Comments on the ELSV compactification of Hurwitz stacks

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Comments on the ELSV compactification of Hurwitz stacks

Bashar Dudin

To cite this version:

Bashar Dudin. Comments on the ELSV compactification of Hurwitz stacks. 2014. �hal-00705862v2�

BASHAR DUDIN

Abstract. We revisit Ekedahl, Lando, Shapiro and Vainshtein’s compactification of the stack of simply ramified covers of the projective line except for a fixed ramification profile above infinity. In particular we draw a connection with the Harris and Mumford stack of admissi- ble covers showing that the boundary of the ELSV compactification appears as a contraction of the boundary of the stack of admissible covers. This lays the needed foundations for a combinatoral interpretation of boundary points of the ELSV compactification.

Contents

1. Introduction 1

2. Toroidal construction of twisted polar parts 3

3. Polar curves 5

4. Cone structure and projectivization 13

5. Residues and the ELSV compactification 16

6. Preliminaries on admissible covers 18

7. Extension of theLLmorphism 19

8. Rigidifications and the Hurwitz projection 23

Appendix A. Compatibility of branch divisors 26

References 28

1. Introduction

Fix integers (g,n)∈ N×N^∗ distinct from (0,1) and (0,2). A genusg cover of profile

~k= (k₁,... ,k_n)∈(N^∗)ⁿis a ramified coverφ:X→Y from a genusgsmoothn-marked curve (X,p=p₁,... ,p_n) on a genus 0 1-marked smooth curve (Y ,q_∞), simply branched away from q∞and which sendsp_ionq∞(and only thep_is) with multiplicityk_i. The Hurwitz number µ_g,~k is the sum over g coversφof profile~k havingr fixed branch points each weighted by 1/Aut₀(φ). The automorphism group Aut₀(φ) is the group of automorphisms of the domain curve that fix the marked points and commute toφ. In case the base field isCthe celebrated ELSV formula [10, 1.1]

(1) µ_g,~k=r!

Yn i=1

k_i^ki k_i!

Z

M_g,n

c E^∨_g,n (1−k₁ψ₁)· · ·(1−k_nψ_n)

relatesµ_g,~kto an intersection number on the stack of stable genusg n-marked curvesM_g,n ([8]). It was announced in [11] with a sign error, corrected and proved in [13] in the case

~k= (1,... ,1) and finally proved in the general case in [10]. Techniques in [13] and later on [15] depend on a fine understanding of Gromov-Witten theory ofP¹, those in [10] build up a geometry reflecting the right-hand side of 1. This geometry is the main interest of this paper. It involves constructing compactifications of stacks of covers ofP¹ that have a projective cone structure overM_g,n. The question we’re interested in is how are these

“ELSV” compactifications related to more standard compactifications of covers ofP¹such

Date: February 14, 2014.

1

as [17], [25], [1] or [6]. The answer to this question is the content of theorem 8.4. In order to give a smooth functorial treatment of the subject we had to give a satisfactory algebro- geometric understanding of the cone of twisted polar parts [10, 3.4]. In sections 2 and 3 we give two constructions of this cone, they are isomorphic if the characteristic of the base field is big enough. The present work makes clear the fact that the ELSV formula is still valid as long as the characteristic of the base field doesn’t divide any of thek_is and the number of branch points away fromq∞. Lastly we mention that theorem 8.4 is the starting point of the combinatorial description of boundary points of the ELSV compactification announced in [9]. It is the subject of a forthcoming paper by the author.

Throughout the paper k is an algebraically closed field of characteristic p≥ 0 and schemes are algebraic schemes over k. We denote H_g,~k the stack of g covers of profile

~k. An automorphism of a pointφ:X →Y inH_g,~k is a couple of isomorphisms (α,β) of the domain and target, fixing the marked points and such thatβφ =φα, details can be found in [17], [12] and [6]. More generally, following [6], if given a set{ǫ_j}^r_j=1^¯ of partition ofd=P

i=1k_i one can define the stackH^ǫ_g,~k ofg covers that have profile~kabove the only marked point of the target and ramification indices given by the partitionsǫ_j over each branch point. DenoteH^•_g,~kthe finite disjoint union of all suchH^ǫ_g,~k.

The starting point of the ELSV compactification comes from the following understand- ing of the moduli stack ofgcovers of profile~k: denotePg,~k the space of couples ((X,p),̺) where (X,p) is a smoothn-marked genusg curve and̺= (̺₁,... ,̺_n) is a global section of O_X(P

ik_ip_i)/O_Xcalled a polar part. Locally atp_i we ask for̺_i to be of the form

̺_i=a_ki t_i^ki +a_ki−1

t^k_iⁱ⁻¹ +· · ·+a₁

t_i wherea_ki,0.

We have a naturalG_m-action on the fibers ofP_g,~k overM_g,n. The residue theorem says the zero locus of the residue linear formκ7→P

iRes_pi(κ̺_i) forκ∈H⁰(X,Ω_X) is precisely the set of meromorphic functions (up to the addition of a constant) whose behavior in the neighborhood of p_i is given by̺_i. The quotient of this locus by theG_m-action coming fromP_g,~k isH_g,~k. The ELSV approach is to extendP_g,~k into a cone over the proper stack of stable curves M_g,n for which the residue theorem still makes sense. Notice first that one can still make sense out of polar parts at the marked points of any prestable genusg n-marked curve (see [5]), in particular this is as well the case for stable curves. In [10] the authors construct the desired extension ofP_g,~k locally overM_g,n. Take a stablen-marked curve (X,p). If given a local equationt_iat the neighborhood ofp_ia polar part atp_iis given by ak_i-tuple (a_ki,a_ki−1,... ,a₁) witha_ki , 0. In loc. cit. the authors choose to describe a polar part using coordinates (u_i,c₁,... ,c_ki−1) withu_i,0 corresponding to the polar part

u_i t_i

!k_i

+c₁ u_i t_i

!k_i−1

+· · ·+c_ki−1 u_i t_i

! .

For this to make sense one has to mod out the coordinates by the action of µ_k

i having weights (1,1,2,... ,k_i). ExtendingP_g,~kis to allowu_i = 0. The resulting extension ofP_g,~k to M_g,nwhich shall still be writtenP_g,~kis locally overM_g,nisomorphic to

Yn i=1

[A^ki/µ_k

i].

This is the model we keep in mind through sections 2 and 3 while giving more natural and functorial constructions ofPg,~k.

Conventions on stacks. We use the conventions of [4]. In particular, an algebraic stack is an algebraic stack overkin the sense of [23] that is quasi-separated and locally of finite type onk. A Deligne–Mumford stack is an algebraic stack in the sense of definition [8, 4.6], i.e. an algebraic stack whose diagonal is unramified. By a morphism of stacks we

mean a 1-morphism. A Deligne–Mumford stack comes with the Grothendieck topology of the small étale site. An algebraic stack that is not Deligne–Mumford has the faithfully flat of finite presentation topology. The action of a group scheme on a stack and the resulting quotient is to be understood in the sense of [28]. AnA¹-action on a stack in the sense of [4] is called anA¹-structure in order to avoid confusion with the notion of a group action.

Acknowledgements. The present work is part of the author thesis and he would like to thank his advisor J. Bertin for his guidance and support.

2. Toroidal construction of twisted polar parts

We give an alternative description of the stack of twisted polar partsP_g,~k making use of toroidal techniques due to [20]. This approach is of local nature on the base. We start by focusing on the 1-marked case.

2.1. The group of formal diffeomorphisms of a marked point. Let (π:X →S,D) be a prestable 1-marked curve over an affine local schemeS= Spec(A). Letpbe a point in the support ofDhaving images=π(p) andta local equation ofDatp. Sinceπis smooth atp the formal completion ofO_prelative to the maximal ideal is isomorphic toAJtK. The ring P of Laurent tails atpcan be written

P=AJtK[t⁻¹]/AJtK.

An element̺∈Pcan be uniquely written as

̺= X

0<ℓ≤k

a_ℓ

t^ℓ withk >0 anda_k,0

The integerk is the order of̺. The set of Laurent tails atp of orderk is writtenP_k, it corresponds to the set of polar parts atpof orderk.

Define Diff_A as the set ofA-automorphisms ofAJtKfixing the sectiont= 0. It is com- posed of elementsf(t) =α₀t¹+α₂t²+· · ·such thatα₀∈A^∗and has a group structure given byf(t)h(t) =h(f(t)). One has a left action of Diff_A onP defined byf(t)·̺(t) =̺(f(t))_<

where the right-hand side denotes the Laurent tail of ̺(f(t)). It is clear that ̺(t) is in- variant under the action of an element of the formf(t) =t+α_kt^k+1+α_k+1t^k+2+· · ·. Such elements define a normal subgroup Diff_A(k) of Diff_A. The elements of the quotient group Diff_A,k= Diff_A/Diff_A(k) are of the formf(t) =α₀t¹+α₁t²+· · ·+α_k−1t^k.

Lemma 2.1. Diff_A,kacts transitively on the set of Laurent tails of orderkwith isotropy group µ_k.

Proof. Transitivity results from the fact any such Laurent tail can be written̺(t) =t^−kϕ(t)_<

for a truncated invertible formal seriesϕ(t)_<. The stabilizer oft^−k is clearlyµ~k. To look for affine embeddings ofP_k we study affine equivariant embeddings of Diff_A,k. 2.2. The group structure ofDiff_k. Recall the algebraic groupG_m,Ais the spectrum of the gradedA-algebra

A[M] =M

γ∈Z

Aχ^γ.

The set of variables M is the group of characters of G_m,A. Denote N the group of 1- parameter subgroups ofG_m,A.

In the sequel we drop the indexArelative to the base scheme we work on. We thus writeG_minstead ofG_m,Aand Diff_k instead of Diff_A,k.

The group Diff(2) defines a normal subgroup of Diff_k. The corresponding quotient group is equal to Diff₁which is the automorphism group ofOb_p/Ob_p(−p) and is thus equal toG_m. This gives an extension of groups

(2.1) 1 U_k Diff_k G_m 1.

Given a local coordinatetatpan elementf(t)∈ U_kis of the formf(t) =t+α₁t²+· · ·+α_k−1t^k and henceU_k is unipotent. The coordinate choicetgives sections of Diff_k+1 →Diff_k for k >0 sendingf(t) =α₀t¹+· · ·+α_k−1t^kon the corresponding element in Diff_k+1withα_k= 0.

We get an isomorphism ofU_k+1/U_k= Diff_k+1/Diff_konG_ashowing Diff_kis a solvable group of unipotent radicalU_k, and a splitting of 2.1 sendingλ∈G_monθ_t(λ) =λt. The image of θ_tin Diff_kis a maximal torus and each maximal torus in Diff_k appears this way.

For Diff_k=G_m⋉U_kthe action ofG_monf(t) =t+α₁t²+· · ·+α_k−1t^k∈ U_k is written θ_t(λ)⁻¹f(t)θ_t(λ) =t+χ⁻¹(λ)α₁t²+χ⁻²(λ)α₂t³+· · ·+χ^−(k−1)(λ)t^k.

The weights of theG_m-action onU_k areχ⁻¹,χ⁻²,...,χ^−(k−1). For Diff_k=U_k⋊ G_mone gets the weightsχ¹,χ²,...,χ^(k−1). More generally ifu₁,u₂,...,u_kare coordinates ofU_ksuch that u_ℓis non-zero inG_a=U_ℓ/U_ℓ−1the group law onG_m⋉U_k is given by the co-multiplication µ^∗onO_s[M]⊗O_s[u₁,... ,u_k] such that

(2.2) µ^∗|O_s[M] = 1⊗1 and µ^∗(u_ℓ) =u_ℓ⊗1 +χ^−ℓ⊗u_ℓ+ (⋆⋆)_ℓ

where (⋆⋆)_ℓis a polynomial inu_m⊗1, 1⊗u_mfor 1≤m < ℓandχ^−mfor 1≤m≤ℓ. This is a special case of [20, 4.2 (3)] for a solvable group. If we write Diff_k =U_k⋊ G_m we have to replaceχ^−ℓbyχ^ℓin the group law expressions of 2.2.

2.3. Toroidal embeddings ofDiff_k. There are two toric embeddings ofG_mcorresponding to the subcones ofN given byσ+= [0,+∞[ andσ−=]− ∞,0]. IfSσ ={m∈M;m|σ≥0}

these embeddings are respectively given by the spectra of A[S_σ+] =M

γ∈σ₊

Aχ^γ and A[S_σ−] =M

γ∈σ−

Aχ^γ.

Following techniques of [20, 4] one can give equivariant embeddings of Diff_k in affine schemes. Let σ be either σ₊ or σ− and write G_m ֒→ A_σ = A¹ for the attached toric embedding. One can get an affine equivariant embedding of Diff_k by looking at C_σ = A_σ×^GmDiff_k. By definition this is the quotient ofA_σ×(G_m⋉U_k) by the relation [λ·x,f(t)] = [x,θ_t(λ)f(t)]. As an algebraic varietyC_σ=A_σ× U_k. It is clear that the right action of Diff_k on itself extends toC_σ. The left action extends toC_σ if and only ifσ =σ− : the affine embedding Diff_k⊂C_σis given by

A[S_σ]⊗O_s[u₁,... ,u_k]⊂A[M]⊗O_s[u₁,... ,u_k],

looking at the group law 2.2 one sees the left action of Diff_k on itself extends if and only ifS_σ∋χ^−ℓ and thusσ=σ−. Proceeding the same way (exchanging rights and lefts) with Eσ = Diff_k×^GmA_σ we get two affinebi-equivariantembeddingsCσ₋ and Eσ₊ (for short C andE) of Diff_k.

These constructions depend on the coordinatet. Let’s focus on the case of the embed- dingC, theEcase is identical. Any other coordinate atpis writtenϑtforϑ∈Diff_k. IfT is the maximal torus in Diff_kdefined bytthenϑTϑ⁻¹is the one defined byϑt. LetC_tand C_ϑt be the affine bi-equivariant embeddings of Diff_kdefined byσ−. InC_ϑtwe have the re- lation [λ⁻¹x,f(t)] = [x,ϑλϑ⁻¹f(t)]. By [20, 4 (iv)] page 183, there is a unique bi-equivariant isomorphismΨ_ϑ:C_t≃C_ϑt commuting to the embeddings of Diff_kinC_tandC_ϑt. The iso- morphismΨ_ϑ is the composition of the left actionL_ϑ⁻¹ :C_t →C_t with the isomorphism δ_ϑ:C_t→C_ϑtgiven by [x,f(t)]7→[x,ϑf(t)]. NoticeΨ_ϑis the identity on Diff_k.

2.4. Back to twisted polar parts. Let (π:X →S,D) be a stable genus g marked curve over a schemeS. Following the previous procedure one gets two affine cones locally on the base. To glue these into a stack overM_g,n one needs to specify transition functions on the intersection of two open affine subsets of the base. LetU₁₂be the intersection of two such affine subsets U₁ andU₂and lett₁andt₂ be coordinates atD respectively overU₁ andU₂. Looking att₁andt₂as elements of Diff_k there existsϑ∈Diff_k such thatt₂=ϑt₁.

The transition function from C₁ on C₂ should correspond to the polar part coordinate change

X

0<ℓ≤k

a_ℓ t₁^ℓ = X

0<ℓ≤k

a_ℓ (ϑ⁻¹t₂)^ℓ.

This is exactly the right multiplication R_ϑ⁻¹ on Diff_k. This isomorphism extends onC₁ andC₂byG_ϑ=Ψ_ϑR_ϑ⁻¹. To keep track of the polar parts we need to mod out by the left action of µ_k ⊂G_m on both C₁ and C₂. Fortunately, G_ϑ isµ_k-equivariant and this local construction together with the transition functionsG−give a stack overS. This procedure still makes sense in the case ofn-marked points and gives a locally affine stack C~k over M_g,n.

Definition 2.1. The stack of twisted polar parts over M_g,n is the locally affine coneC~k

defined previously.

Using the same transition functions we can glue the local coneE into a stack E over M_g,n. However,Edoesn’t correspond to the stack Pg,~k of twisted polar parts introduced in [10]. The reason is simple, locally at the i-th marked point the right action of G_m onC =A_σ−× U_k has weights χ⁻¹,χ⁻¹,χ⁻²,... ,χ^−(ki−1) which is (up to isomorphism) the expected behavior for the right action ofG_m on twisted polar parts at the same marked point. This is not the case for the right action ofG_monEwhich has weights equal toχ.

3. Polar curves

3.1. Stable polar curves. Let (π:X →S,D) be a prestablen-marked curve of genusg.

The sheaf

O_X( Xn i=1

k_iD_i)/O_X= Mn

i=1

O_X(k_iD_i)/O_X

has support on the scheme defined byD. Base change in cohomology for finite morphisms ensures that the pushforward

P_X,~k= Mn

i=1

π∗(O_X(k_iD_i)/O_X) = Mn

i=1

P_X,k

i

is locally free onS of rankd=Pn

i=1k_i. Letpbe a point in the support ofD_i andta local coordinate atp. Fors=π(p) any section̺ofP_X,k

i can be locally written in the form

(3.1) ̺_i =a_ki

t^ki +· · ·a₁

t witha_ℓ∈O_s.

With the difference with other coefficientsa_kiis of global nature. The quotient morphism O_X(k_iD_i)/O_X O_X(k_iD_i)/O_X((k_i−1)D_i)

has kernel supported onD_i. Its higher pushforwards are 0 and one gets an onto map P_X,k

i L^−ki

i ϑ_X,ki

on the (−k_i)-th tensor power of the restriction of the cotangent bundle at thei-th marked point. The morphismsϑ_−,kicommute to base change for each domain and target does and the quotient morphisms do. The global picture we get is that of a locally free sheafP_g,~k given by the collectionP_−,~kand morphismsϑ_ki:P_g,k

i →L^−ki

i over the stack of prestable marked curvesM^ps_g,n.

Definition 3.1. A polar S-curve ((π:X →S,D),̺) is composed of a marked prestable curve and a global section̺= (̺₁,... ,̺_n) ofP_X,~ksuch thatϑX,k_i(̺_i) trivializesL^−ki

i .

Most of the time we will abuse the notation by writing (π:X →S,̺) the previous polar curve. The condition “ϑ_X,ki(̺_i) trivializesL^−ki

i ” meansa_ki(in expression 3.1) is invertible in any local representation of̺_i atp_i. We say̺_i is of orderk_i atp_i.

Definition 3.2. A morphism from a polar curve (π: X →S,̺) on (ψ :Y → T ,ε) over f :S→Tis an isomorphism of the underlying marked curvesψandf^∗πsending̺onf^∗ε.

It is straightforward and formal to show that the functor in groupoids defined by the collection of polar curves over Sch is an algebraic stack. It is not separated, we do the obvious thing : restrict ourselves to objects having finite automorphism groups.

Lemma 3.1. Let(X,̺)be a polar curve overk. The following are equivalent (1) the automorphism group of(X,̺)is finite,

(2) the sheafω_X(Pn

i=1m_ip_i)is ample form_i≥2, (3) the sheafω_X(P_n

i=12p_i)is ample,

(4) each irreducible component ofXhas at least3nodal or marked points except forpossi- blesmooth rational components having only1marked point and1nodal point.

Proof. (3)⇒(2) is clear. Proving (2)⇒(4) and (4)⇒(3) boils down to a simple degree count of the restrictionω_ℓ(Pn

i=1m_ip_i) ofω_X(Pn

i=1m_ip_i) to each irreducible componentX_ℓ. Both result from the relation

(3.2) deg ω_ℓ(

Xn i=1

m_ip_i)

= 2g_Xℓ−2 + X

{i|p_i∈X_ℓ}

m_i

+♯{e∈X_ℓ|enodal point ofX}.

Let us focus on (1)⇔(4). Assume (X,p) satisfies (4). The only components of (X,p) that have infinite automorphism groups are genus 0 components (R,p,e) having marked point p and nodal pointe. It is enough to show the automorphism group of (R,p,e) endowed with a polar part̺ at pare finite. Identify (R,p,e) with (P¹,∞,0) and lett∞,t₀ be the coordinates ofP¹. An automorphism of (P¹,∞,0) is an element inG_msendingt₀onλt₀. The polar part̺atpis written in the chartP¹\ {0}as

̺= Xk

ℓ=1

a_ℓ

t^ℓ, witha_k,0 andt=t∞

t₀.

Ifλ∈G_m is an automorphism of (P¹,∞,0,̺) we have that for allℓ∈ {1,... ,k},a_ℓ =a_ℓλ^ℓ. For ℓ =k we get a_k = λ^ka_k and λ is therefore a root of unity. Conversely, assume the automorphism group of (X,̺) is finite. There is nothing to prove if (X,p) is stable. Let (R,q₁,q₂) be an unstable component ofX withq₁ andq₂ either marked or nodal points.

Then (R,q₁,q₂) has exactly 1 marked point and 1 nodal one. If this wasn’t the case then eitherq₁andq₂are both nodal points and the infinite automorphism group of (R,q₁,q₂) is a subgroup of Aut(X,̺) orq₁andq₂are both marked points and for connectivity reasons (X,p) = (R,q₁,q₂). This case is excluded for (g,n),(0,2). Other cases with less marked or

nodal points are treated similarly.

Definition 3.3. A stablepolar curve (X,̺) over kis a polar curve satisfying one of the conditions of lemma 3.1.

We show that stable polar curves form an open substack of the one of polar curves. We first make a statement onprestable curvesunderlying a stable polar curve.

Definition 3.4. A bubbly curve is a prestable curve (X,p) satisfying the conditions (2) to (4) of lemma 3.1. An unstable component of (X,p) is called a bubble.

Let (π:X →S,D) be a prestable curve whose fibers are bubbly curves. Having fibers that are bubbly curves is stable under base change. Indeed, this means the invertible sheaf ωX_s(Pn

i=12D_i,s) is ample on each fiber X_s for s∈ S. Now ampleness and ωX(Pn i=12D_i)

commute to base change. Furthermore, relative ampleness is an étale-local property on the base ([24, III.3.7]). Thus having bubbly fibers is étale-local on the base. Both remarks ensure that the functor in groupoidsB_g,nsending a schemeSon the groupoid of prestable curves having bubbly fibers is a substack ofM^ps_g,n.

Proposition 3.2. B_g,nis an open (and thus algebraic) substack ofM^ps_g,n.

Proof. Let (X,p) be ak-point ofB_g,nand (π:X →S,D) be a versal deformation of (X,p) as a prestable curve. Since (X,p) is a bubbly curveωX(Pn

i=12D_i)_|X=ω_X(Pn

i=12p_i) is am- ple. By Grothendieck existence theorem [16, 5.4.5] a formal versal deformation of (X,p) is algebraizable and we get a family (π^a :X^a → S^a,D^a) over a complete henselian local ring extendingπ. Furthermore, the invertible sheafωX^a(P_n

i=12D_i^a) corresponding to this algebraization is relatively ample. We thus get a family of bubbly curves over an étale- local neighborhood of (X,p) in the stack of prestable curves. This meansB_g,n is open in

M^ps_g,n.

The previous definition implies families of polar curves having fibers that are stable polar curves define an open (algebraic) substackP_g,~kof the stack of polar curves.

Proposition 3.3. P_g,~k is a tame stack in the sense of[2]. It is a Deligne–Mumford stack if the characteristic of the base fieldpis either0or does not divide any of thek_is.

Proof. Following [23, 8.1] to showP_g,~kis a Deligne–Mumford stack it is enough to show its diagonal is formally unramified ([8, 4]). Since B_g,n and P_g,~k → B_g,n are locally of finite presentation it is the case forP_g,~k as well. It is therefore enough to show fibers of closed points on the diagonal are discrete and reduced. These fibers are automorphism groups of points ofP_g,~k, they are quasi-finite. To show they are reduced it is enough to check automorphism groups overS_ǫ= Spec(k[ǫ]/(ǫ²)) are trivial. Let (X →S_ǫ,̺) be such an object. Automorphisms of (X,D) are given by infinitesimal vector fields on the central fiber (X,p) fixing the marking. There are no such vector fields on stable marked curves ([8, 1.4]), we only have to check the case of a bubble (R,p,e). Vector fields onRfixingpande are classified by H⁰(T_R(−p−e)) which is of dimension 1. Identify (R,p,e) with (P¹,∞,0). A vector field on (P¹,∞,0) is of the formλt_∂t^∂ wheretis the local parameter at∞andλ∈k.

A polar part of orderkatpis locally of the form̺=Pk

ℓ=1aℓt^−ℓ. The vector field acts on̺ through the automorphismα= id +ǫ∂. Asking thatαfixes̺means

k_p

X

ℓ=1

aℓ

t^ℓ =

k_p

X

ℓ=1

aℓ

t^ℓ+ǫ∂(t^ℓ)=

k_p

X

ℓ=1

aℓ

t^ℓ(1 +λℓǫ)=

k_p

X

ℓ=1

aℓ

t^ℓ −λǫ

k_p

X

ℓ=1

ℓaℓ

t^ℓ .

Hence∂,0 if and only ifp>0 andℓis a multiple pfor everya_ℓ ,0. In other words

̺ is a p power. In this case the automorphism group scheme of (P¹,̺) is a µ_pm. The automorphism group of a polar curve having vector fields is thus a product of finite tame groups by a number ofµ_pℓ. This is a linear reductive group and tameness follows from [2, 3.2]. If eitherp∤k orp= 0 then∂= 0 becausea_k ,0. If this is the case for everyk_i then

P_g,~kis Deligne–Mumford.

3.2. Structure ofP_g,~koverM_g,n. There is a natural forgetful onto morphismP_g,~k→B_g,n sending a stable polar curve on the underlying bubbly curve. Composing with the stabi- lization morphismρ:B_g,n→M_g,n(see [5]) contracting possible bubbles one gets an onto morphismκ:P_g,~k→M_g,n. Our goal is to studyκ, we start by studyingρ.

Remark1. We shall often restrict our study to the 1-marked case. This is due to the fol- lowing fact: let P_g,ki,n−1 be the fiber productP_g,ki×_Mg,1M_g,n whereP_g,ki →M_g,1 is the forgetful mapκin the 1-marked case andM_g,n→M_g,1keeps only thei-th marked point

and stabilizes the resulting marked curve. Points ofP_g,ki,n−1 are marked bubbly curves having one polar part alongp_i and whose bubbles are marked byp_i. The fact is that (3.3) P_g,~k≃P_g,k1,n−1×_M

g,nP_g,k2,n−1×_M

g,n· · · ×_M

g,nP_g,kn,n−1.

The morphism fromP_g,~kto the right-hand side comes from the forgetful morphisms keep- ing thei-th polar part and contracting bubbles not containingp_i.

Lemma 3.4. Let(π:X →S,D)be a prestable curve and(π^⋆:X^⋆→S,D^⋆)its stable model. For m= (m₁,... ,m_n)∈(N^∗)ⁿthe stabilization morphism gives natural mapsα_i :N_miDi→ N_m

iD^⋆_i. Proof. Sinceρcommutes to the markings defined byD_iandD_i^⋆we haveD_i ⊂ρ⁻¹D_i^⋆. Thus ρdefines anO_X-morphismρ^∗O_X⋆(−m_iD_i^⋆)→O_X(−m_iD_i). Dualizing we get the morphism

O_X(m_iD_i) HomO_X(ρ^∗O_X⋆(−m_iD_i^⋆),O_X) which gives by adjunction

ρ∗O(m_iD_i) HomO_X^⋆(O_X⋆(−m_iD_i^⋆),ρ_∗O_X).

The term on the right is isomorphic toO_X⋆(m_iD_i^⋆) forρ∗O_X =O_X⋆. We get the desired map by moding out byO_X⊂O_X(m_iD_i) andO_X⋆⊂O_X^⋆(m_iD_i^⋆).

Following [21, 2, (II)] of proof 2.4, the bubbly curve (π:X →S,D) can be recovered as the relative Proj overX^⋆ofρ∗O_X(Pn

i=1D_i). ThenmarkedS-points are given by (ρ_∗O_X(D_i))|_Di։ρ∗N_Di

whereρ∗N_Di is invertible because it is supported onD_i and thus is equal to (ρ|_Di)∗N_Di. The starting point to the study ofρis the following fact: we have a commutative diagram

(3.4)

0 O_X⋆ ρ∗O_X(Pn

i=1D_i) Ln

i=1ρ∗N_Di 0 0 O_X⋆ O_X⋆(Pn

i=1D_i^⋆) Ln

i=1N_D⋆

i 0

ˆ

α α

where ˆαis induced byρandαis the direct sum of theα_is of lemma 3.4 form= (1,... ,1).

This diagram meansρ∗O_X(P_n

i=1D_i) is the Yoneda pullback of the low horizontal extension byα, i.e.

ρ∗O_X Xⁿ

i=1

D_i

≃O_X⋆ Xⁿ

i=1

D_i^⋆

×_NPn

i=1D⋆i

ρ∗N^Pn

i=1D_i

and ˆαis given by the projection on the first factor. The only point to clarify about diagram 3.4 is the right exactness of the first row. The morphismρ satisfiesρ∗O_X =O_X⋆ and its fibers are rational 1-dimensional curves. In particular H¹(ρ⁻¹(x),O_ρ−1(x)) = 0 for every pointx∈ X^⋆and following [21, 1.5] we get that R¹ρ∗O_X= 0.

Notation. Given morphismsα_i:R_i →Ln i=1N_D⋆

i of invertibleO_X⋆-modules write E_α=O_X⋆

Xⁿ

i=1

D_i^⋆

×_NPn

i=1D⋆i

Mⁿ

i=1

R_i

andD_α,ifor the markedS-point defined byτ_α,i: (E_α)|_D⋆ i։N_D⋆

i.

Proposition 3.5. The fiber ofρover(π^⋆:X^⋆→S,D^⋆)is isomorphic to the stack whose (1) objects overT∈Sch_Sare morphisms(α_i:R_i → N_D⋆

i)ⁿ_i=1of invertible sheaves, (2) morphisms from(α_i:R_i → N_D⋆

i)ⁿ_i=1et(β_i :S_i→ N_D⋆

i)ⁿ_i=1overf ∈Hom_S(T ,T^′)are isomorphismsR_i≃f^∗S_i sendingβ_ionα_i.

Proof. We restrict the proof to the 1-marked case. It is clear that points (1) and (2) define a category fibered in groupoids. For each T-object (π :X → T ,D) andT^′-object (π^′ : X^′→T^′,σ”) in the fiber ofρ⁻¹(π^⋆ :X^⋆→S,D^⋆) and each morphismψfrom (π^′,D^′) on (π,D) overf ∈Hom_S(T^′,T) (i.e. isomorphismψ: (π^′,D^′)≃f^∗(π,D)) define the functor (of categories fibered in groupoids)Fas follows

• F(π:X →T ,D) is the morphismα_π:ρ∗N_D→ N_D⋆of lemma 3.4,

• sinceψfixes the marked point we get an isomorphismµ_ψ:ρ∗N_D′≃ρ∗f^∗N_Dsend- ingα_π^′ onψ^∗α_π. Sinceρ∗N_Dcommutes to base change we getρ∗f^∗N_D≃f^∗ρ∗N_D. We defineF(ψ) as the morphism (µ_ψ:ρ∗N_D′≃f^∗ρ∗N_D).

This functor is fully faithfull: anO_T′-isomorphismµ:ρ∗N_D′ ≃f^∗ρ∗N_DforT^′∈Sch_Sde- fines by pulling-back Yoneda extensions an isomorphism

α^∗Yµ:ρ∗O_X′(D^′)≃f^∗ρ∗O_X(D)

such thatµ⁻¹τ_α^′=τ_αα^∗Yµ. This is exactly saying the (X^⋆,D^⋆)-isomorphism P(ρ_∗O_X′(D^′))≃P(f^∗ρ∗O_X(D)) =f⁻¹P(ρ_∗O_X(D)) attached toα^∗Yµfixes the marked points.

Let’s show F is essentially surjective. Let (π_α :X_α → T ,D_α) be the bubbly T-curve (π_α :P(E_α)→T ,D_α) attached toα:R→ N_D⋆. We are going to show we have canonical isomorphisms E_α ≃ρ∗O_X

α(D_α) andR ≃ ρ∗N_Dα commuting to the quotient morphisms E_α→R andρ∗O_X

α(D_α)→ρ∗N_Dα. The first isomorphism gives isomorphisms from (π_α : X_α→T) on (P(ρ∗O_X

α(D_α))→T) and the second says these fixD_α. Letσ_αbe theS-section definingD_αand given byτ_α:E_α։(σ^⋆)∗R. Consider the exact sequence

(3.5) 0 O_X

α ρ^∗E_α ρ^∗ρ∗σ_α∗R 0.

By definitionρ^∗E_α is the invertible sheafO_P(E

α)(1) onX_α =P(E_α). Becauseσ_α∗R is sup- ported onD_α we have thatρ^∗ρ∗σ_α∗R=σ_α∗R. Now locallyσ_α∗R=O_D

α and the 0-Fitting ideal of the left morphism of the exact sequence 3.5 describes the equations ofD_α inX_α. By definition of Fitting ideals these equations are given by the image of the morphism

O_X

α⊗

ρ^∗E_α∨

= ρ^∗E_α∨

O_X

α. attached toO_X

α →ρ^∗E_α. This morphism is locally given by a generator of O_P(E

α)(−1) = ρ^∗E_α^∨and has imageO_X

α(−D_α)⊂O_X

α. We thus get an isomorphismρ^∗E_α=O_X

α(D_α) and the injectionO_X

α →ρ^∗E_αis the canonical mapO_X

α →O_X

α(D_α). Following the projection formula and the fact thatρ∗O_X

α=O_X⋆ we get thatE_α≃ρ∗O_X

α(D_α). To conclude, we have got a commutative diagram giving the needed material to end the proof

0 O_X⋆ E_α R 0

0 O_X⋆ ρ∗O_X

α(D_α) ρ∗N_Dα 0.

Corollary 3.6. We have an isomorphism

ρ⁻¹(π^⋆:X^⋆→S,D^⋆)≃ Yn

i=1

[V_S(N_D⋆ i)/G_m]

Proof. Proposition 3.5 is the definition of the right-hand side of this isomorphism.

Proposition 3.7. The fiber ofκover(π^⋆:X^⋆→S,D^⋆)is isomorphic to the stack whose