Degeneration and existence of Raynaud-Tamagawa theta di- di-visor

We introduce a condition concerning degeneration.

Condition 6.1. Let v ∈ v(ΓX^•). We shall call that Xe_v^• satisfies (DEG) if there exist a complete discrete valuation ring R_v with an algebraically closed residue field k_Rv and a pointed stable curve Xv^• = (Xv, D_Xv) of type (g_v, n_v) over R_v satisfying the following conditions:

(i)k contains the quotient fieldK_Rv of R_v.

(ii) WriteK_Rv for the algebraic closure ofK_Rv ink. ThenXe_v^• isk-isomorphic toXv^•×Rvk. Moreover, thek-isomorphism induces a bijectionι_v :D_Xv →^∼ D_Xe

v. For each C ∈π₀(v), writeD_v,C for ι⁻_v¹({x_e}e∈Ev,C). Then we have

D_Xv = ∪

C∈π0(v)

D_v,C.

(iii) Write Xv,η^• = (Xv,η, D_Xv,η) for the geometric generic fiber Xv^• ×Rv K_Rv of Xv and Xv,s^• = (Xv,s, D_Xv,s) for the special fiber Xv^•×Rv k_Rv of Xv. For each C ∈π₀(v), writeD^η_v,C for D_v,C ×RvK_Rv and D^s_v,C for D_v,C ×Rvk_Rv. Then we have

D_Xv,η = ∪

C∈π0(v)

D_v,C^η , D_Xv,s = ∪

C∈π0(v)

D_v,C^s .

Moreover, we have

Xv,s= ( ∪

C∈E^>1v

P_v,C)∪Z_v

such that the following conditions hold: (1)D_v,C^s is contained in P_v,C for each C ∈ E_v^>1; (2) P_v,C ∼= P¹_kRv; (3) the dual semi-graph of Xv,s^• is a tree; (4) if

#E_v^>1 ̸= 0, then Zv is either a smooth projective curve over kRv of genus gv

when g_v ̸= 0 or an empty set when g_v = 0; (5) if #E_v^>1 = 0, then Z_v is a smooth projective curve over k_Rv of genusg_v.

Let Fp be an algebraic closure of Fp in k. For each v ∈ v(ΓX^•), write Mgv,nv for the moduli stack Mgv,nv,Z×ZFp. For each 0 ≤σ ≤g_v, we denote by

M^σgv,nv

the p-rank strata of Mgv,nv with p-rank σ (i.e., the locally closed reduced substack of Mgv,nv whose geometric points corresponding to pointed stable curves with p-rank σ).

Note thatM^σgv,nv is not irreducible in general.

Definition 6.2. For each v ∈ v(ΓX^•), write M^σ_gv,nv for the coarse moduli space of the substackM^σgv,nv. Let q_v^σ-gen be a generic point of M^σ_gv,nv and k(q_v^σ-gen) the residue field of q^σ-gen_v . Suppose thatk(q^σ-gen_v )⊆k for eachv ∈v(Γ_X•). Letk_qv^σ-gen be the algebraic closure

of the residue field ofk(q_v^σ-gen) inkandX_q^•σ-gen

v the geometric generic curve corresponding to the geometric generic point Speck_q^σ-gen_v →Speck(q_v^σ-gen)→M^σ_gv,nv. We shall call thatX^• is acomponent-generic pointed stable curveoverk ifXe_v^•isk-isomorphic toX_q^•σ-gen

v ×k_q^σ-gen

v k

for each v ∈v(Γ_X•).

We have the following proposition.

Proposition 6.3. Suppose that X^• is a component-generic pointed stable curve over k.

Then Xe_v^• satisfies (DEG) for each v ∈v(Γ_X•).

Proof. IfE_v^>1 =∅, then the proposition is trivial. We may assume thatE_v^>1 ̸=∅, and let E_v^>1 :={C₁, . . . , C_q}. For each C_i ∈E_v^>1, we put

E_v,Ci ={e₍^∑

j<in_v,Cj)+1, . . . , e^∑

j≤in_v,Cj}. Moreover, we put ∪

C∈E_v⁼¹

E_v,C ={e₍^∑

C∈E>1

v nv,C)+1, . . . , e_nv}.

Then the order ofe^op(Γ_v) defined above induces an order of the set of marked pointsD_Xe

v. Suppose that g_v = 0. Then the definition of component-generic pointed stable curves implies that Xe_v^• is a geometric generic curve ofM_0,nv. Then Xe_v^• satisfies (DEG).

Suppose thatg_v = 1 andσ = 1. ThenXe_v^• is a geometric generic curve ofM_1,nv. Then Xe_v^• satisfies (DEG).

Suppose that g_v = 1 and σ = 0. Write π_1,nv,1 : M1,nv → M1,1 for the morphism induced by forgetting the marked points except the first marked point and c_v : Speck→ M1,nv for the classifying morphism determined by Xe_v^•. Then the composite morphism

π_1,nv,1◦c_v : Speck → M1,1

determines a supersingular elliptic curve Z_v^∗,• = (Z_v^∗, DZ_v^∗) over k. Since Z_v^∗,• can be defined over Fp, there exists a supersingular elliptic curve

Z_v^∗∗,• = (Z_v,{z_v})

over Fp such that Z^∗,• ∼= Z_v^• ×_Fp k. Let P_v,Ci ∼= P_Fp for each i ∈ {1, . . . , q}, D_Pv,Ci a set of distinct closed points {x_1,Ci, x_2,Ci} ∪ {x₍^∑

j<in_v,Cj)+1, . . . , x^∑

j≤in_v,Cj} of P_v,Ci if i∈ {1, . . . , q−1},D_Pv,Cq a set of distinct closed point{x_1,C1}∪{x₍^∑_j<qnv,Cj)+1, . . . , x^∑_j≤qnv,Cj} of P_v,Cq. Then we obtain a pointed stable curve

P_v,C^• _i = (P_v,Ci, D_Pv,Ci), i∈ {1, . . . , q}, and a pointed stable curve

Z_v^• = (Z_v, D_Zv :={z_v} ∪ {x₍^∑

C∈E>1

v nv,C)+1, . . . , x_nv}) over Fp, where {zv} ∩ {x₍^∑

C∈E>1

v nv,C)+1, . . . , xnv} = ∅. We glue {P_v,C^• _i}i∈{1,...,q} and Z_v^• by identifying z_v, x_2,Ci, i ∈ {1, . . . , q−1}, with x_1,C1, x_1,Ci, i ∈ {2, . . . , q}, respectively.

Thus, we obtain a pointed stable curve

Xv,s^• = (Xv,s, D_Xv,s)

of type (1, n_v) over Fp which determines a classifying morphism c_v,s : SpecFp → M1,nv. Moreover, we write q_v,s for the image of the composite morphism

SpecFp → M1,nv →M_1,nv.

Note that the construction ofXv,s^• implies that the curve corresponding to the composite morphism

π_1,nv,1◦c_v,s : SpecFp → M1,1

isFp-isomorphic toZ_v^•. This means thatq_v,s is contained in the closure ofq_v^0-gen inM_1,nv. Then the proposition holds when gv = 1 andσ = 0.

Suppose thatg_v ≥2. We denote byS_g^σ_v,nv the set of irreducible components of M^σgv,nv. Write π_gv,nv,0 : Mgv,nv → Mgv,0 for the morphism induced by forgetting the marked points. We note that π_g⁻_v¹_,nv,0(S)∈S_g^σ_v,nv for each S ∈S_g^σ_v,0, and that

∪

S∈S_{gv ,nv}^σ

π⁻¹_gv,nv,0(S) =M^σgv,nv.

Then by applying [AP, Proposition 3.5], we see thatXe_v^• admits a pointed stable reduction W^• = (W, D_W)

such that W is a chain of nonsingular projective curves of genus 1. Moreover, without loss of generality, we may assume that W^• is component generic. Write Γ_W• for the dual semi-graph of W^•. Let

v(Γ_W•) ={u₁, . . . , u_gv}.

We may assume that for eachi∈ {1, . . . , g_v−1},W_ui∩W_ui+1 ̸=∅. For eachi∈ {1, . . . , g_v}, we define a smooth pointed stable curve to be

W_u^•

i = (W_ui, D_Wui := (W_ui ∩W^sing)∪(D_W ∩W_ui)).

Moreover, we may choose W^• such that D_W is contained in W_u1. This means that W_ui ∩D_W = ∅ if u_i ̸= u₁. Let W_ui ∩W^sing = {x_ui,1} if i ∈ {1, g_v} and W_ui ∩W^sing = {x_ui,1, x_ui,2} if i∈ {2, . . . , g_v−1}.

The proposition in the case where g_v = 1 implies that W_u^•

1 satisfies (DEG). Let W_u^•_1,s = (W_u1,s, D_Wu

1,s) be such a reduction of W_u^•₁ and x_u1,s,1 ∈ D_Wu

1,s the reduction of the point of W_u1 ∩W^sing. We may glue W_u^•_1,s and {W_u^•_i}i∈{2,...,gv} by identifying x_u1,s,1, x_ui,2, i∈ {2, . . . , g_v −1} with x_u2,1, x_ui,1, i∈ {3, . . . , g_v}, respectively. Then we obtain a pointed stable curve

Ws^• = (Ws, D_Ws)

of type (g_v, n_v) which is a pointed stable reduction ofW^•. WriteV_q^σ-gen_v for the topological closure of {q^σ-gen_v } inM_gv,nv. ThenWs^• corresponds to a geometric point of M_gv,nv whose image is contained in V_q^σ-gen_v . Write N for the set of reduction of the points of W^sing in Ws. Then N ⊆ W^sing. Thus, there exists a deformation of the pointed stable curve Ws

along N (cf. Section 2), and we obtain a pointed stable curve Xs^•

of type (g_v, n_v) such that Xv,s^• = (Xv,s, D_Xv,s) corresponds to a geometric point of M_gv,nv whose image is contained in V_qv^σ-gen. Note that Xv,s^• satisfies (iii) of Condition 6.1. Then Xe_v^• satisfies (DEG). This completes the proof of the proposition.

In the remainder of the present paper, we assume that X^• is a component-generic pointed stable curve over k. Then Proposition 6.3 implies that, for each v ∈ v(Γ_X•), Xe_v^• satisfies (DEG). Moreover, we denote by Π_v,η and Π_v,sthe admissible fundamental groups of Xv,η^• and Xv,s^• , respectively. Then Π_v,η is naturally isomorphic to Π_v, and there is a specialization map

sp_Rv : Π_v,η↠Π_v,s.

Then we obtain a continuous surjective homomorphism of maximal pro-p quotients sp^p_Rv : Π^p_v,η↠Π^p_v,s,

where (−)^p denotes the maximal pro-pquotient of (−). On the other hand, the specializa-tion theorem of maximal prime-to-p quotients of admissible fundamental groups implies that

sp^p_R^′_v : Π^p_v,η^′ →^∼ Π^p_v,s^′ .

Let Q_v be an effective divisor on Xv of degree (#E_v^>1)n such that Supp(Q_v) ⊆ D_Xv and

∑

x∈Dv,C

ord_x(Q_v) =

{ n, if C ∈E_v^>1, 0, if C ∈E_v⁼¹.

Write Q^η_v for Q_v ×RvK_Rv, Q^s_v for Q_v ×Rv k_Rv, and Q^s_v,C, C ∈ E_v^>1, for Q^s_v ∩P_v,C. Then we have deg(Q^s_v,C) = n, C ∈E_v^>1.This means that

s(Q^s_v,C) = 1, C ∈E_v^>1.

LetLv,η be a line bundle onXv,η such thatLv,η^⊗n ∼=OXv,η(−Q^η_v). We put E_Qη

v :=B^t

Q^ηv ⊗Lv,η. Then we have the following proposition.

Proposition 6.4. The Raynaud-Tamagawa theta divisor ΘE

Qη v

associated to E_Q^η

v exists.

Proof. If #E_v^>1 ≤ 1, then the proposition follows immediately from Theorem 4.3. We may assume that #E_v^>1 ≥ 2. To verify the proposition, it is sufficient to prove that E_Q^η

v

satisfies (⋆). This is equivalent to prove that there exists a line bundle Iv,η on Xv,η of degree 0 such that

γ_([L

v,η⊗Iv,η],Q^ηv) = dim_kRv(H¹(Xv,η,Lv,η⊗Iv,η)) =g_v + #E_v^>1−1.

For each C ∈E_v^>1, let Lv,C be a line bundle on P_v,C such thatLv,C^⊗n ∼=OPv,C(−Q^s_v,C), and let

f_v,C^• :Y_v,C^• = (Y_v,C, D_Yv,C)→P_v,C^• = (P_v,C, D_Pv,C), C ∈E_v^>1,

be the connected Galois admissible covering corresponding to Lv,C over k_Rv with Galois group Z/nZ, where D_Pv,C := D_Xv,s ∩P_v,C. Then the k_Rv[µ_n]-module H_´¹_et(Y_v,C,Fp)⊗k_Rv admits the following canonical decomposition

H¹_´et(Y_v,C,Fp)⊗k_Rv = ⊕

i∈Z/nZ

M_v,C(i),

where ζ ∈ µ_n acts on M_v,C(i) as the ζⁱ-multiplication. By applying Theorem 4.3 and Theorem 4.5, we may choose Lv,C, C ∈E_v^>1, such that

dimk_Rv(Mv,C(1)) = 0.

Ifg_v ̸= 0, letLZv be a non-trivial line bundle onZ_v of degree 0 such thatLZ^⊗vⁿ∼=OZv. We denote by

f_Z^•_v :Y_Z^•_v = (YZv, DY_Zv)→Z

the connected Galois ´etale covering corresponding to LZv over k_Rv with Galois group Z/nZ, where DY_Zv :=D_Xv,s∩Zv. Then the kRv[µn]-module H¹_´et(YZv,Fp)⊗kRv admits the following canonical decomposition

H_´¹_et(Y_Zv,Fp)⊗k_Rv = ⊕

i∈Z/nZ

M_Zv(i),

where ζ ∈ µ_n acts on M_Zv(i) as the ζⁱ-multiplication. By applying Theorem 4.3 and Theorem 4.5, we may choose LZv such that

dim_kRv(M_Zv(1)) =g_v −1.

We glue {Y_v,C^• }C∈Ev^>1 if g_v = 0, and glue {Y_v,C^• }C∈E^>1v and Y_Z^•

v if g_v ̸= 0. Then we obtain a connected Galois admissible covering

f_v,s^• :Yv,s^• = (Yv,s, D_Yv,s)→ Xv,s^•

over k_Rv with Galois group Z/nZ. Write Γ_Y•

v,s for the dual semi-graph of Yv,s^• and r_Yv,s for the Betti number of Γ_Y•

v,s. The construction of Yv,s^• implies that r_Yv,s =

{ (#E_v^>1−1)(n−1), if g_v = 0,

#E_v^>1(n−1), if gv ̸= 0.

The k[µ_n]-module H_´¹_et(Yv,s,Fp)⊗k_Rv admits the following canonical decomposition H¹_´et(Yv,s,Fp)⊗k_Rv = ⊕

i∈Z/nZ

M_v,s(i),

where ζ ∈ µ_n acts on M_v,s(i) as the ζⁱ-multiplication. Moreover, we have a natural k[µ_n]-submodule

H¹(Γ_Y•

v,s,Fp)⊗k_Rv ⊆H_´¹_et(Yv,s,Fp)⊗k_Rv

which admits a canonical decomposition

H¹(Γ_Yv,s• ,Fp)⊗k_Rv = ⊕

i∈Z/nZ

M_ΓY•

v,s(i), where ζ ∈µn acts onMΓ_Y•

v,s(i) as the ζⁱ-multiplication. Then we have M_v,s(1) =M_Zv(1)⊕M_ΓY•

v,s(1).

We see immediately that dim_kRvM_ΓY•

v,s(i) =





0, if i= 0,

#E_v^>1−1, if i̸= 0 andg_v = 0,

#E_v^>1, if i̸= 0 andg_v ̸= 0.

Thus, we obtain that

dim_kRv(M_v,s(1)) =g_v+ #E_v^>1−1.

On the other hand, since (p, n) = 1, the isomorphism sp^p_R^′_v : Π^p_v,η^′ →^∼ Π^p_v,s^′ implies that, by replacing R_v by a finite extension of R_v, there exists a finite morphism of pointed stable curves

f_v^• :Yv^• = (Yv, D_Yv)→ Xv^•

overR_v such that the restriction of f_v^• on the special fibers is k_Rv-isomorphic to f_v,s^• , and that the restriction of f_v^• on the geometric generic fibers is a connected Galois admissible covering

f_v,η^• :Yv,η^• = (Yv,η, D_Yv,η) := Yv^•×RvK_Rv → Xv,η^•

withe Galois group Z/nZ over K_Rv. The k_Rv[µ_n]-module H_´¹_et(Yv,η,Fp)⊗k_Rv admits the following canonical decomposition

H¹_´et(Yv,η,Fp)⊗k_Rv = ⊕

i∈Z/nZ

M_v,η(i), whereζ ∈µ_nacts onM_v,η(i) as theζⁱ-multiplication. Write Π_Y•

v,η ⊆Π_v,η and Π_Yv,s• ⊆Π_v,s for the open normal subgroups corresponding to Y_v,η^• and Y_v,s^• , respectively. Then the surjection sp_v : Π_v,η ↠Π_v,s induces a surjection sp_v,Y : Π_Y^•

v,η ↠Π_Y•

v,s. Thus, we obtain a surjection

sp^p_v,Y : Π^p_Y•

v,η ↠Π^p_Y• v,s.

Since H¹_´et(Yv,η,Fp)⊗ k_Rv and H¹_´et(Yv,s,Fp)⊗ k_Rv are semi-simple k_Rv[µ_n]-modules, the surjection sp^p_v,Y induces an injection Mv,s(1),→Mv,η(1). This implies that

dim_kRv(M_v,η(1))≥g_v+ #E_v^>1−1.

Write Lv,η^′ for the line bundle on Xv,η corresponding to Yv,η^• . Then Lemma 4.8 implies that (L_v,η^′ )^⊗n∼=O_Xv,η(−Q^η_v). Moreover, we have

dim_kRv(M_v,η(1)) =γ_([L′

v,η],Q^ηv)≤dim_k(H¹(Xv,η,Lv,η^′ )) =g_v + #E_v^>1−1.

Then we obtain that

dim_kRv(M_v,η(1)) =γ_([L′

v,η],Q^ηv) = dim_kRv(H¹(Xv,η,Lv,η^′ )) = g_v+ #E_v^>1 −1.

We define Iv,η := Lv,η⁻¹⊗Lv,η^′ . Note that Iv,η is a line bundle on Xv,η of degree 0.

Then we have

γ_([L

v,η⊗Iv,η],Q^ηv)=γ_([L′

v,η],Q^ηv)

= dim_kRv(H¹(Xv,η,Lv,η^′ )) = dim_kRv(H¹(Xv,η,Lv,η⊗Iv,η)) =g_v+ #E_v^>1−1.

This completes the proof of the proposition.

Remark 6.4.1. Proposition 6.4 gives a positive answer of Problem of Remark 4.5.2 under certain assumptions of divisors. On the other hand, we may pose a generalized version of Tamagawa’s problem as follows.

Problem . We maintain the notation introduced in Remark 4.5.2. Suppose that X^• is a component-generic smooth pointed stable curve over k. Let ([L], D) be an arbitrary element of Pe_X•,n. Does the Raynaud-Tamagawa theta divisorΘ_ED associated toE_D exist?

ドキュメント内 On the Averages of Generalized Hasse-Witt Invariants of Pointed Stable Curves in Positive Characteristic (ページ 36-42)