whose exceptional locus is not contained in the boundary and this contradicts Theorem [GKM,Theorem 0.9]. The boundary of M 2,1 has two codimension one components. The component ∆ irr,1 parametrizing irreducible nodal curves, and the component ∆ 1,1 parametrizing curves with two genus one components. Clearly ∆ irr,1 and ∆ 1,1 are mapped to the respective components of ∂M 2 by the forgetful morphism π : M 2,1 → M 2 . Let φ be an automorphism of M 2,1 . By [GKM,Theorem 0.9] we have the commutative diagram Now, if φ does not preserve the boundary then φ does not preserve the boundary as well. This contradicts the first part of the proof.
The main results of [MM1] and [MM2] on the automorphism groups of Hassett's moduli spaces can be generalized on fields different from C as well. Let us begin by the case g = 0. Since Constructions 1.3, 1.5, 1.6, [MM1, Theorem 2.6, Lemma 3.2] and [MM2, Theorem 2.6] hold over any algebraically closed field of characteristic zero we have the following. Now, let us move to the case g ≥ 1. Notice that [MM1, Proposition 2.7 and Lemma 2.8] hold on any algebraically closed field K with char(K) = 2 because they are both consequences of [GKM,Theorem 0.9]. In the case of M g,n any permutation of the markings induces an automorphism. However, in the more general setting of Hassett's spaces this is not true [MM1,Example 3.6]. To avoid this problem we consider just particular permutations.
Furthermore we define A A[n] ⊆ S n to be the sub-group generated by admissible transpositions. Now, with the same proofs of [MM1, Theorems 3.12 and 3.15] we have the following.
Theorem A.7. Let K be an algebraically closed field with char(K) = 2. Let M g,A[n] be Hassett's moduli stack parametrizing weighted n-pointed genus g stable curves, and let M g,A[n] be its coarse moduli space. If g ≥ 1, (g, n) = (2, 1) and 2g − 2 + n ≥ 3 then Remark A.8. We have that M 2,A[1] ∼ = M 2,1 for any vector of weights. By [Ma,Theorem 3.10], over an algebraically closed field of characteristic zero, we have that Aut(M 2,A[1] ) is trivial.