4 Maximum generalized Hasse-Witt invariants of cyclic admissible coverings of pointed stable curves

In the present section, we discuss the maximum generalized Hasse-Witt invariants of cyclic admissible coverings of an arbitrary pointed stable curve. Let us return to the case where X^• is an arbitrary pointed stable curve over k, and we maintain the notation introduced in Section 2.2. First, by applying Theorem 2.9, we have the following lemma (cf. [T2, Corollary 2.6 and Lemma 2.12 (ii)]).

Lemma 4.1. (i) Let Q∈Z[D_X] be an effective divisor on X of degree deg(Q) =s(Q)n, LQ a line bundle on X such that L^⊗_Qⁿ ∼= OX(−Q), and LQ,t the pull-back of LQ by the natural morphism X_t→X. Suppose that X^• is smooth over k, and that

#{x∈X | ord_x(Q) = n} ≥s(Q)−1.

Then the Raynaud-Tamagawa theta divisor associated to B^tQ⊗ LQ,t exists.

(ii) Let t_i, i ∈ {1,2}, be an arbitrary positive natural number and n_i ^def= p^ti −1. Let Q_i ∈Z[D_X] be an effective divisor on X of degree deg(Q_i) = s(Q_i)n_i, LQi a line bundle onX such thatL^⊗_Qⁿ_iⁱ ∼=OX(−Qi), andLQi,ti the pull-back ofLQi by the natural morphism X_ti →X. Suppose that s^def= s(Q₁) =s(Q₂). Let t^def= t₁+t₂, n^def= n₁+p^t1n₂,

Q^def= Q₁+p^t1Q₂ ∈Z[D_X]

an effective divisor on X of degree deg(Q) = sn, LQ a line bundle on X such that L^⊗_Qⁿ∼= OX(−Q), and LQ,t the pull-back of LQ by the natural morphism X_t → X. Suppose that X^• is smooth over k. Then the Raynaud-Tamagawa theta divisor associated to B^t_Q⊗ LQ,t

exists if and only if the Raynaud-Tamagawa theta divisor associated to B_Q^ti_i⊗ LQi,ti exists for each i∈ {1,2}.

Lemma 4.1 implies the following proposition.

Proposition 4.2. Suppose that X^• is irreducible. Then there exist a positive natural number n ^def= p^t −1 ∈ N, an effective divisor D ∈ (Z/nZ)^∼[D_X]⁰, and an element α ∈ Rev^adm_D (X^•) such thatα ̸= 0, and that the generalized Hasse-Witt invariant γ_(α,D) attains the maximum

γ_X^max• =

{ g_X −1, if n_X = 0, gX +nX −2, if nX ̸= 0.

Proof. We write Xe for the normalization of X and norm :Xe →X for the normalization morphism. We define

Xe^• = (X, De _Xe ^def= norm⁻¹(D_X ∪X^sing))

to be a pointed stable curve of type (g_Xe, n_X) over k. Note that g_Xe = g_X −#X^sing. Moreover, we putDe_X ^def= norm⁻¹(D_X). By applying Lemma 3.1, to verify the proposition, it is sufficient to prove that there exist a positive natural number n ^def= p^t−1 ∈ N, an effective (Weil) divisor De ∈ (Z/nZ)^∼[De_X]⁰, and an element αe ∈ Rev^adm_De (Xe^•) such that e

α̸= 0, and that the generalized Hasse-Witt invariant γ_(α,eD)e attains the maximum γ_X^max_e• =

{ g_Xe −1, if n_X = 0, g_Xe +n_X −2, if n_X ̸= 0.

Suppose that n_X ≤ 2. Then s(D)e ≤ 1 for each De ∈ (Z/nZ)^∼[De_X]⁰. Thus, the proposition follows immediately from Proposition 2.8 and Theorem 2.9.

Suppose that n_X ≥3. LetDe_X ^def= {x₁, . . . , x_nX},n_i ^def= p^ti−1 for eachi∈ {1, . . . , n_X− 1}such thatn_i >max{C(g_X) + 1,#(e^cl(Γ_X•)∪e^op(Γ_X•))}, and 0< a_i,1, a_i,2 < n_i for each i∈ {1, . . . , nX −1} such that ai,1+ai,2 =ni. We put

D_i ^def= a_i,1x_i+a_i,2x_i+1+ ∑

x∈DeX\{xi,xi+1}

n_ix, i∈ {1, . . . , n_X −1},

which is an effective divisor on Xe with degree deg(D_i) = (n_X −1)n_i. Moreover, we put De ^def=

n∑X−1 i=1

p^{∑ij=0−1tj}D_i and

n^def= p^{∑nX−i=0 1tj}−1 =

n∑X−1 i=1

p^{∑ij=0−1tj}(p^ti−1),

wheret₀ ^def= 0. We see immediately that deg(D) = (ne _X−1)n, and thatDe ∈(Z/nZ)^∼[D_X]⁰. Let LDe a line bundle on Xe such that L^⊗_Deⁿ ∼= OX(−D), ande LD,te the pull-back of LDe by the natural morphism Xet→X. Then Lemma 4.1 implies the Raynaud-Tamagawa thetae divisor associated toB^t_De⊗ LD,te exists. Moreover, Proposition 2.8 implies that there exists a line bundle Ie of degree 0 on Xe such that [Ie]̸= [OXe], that [Ie^⊗n] = [OXe], and that

γ_([L

De⊗Ie],D)e =

{ g_Xe−1, if n_X = 0, g_Xe+n_X −2, if n_X ̸= 0.

Let αe ∈ Rev^adm_De (Xe^•) be the element corresponding to the pair ([LDe ⊗Ie],D)e ∈ PfXe^•,n. This completes the proof of the proposition.

Remark 4.2.1. We maintain the notation introduced in the proof of Proposition 4.2.

By choosing a suitable a_i,2 and a_i,2 for eachi ∈ {1, . . . , n_X −1}, we may obtain that the Galois multi-admissible covering induced by α is connected.

In the remainder of this section, we will generalizes Proposition 4.2 to the case where X^• is an arbitrary pointed stable curve over k.

Definition 4.3. Let G be a connected semi-graph and v ∈ v(G) an arbitrary vertex.

Moreover, we suppose that G is a tree. For each v^′ ∈ v(G), there exists a path p_v,v′

connecting v and v^′ in G. We define

leng(p_v,v′)^def= #{p_v,v′∩v(G)} −1

to be the length of the path p_v,v′. Moreover, since G is a tree, there exists a unique path connecting v and v^′ whose length is equal to min{leng(p_v,v′)}pv,v′. We shall write

p(G, v, v^′)

for this unique path connectingv andv^′ inG, and say that p(G, v, v^′) is theminimal path connecting v and v^′ in G.

Lemma 4.4. Let Γ^def= ΓDX be a minimal quasi-tree associated to DX, X_Γ^• = (X_Γ, D_XΓ)

the pointed stable curve of type (g_XΓ, n_XΓ) associated to Γ, and Π_X^•

Γ the admissible fun-damental group of X_Γ^•. Suppose that n_X ≥2. Then there exist a positive natural number n^def= p^t−1∈N, an effectiv divisorDΓ∈(Z/nZ)^∼[DX]⁰, and an elementαΓ∈Rev^adm_DΓ (X_Γ^•) such that α_Γ ̸= 0, and that the generalized Hasse-Witt invariant

γ_(αΓ,DΓ)=g_XΓ+n_X −2.

Proof. Since Γ is a minimal quasi-tree associated toD_X, we obtain that Γ^{′ def}= Γ\e^lp(Γ) is a tree. Then we have v(Γ) =v(Γ^′). Note that D_X ⊆ D_XΓ. Let v ∈v(Γ) be an arbitrary vertex and n₀ =p^t0 −1∈N a positive natural number such that

n₀ >max{C(g_X) + 1,#(e^cl(Γ_X•)∪e^op(Γ_X•))}. We put

D_v^{′ def}= D_X ∩X_v, m_v ^def= #D^′_v, and D^′_v ^def= {x_v,1, . . . , x_v,mv} if m_v ̸= 0.

Moreover, we put

D_v ^def= D_v^′ ∪(X_v∩X_Γ\X_v),

whereX_Γ\X_v denotes the topological closure ofX_Γ\X_v inX_Γ. Note that sincen_X >0, we have #D_v >0. Letw∈v(Γ) be an arbitrary vertex distinct fromv. Since Γ^′ is a tree, there exists a unique node

x⁻_v,w

such that the closed edge of Γ^′ corresponding to x⁻_v,w is contained in the minimal path p(Γ^′, v, w) connectingv and w in Γ^′. On the other hand, we define a set of nodes to be

Node⁺_v,w ^def= {X_w∩X_w′, w^′ ∈v(Γ) |leng(p(Γ^′, v, w^′)) = leng(p(Γ^′, v, w)) + 1}. Note that Node⁺_v,w may possibly be an empty set, and that D_w ={x⁻_v,w} ∪Node⁺_v,w∪D_w^′ .

First, we define two sets of effective divisors Div^irr-st_v , Div^st_v

associated to v as follows, where “st” means that “standard”, and “irr” means that

“irreducible components”. Let i ∈ {1, . . . , mv −1} and 0 < av,i,1, av,i,2 < n0 such that a_v,i,1+a_v,i,2 =n₀. Suppose that m_v ≤1. Then we put

Div^st_v ^def= ∅, Div^st_v ^def= ∅. Suppose that m_v ≥2. We define

Q_v,v,i^def= a_v,i,1x_v,i+a_v,i,2x_v,i+1+ ∑

x^′∈D_v^′\{xv,i,xv,i+1}

n₀x^′+ ∑

x∈Dv\D^′_v

n₀x

to be an effective divisor on X_v whose support is D_v, and whose degree is equal to (#D_v−1)n₀. We define

Qv,w,i

def= ∑

x∈Dw\{x⁻v,w}

n0x, w ∈v(Γ)\ {v},

to be an effective divisor on X_w whose support isD_w\ {x⁻_v,w}, and whose degree is equal to (#D_w−1)n₀. Moreover, we define

Q^v_i ^def= av,i,1xv,i+av,i,2xv,i+1+ ∑

x∈DX\{xv,i,xv,i+1}

n0x,

to be an effective divisor on X_Γ whose support is D_X, and whose degree is (n_X −1)n₀. Then we put

Div^irr-st_v,i ^def= ∪

u∈v(Γ)

{Q_v,u,i}, Div^irr-st_v ^def=

m∪v−1 i=1

Div^irr-st_v,i , Div^st_v ^def=

m∪v−1 i=1

{Q^v_i}.

Next, we define two sets of effective divisors Div^irr-md_v , Div^md_v

associated to v as follows, where “md” means that “modification”. Letz ∈D_X \D^′_v and 0< b_v,z,1, b_v,z,2 < n₀ such thatb_v,z,1+b_v,z,2 =n₀. Suppose that m_v = 0. Then we put

Div^irr-md_v ^def= ∅, Div^md_v ^def= ∅.

Suppose that m_v ̸= 0. Let w_z be the vertex such that the irreducible component X_wz corresponding towz containsz (i.e., z ∈D_w^′ _z),p(Γ^′, v, wz) the minimal path connectingv andw_z in Γ^′, andw∈v(Γ) an arbitrary vertex distinct fromw_zsuch thatw⊆p(Γ^′, v, w_z).

Since Γ^′ is a tree, we have that #(Node⁺_v,w∩p(Γ^′, v, w_z)) = 1. We put x⁺_v,w ^def= Node⁺_v,w ∩p(Γ^′, v, w_z).

We define

Q_v,v,z ^def= b_v,z,1x_v,mv+b_v,z,2x⁺_v,v+ ∑

x∈Dv\{x_v,mv,x⁺v,v}

n₀x

and

Qv,wz,z

def= bv,z,1x⁻_v,wz +bv,z,2z+ ∑

x∈D_wz\{x_v,wz,z}

n0x

to be effective divisors onXv andXwz whose supports areDv andDwz, and whose degrees are equal to (#D_v−1)n₀ and (#D_wz −1)n₀, respectively. Let w∈ v(Γ)\ {v, w_z} be an arbitrary vertex such thatw⊆p(Γ^′, v, w_z). Then we define

Q_v,w,z ^def= b_v,z,1x⁻_v,w +b_v,z,2x⁺_v,w+ ∑

x∈Dw\{x⁻v,w,x⁺v,w}

n₀x

to be an effective divisor on X_w whose support is D_w, and whose degree is equal to (#Dw−1)n0. Let w^′ ∈v(Γ) be an arbitrary vertex such that w^′ ̸⊆p(Γ^′, v, wz). Then we define

Q_v,w′,z

def= ∑

x∈D_w′\{x⁻_v,w′}

n₀x

to be an effective divisor onX_w′ whose support isD_w′\{x⁻_v,w′}, and whose degree is equal to (#D_w′ −1)n₀. Moreover, we define

Q^v_z ^def= b_v,z,1x_v,mv+b_v,z,2z+ ∑

x∈DX\{xv,mv,z}

n₀x

to be an effective divisor on X_Γ whose support is D_X, and whose degree is equal to (n_X −1)n₀. Then we put

Div^irr-md_v,z ^def= ∪

u∈v(Γ)

{Q_v,u,z}, Div^irr-md_v ^def= ∪

z∈DX\D^′_v

Div^irr-md_v,z , Div^md_v ^def= ∪

z∈DX\D^′_v

{Q^v_z}.

We put

Div^irr_X ^def= ∪

v∈v(Γ)

(Div^irr-st_v ∪Div^irr-md_v ) and

Div_X ^def= ∪

v∈v(Γ)

(Div^st_v ∪Div^md_v ).

We denote by Div^irr_X(X_u), u ∈ v(Γ), the subset of Div^irr_X whose elements are effective divisors on X_u. Note that d ^def= #Div^irr_X(X_u1) = #Div^irr_X(X_u2) = #Div_X for each u₁, u₂ ∈ v(Γ). Moreover, let

Div^irr_X(X_u)^def= {P_u,1, . . . , P_u,d}, u∈v(Γ),

be an order of Div^irr_X(X_u) such that, for each u₁, u₂ ∈ v(Γ) and each j ∈ {1, . . . , d}, one of the following conditions is satisfied: (i) if P_u1,j ∈ Div^irr-st_v,i for some v ∈ v(Γ) and some i ∈ {1, . . . , mv −1}, then Pu2,j ∈ Div^irr-st_v,i ; (ii) if Pu1,j ∈ Div^irr-md_v,z for some v ∈ v(Γ) and some z ∈ DX \D^′_v, then Pu2,j ∈ Div^irr-md_v,z . Then, by the construction of DivX, the order of Div^irr_X(X_u), u∈v(Γ), induces an order of Div_X. We may put

Div_X ^def= {P₁, . . . , P_d}. Let t^def= dt₀ and n ^def= ∑d

j=1p^(j−1)t0(p^t0 −1) = p^t−1. We define P_u ^def=

∑d j=1

p^(j−1)t0P_u,j ∈(Z/nZ)^∼[D_u]⁰, u∈v(Γ), and

P_Γ^def=

∑d j=1

p^(j−1)t0P_j ∈(Z/nZ)^∼[D_X]⁰

to be effective divisors on X_u and X_Γ, respectively. We see immediately that the support of Pu, u ∈ v(Γ), is Du, that the support of PΓ is DX, that deg(Pu) = (#Du −1)n, and that deg(P_Γ) = (n_X −1)n.

Let u ∈v(Γ) be an arbitrary vertex and Pe_u ^def= norm^∗_v(P_u) an effective divisor on Xe_u. By applying similar arguments to the arguments given in the proof of Proposition 4.2, there exists αe_u ∈Rev^adm_Pe

u (Xe_u^•) such that γ_(αe

u,Peu) =g_u+ #D_u−2.

We define

X_u^• = (X_u, D_Xu ^def= D_u)

to be a pointed stable curve over k. Then Lemma 3.1 implies that the element α_u ∈ Rev^adm_P

u (X_u^•) induced byαe_u such that γ_(αu,Pu) attains the maximum γ_X^maxu• =gXu+ #Du−2,

where g_Xu denotes the genus of X_u. Write

f_u^• :Y_u^• →X_u^•

for the Galois multi-admissible covering over k with Galois group Z/nZ induced by α_u. By gluing {Y_u^•}u∈v(Γ) along {f_u⁻¹(D_u \D^′_u)}u∈v(Γ) that is compatible with the gluing of {X_u^•}u∈v(Γ) that gives rise to X_Γ^•, we obtain a Galois multi-admissible covering

f_Γ^• :Y_Γ^• →X_Γ^•

overk with Galois groupZ/nZ. Note that the construction of f_Γ^• implies thatf_Γ^• is ´etale overD_XΓ\D_X. We denote by α_Γ∈Hom(Π^ab_X•

Γ,Z/nZ) an element induced byf_Γ^•. We put D_Γ^def= P_Γ. By the construction of D_Γ, we see immediately that

α_Γ ∈Rev^adm_D

Γ (X_Γ^•).

Moreover, Lemma 3.1 implies that

γ_(αΓ,DΓ)=g_XΓ+n_X −2.

We complete the proof of the lemma.

Next, we prove the main result of the present section.

Theorem 4.5. There exist a positive natural number n^def= p^t−1∈N, an effective divisor D ∈ (Z/nZ)^∼[D_X]⁰, and an element α ∈ Rev^adm_D (X^•) such that α ̸= 0, and that the generalized Hasse-Witt invariant γ(α,D) attains the maximum

γ_X^max• =

{ g_X −1, if n_X = 0, g_X +n_X −2, if n_X ̸= 0.

Proof. Lett ∈Nbe an arbitrary positive natural number and n^def= p^t−1 such that n >max{C(g_X) + 1,#(e^cl(Γ_X•)∪e^op(Γ_X•))}.

First, we suppose that n_X ≤ 1. We denote by v(Γ_X•)^>0 ⊆ v(Γ_X•) the set of vertices such thatg_v >0 for eachv ∈v(Γ_X•)^>0. Suppose thatv(Γ_X•)^>0 =∅. Thenn_X ≤1 implies that Γ_X• is not a tree. This means that Π^top_X• is not trivial. Let α^′ : Π^top,ab_X• ↠Z/nZ be a surjection and

α: Π^ab_X• ↠Z/nZ the composite morphism Π^ab_X• ↠Π^top,ab_X• ^α

↠′ Z/nZ. Then the theorem follows immediately from Lemma 3.1.

Suppose that v(Γ_X•)^>0 ̸= ∅. Let v ∈ v(Γ_X•)^>0. Then Proposition 2.8 and Theorem 2.9 imply that there exists an element αe_v ∈Rev^adm₀ (Xe_v^•) such that αe_v : Π^ab_e

X_v^• ↠Z/nZ is a surjective, and that

γ_(eαv,0) =g_v−1.

Write fe_v^• : Ye_v^• → Xe_v^• for the connected Galois ´etale covering with Galois group Z/nZ induced by αe_v. Let

π₀(X\ ∪

v∈v(ΓX•)^>0

X_v) be the set of connected components ofX\∪

v∈v(ΓX•)^>0X_vandC ∈π₀(X\∪

v∈v(ΓX•)^>0X_v), whereX\∪

v∈v(ΓX•)^>0X_v denotes the topological closure ofX\∪

v∈v(ΓX•)^>0X_v inX. We define

C^• = (C, D_C ^def= (C∩( ∪

v∈v(ΓX•)^>0

X_v))∪(D_X ∩C))

to be a pointed stable curve over k. Note that the normalization of each irreducible component of C is a rational curve over k. Let

Y_C^{• def}= ⨿

i∈Z/nZ

C_i^•,

where C_i^• is a copy of C^•. Then we obtain a Galois multi-admissible covering f_C^• :Y_C^• →C^•

over k with Galois group Z/nZ, where the restriction morphismf_C^•|Ci is an identity, and the Galois action isj(Ci) = Ci+j for each i, j ∈Z/nZ. By gluing

{Ye_v^•}v∈v(ΓX•)^>0 and {Y_C^•}_C∈π0(X\^∪

v∈v(ΓX•)>0Xv)

along {D_Xe

v}v∈v(ΓX•)^>0 and {D_C}_C∈π0(X\^∪

v∈v(ΓX•)>0Xv) that is compatible with the gluing of {Xe_v^•}v∈v(ΓX•)^>0 ∪ {C^•}_C∈π0(X\^∪

v∈v(ΓX•)>0Xv) that gives rise to X^•, we obtain a Galois (´etale) admissible covering

f^• :Y^• →X^•

over k with Galois group Z/nZ. Then theorem follows immediately from Lemma 3.1.

Next, we suppose that n_X ≥ 2. Let Γ ^def= Γ_DX be a minimal quasi-tree associated to D_X, Γ^im the image of the natural morphism ϕ_Γ : Γ→Γ_X•, and

X_Γ^• = (X_Γ, D_XΓ), X_Γ^•im = (X_Γ^im, D_X

Γim) the pointed stable curves over k associated to Γ, Γ^im, respectively.

Lemma 4.4 implies that there exist a natural number n ^def= p^t−1 ∈ N, an effective divisor D ^def= D_Γ ∈ (Z/nZ)^∼[D_X]⁰ on X_Γ whose degree is (n_X − 1)n, and an element α_Γ ∈Rev^adm_D (X_Γ^•) such that

γ_(αΓ,D) =g_XΓ+n_X −2,

where g_XΓ denotes the genus of X_Γ. We denote by f_Γ^• :Z_Γ^• →X_Γ^•

the Galois multi-admissible covering overk with Galois group Z/nZinduced byα_Γ. Note that f_Γ^• is ´etale over DXΓ\DX. By gluingZ_Γ^• alongf_Γ⁻¹(DXΓ\(DX∪ {xe}_e∈ϕ⁻¹

Γ (e^op(Γ^im)))) that is compatible with the gluing of X_Γ^• that gives rise to X_Γ^•im, we obtain a pointed stable curve Z_Γ^•im over k. Moreover, f_Γ^• induces a Galois multi-admissible covering

f_Γ^•im :Z_Γ^•im →X_Γ^•im

over k with Galois group Z/nZ. Write Π_X^•

Γim for the admissible fundamental group of X_Γ^•im and α_Γ^im for an element of Hom(Π^ab_X•

Γim

,Z/nZ) induced by f_Γ^•im. Note that we have D_α

Γim =D. Then Lemma 3.1 implies thatγ_(α

Γim,D)=g_X

Γim+n_X−2, whereg_X

Γim denotes the genus of X_Γ^im.

On the other hand, we write π₀(X\X_Γ^im) for the set of connected components of X\X_Γ^im. We define the following pointed stable curve

E^• = (E, D_E ^def= E∩X_Γim), E∈π₀(X\X_Γim),

over k. We denote by π0(X\X_Γ^im)^>0 the set of curves of π0(X\X_Γ^im) such that the genus of curves are>0.

LetE ∈π₀(X\X_Γ^im)^>0. Similar arguments to the arguments given in the proof of the case where nX ≤1 and v(ΓX^•)̸=∅ above imply that there exists a Galois ´etale covering

f_E^• :Z_E^• = (ZE, DZE)→E^• over k with Galois group Z/nZ such that

γ_(αE,0) =gE−1,

where g_E denotes the genus of E, and α_E ∈Rev^adm₀ (E^•) is an element induced by f_E^•. Let E ∈π₀(X\X_Γ^im)\π₀(X\X_Γ^im)^>0. We put

Z_E^{• def}= ⨿

i∈Z/nZ

E_i^•,

where E_i^• is a copy of E^•. Then we obtain a Galois multi-admissible covering f_E^• :Z_E^• →E^•

overk with Galois groupZ/nZ, where the restriction morphismf_E^•|Ei is an identity, and the Galois action isj(Ei) =Ei+j for each i, j ∈Z/nZ.

We may glue Z_Γ^•im and {Z_E^•}_E∈π0(X\X

Γim) along f_Γ⁻¹(X_Γ^im ∩(∪

E∈π0(X\X_Γim)E)) and {f_E⁻¹(X_Γ^im∩E)}_E∈π0(X\X

Γim)that is compatible with the gluing of{X_Γ^•im}∪{E^•}_E∈π0(X\X

Γim)

that gives rise to X^•, then we obtain a Galois multi-admissible covering f^• :Z^• →X^•

over k with Galois group Z/nZ. Moreover, we write α∈Hom(Π^ab_X•,Z/nZ)

for an element induced by f^•. We see immediately that α ∈ Rev^adm_D (X^•). By applying Lemma 3.1, we obtain that

γ_(α,D)=g_X +n_X −2.

This completes the proof of the theorem.

5 A group-theoretical formula for topological types

ドキュメント内 Maximum generalized Hasse-Witt invariants and an anabelian formula for topological types of pointed stable curves in positive characteristic (ページ 34-43)