On the Ordinariness of Coverings of Stable Curves

Transcription

1 On the Ordnarness of Coverngs of Stable Curves Yu Yang Abstract In the present paper, we study the ordnarness of coverngs of stable curves. Let f : Y X be a morphsm of stable curves over a dscrete valuaton rng R wth algebracally closed resdue feld of characterstc p > 0. Wrte S for Spec R and η (resp. s) for the generc pont (resp. closed pont) of S. Suppose that the generc fber X η of X s smooth over η, that the morphsm f η : Y η X η over η on generc fber nduced by f s a Galos étale coverng (hence Y η s smooth over η too) whose Galos group s a solvable group G, that the genus of the normalzaton of each rreducble component of the specal fber X s s 2, and that Y s s ordnary. Then we have the morphsm f s : Y s X s over s nduced by f s an admssble coverng. Ths result extends a result of M. Raynaud concernng the ordnarness of coverngs to the case where X s s a stable curve. If, moreover, suppose that G s a p-group, and the p-rank of the normalzaton of each rreducble component of X s s 2, we gve a numercal crteron for the admssblty of f s. Keywords: stable curve, admssble coverng, p-rank, ordnary, p-new-ordnary. Mathematcs Subect Classfcaton: Prmary 14H30; Secondary 11G20. Introducton Let R be a dscrete valuaton rng wth algebracally closed resdue feld k of characterstc p > 0 and K the quotent feld. We use the notaton S to denote Spec R. Wrte η and s for the generc pont of S and the closed pont of S correspondng to the natural morphsms Spec K S and Spec k S, respectvely. Let G be a fnte group, and let X be a stable curve of genus g(x) (n the present paper, the genus of a curve means the arthmetc genus of the curve) over S. Wrte X η and X s for the generc fber of X and the specal fber of X, respectvely. Moreover, we suppose that X η s smooth over η. We are nterested to understand the reducton of an étale coverng of X η. Let Y η be a smooth, geometrcally connected curve over η and f η : Y η X η a Galos étale coverng over η whose Galos group s G. By replacng S by a fnte extenson of S, we have that Y η admts a stable model over S, and f η extends to a unque G-stable coverng f : Y X over S (cf. Defnton 1.5 and Remark 1.5.1). In the present paper, we focus on a geometrc nvarant σ(y s ) of the specal fber Y s whch s called the p-rank of Y s (cf. Defnton 1.2). Let us recall some known results concernng the p-rank of the specal fber Y s. Let x be a closed pont of X s and G an arbtrary p-group. M. Raynaud (cf. [R1, Théorème 1]) 1

2 proved that, f x s a smooth pont, the p-rank of f 1 (x) s equal to 0 (note that f 1 (x) s not a fnte set n general). Afterwards, M. Saïd (cf. [S1, Theorem 1 and Proposton 1]) treated the case where x s a sngular pont of X s. Saïd obtaned an explct formula and a bound for the p-rank of f 1 (x) under the assumpton that G s a cyclc p-group. Recently, the author generalzed the formula for the p-rank of f 1 (x) to the case where G s an arbtrary p-group and obtaned a bound for the p-rank of f 1 (x) n the case where G s an arbtrary abelan p-group (cf. [Y2, Theorem 4.8], [Y3, Theorem 3.4]). On the other hand, f G s an arbtrary fnte group, and X s s smooth over s, Raynaud proved that, f the morphsm f s on specal fbers nduced by f s not an étale coverng, then Y s s not ordnary (cf. [R2, Proposton 3]). In the present paper, we study the ordnarness of stable coverngs. Our man theorem s as follows, see also Theorem 2.6: Theorem 0.1. Let Y be a stable curve over S and f : Y X a Z/pZ-stable coverng over S. Suppose that the genus of the normalzaton of each rreducble component of X s s 2, and the morphsm f s : Y s X s over s nduced by f s p-new-ordnary (cf. Defnton 2.4). Then f s s an admssble coverng (cf. Defnton 1.1). If, moreover, we suppose that the p-rank of the normalzaton of each rreducble component of X s s 2, then f s s an admssble coverng f and only f σ(y s ) 1 = p(σ(x s ) 1). As a corollary, we generalze the man result of [R2] to the case where X s s a stable curve, and G s a solvable group; moreover, f G s a p-group, we obtan a numercal crteron for the admssblty of G-stable coverngs as follows, see also Corollary 2.7. Corollary 0.2. Let G be a fnte solvable group, Y a stable curve over S, and f : Y X a G-stable coverng over S. Suppose that the genus of the normalzaton of each rreducble component of X s s 2, and that Y s s ordnary (.e., σ(y s ) = g(y s ) = (#G)(g(X s ) 1) + 1). Then the morphsm f s : Y s X s over s nduced by f s an admssble coverng. Moreover, suppose that the p-rank of the normalzaton of each rreducble component of X s s 2, and that G s a p-group. Then the morphsm f s : Y s X s over s nduced by f s an admssble coverng f and only f σ(y s ) 1 = (#G)(σ(X s ) 1). Remark Suppose that X s s ordnary, and that f s s an admssble coverng over s. If G s not a p-group, then Y s s not ordnary n general. Fnally, we would lke to menton that Saïd extended the man result of [R2] to the case where f η : Y η X η s a Galos coverng over η (cf. [S2, Thoerem]). More precsely, Saïd proved the followng result: let X be a smooth stable curve over S and f : Y X a morphsm of stable curves over S; suppose that char(k) = p > 0, and η : Y η X η s a Galos coverng whose Galos group s somorphc to Z/pZ (.e., the extenson of functon felds K(Y η )/K(X η ) nduced by f η s a Galos extenson whose Galos group s somorphc to Z/pZ). Saïd proved that, f f s : Y s X s s not genercally étale, then Y s s not ordnary. Note that, f char(k) = 0 and char(k) = p > 0, then ths result follows mmedately from [R1, Théorème 1 ] (.e., a tame verson of [R1, Théorème 1]). 2

3 1 Prelmnares In ths secton, we gve some defntons whch wll be used n the present paper. Defnton 1.1. Let C 1 and C 2 be two sem-stable curves over an algebracally closed feld l and ϕ : C 2 C 1 a morphsm of sem-stable curves over Spec l. We shall call ϕ a Galos admssble coverng over Spec l (or Galos admssble coverng for short) f the followng condtons hold: () there exsts a fnte group G Aut k (C 2 ) such that C 2 /G = C 1, and ϕ s equal to the quotent morphsm C 2 C 2 /G; () for each c 2 C2 sm, ϕ s étale at c 2, where ( ) sm denotes the smooth locus of ( ); () for any c 2 C sng 2, the mage ϕ(c 2 ) s contaned n C sng 1, where ( ) sng denotes the sngular locus of ( ); (v) for each c 2 C sng 2, the local morphsm between two nodes (cf. ()) nduced by ϕ may be descrbed as follows: Ô C1,ϕ(c 2 ) = l[[u, v]]/uv ÔC 2,c 2 = l[[s, t]]/st u s n v t n, where (n, char(l)) = 1 f char(l) = p > 0; moreover, wrte D c2 G for the decomposton group of c 2 ; then τ(s) = ζ #Dc2 s and τ(t) = ζ 1 #D c2 t for each τ D c2, where ζ #Dc2 s a prmtve #D c2 -th root of unt. We shall call ϕ an admssble coverng f there exsts a morphsm of stable curves ϕ : C 2 C 2 over Spec l such that the composte morphsm ϕ ϕ : C 2 C 1 s a Galos admssble coverng over Spec l. For more detals on admssble coverngs and the admssble fundamental groups for (ponted) sem-stable curves, see [M1], [M2]. Remark Note that, f C 2 s smooth over l, then the defnton of admssble coverngs mples that ϕ s an étale coverng. Defnton 1.2. Let C be a proper algebrac curve over an algebracally closed feld of characterstc p > 0. We defne the p-rank σ(c) of C to be σ(c) := dm Fp H 1 ét(c, F p ). Moreover, let C be a noetheran scheme of dmenson 0 over an algebracally closed feld of characterstc p > 0. Then we defne the p-rank of C to be σ(c ) = 0. Remark Suppose that C s a sem-stable curve over an algebracally closed feld of characterstc p > 0. Wrte Γ C for the dual graph of C, v(γ C ) for the set of vertces of Γ C, C v for the rreducble component of C correspondng to v v(γ C ), and C v for the normalzaton of C v, respectvely. Then t s easy to prove that the p-rank σ(c) of C s equal to v v(γ C ) σ( C v ) + rank(h 1 (Γ C, Z)), where rank( ) denotes the rank of ( ) as a free Z-module. 3

4 Defnton 1.3. Let C be a sem-stable curve of genus g(c) over an algebracally closed feld of characterstc p > 0. We shall call C ordnary f σ(c) = g(c). Note that Remark mples that C s ordnary f and only f C v s ordnary for each v v(γ C ). Defnton 1.4. Let ψ : C 2 C 1 be a Galos coverng (possbly ramfed) of smooth proectve curves over an algebracally closed feld of characterstc p > 0, whose Galos group s a fnte p-group G. Wrte g(c 1 ) and g(c 2 ) for the genera of C 1 and C 2, respectvely. We shall call ψ p-new-ordnary f g(c 2 ) σ(c 2 ) = (#G)(g(C 1 ) σ(c 1 )), where #( ) denotes the cardnalty of ( ). Remark Note that, f C 1 s ordnary, then ψ s p-new-ordnary f and only f C 2 s ordnary. Remark For any closed pont c 2 C 2, wrte e c2 for the ramfcaton ndex of ψ at c 2 and δ c2 for the degree of the dfferent of ψ at c 2. Then the genus and the p-rank of C 2 can be calculated by usng the Remann-Hurwtz formula 2g(C 2 ) 2 = (#G)(2g(C 1 ) 2) + c 2 δ c2 and the Deurng-Shafarevch formula (cf. [C, p35], [B, Theorem 3.1]) σ(c 2 ) 1 = (#G)(σ(C 1 ) 1) + c 2 e c2, respectvely. Thus, we have g(c 2 ) σ(c 2 ) (#G)(g(C 1 ) σ(c 1 )) = c 2 (δ c2 2(e c2 1))/2. Let I c2 G be the nerta group of c 2 and I c2, the -th ramfcaton group of c 2. Snce G s a p-group, we obtan that I c2 = I c2,0 = I c2,1. Moreover, we have δ c2 = 0 (#I c2, 1) = 2(#I c2 1) + 2(#I c2, 1). Thus, ψ s p-new-ordnary f and only f δ c2 = 2(e c2 1) (.e., I c2, are trval for all 2 and for all c 2 C 2 ). From now on, we fx some notatons. Let R be a dscrete valuaton rng wth algebracally closed resdue feld k of characterstc p > 0, K the quotent feld of R, and K an algebrac closure of K. We use the notaton S to denote the spectrum of R. Wrte η, η and s for the generc pont of S, the geometrc generc pont of S, and the closed pont of S correspondng to the natural morphsms Spec K S, Spec K S, and Spec k S, respectvely. Let X be a sem-stable curve over S of genus g X 2. Wrte X η := X S η for the generc fber of X, X η := X S η for the geometrc generc fber of X, and X s := X S s for the specal fber of X, respectvely. Moreover, we suppose that X η s smooth over η. 4

5 Defnton 1.5. Let Y be a stable curve over S, f : Y X a morphsm of sem-stable curves over S, and G a fnte group. We shall call f a G-sem-stable coverng over S f the morphsm f η : Y η X η over η nduced by f on generc fbers s an Galos étale coverng whose Galos group s somorphc to G. We shall call f a G-stable coverng over S f f s a G-sem-stable coverng over S, and X s a stable curve over S. Remark Suppose that X s a stable curve over S. Let W η X η be any geometrcally connected Galos étale coverng over η whose Galos group s G. [LL, Proposton 4.4 (a)] mples that, by replacng S by a fnte extenson of S, the morphsm W η X η may extend to a G-stable coverng over S. Remark Let Y be a stable curve over S, f : Y X a G-sem-stable coverng over S, and y any closed pont of Y. Then f nduces a morphsm f y : Spec ÔY,y Spec ÔX,f(y) over S. Suppose that f s : Y s X s over s nduced by f s genercally étale. We clam that f s an admssble coverng. Frst, we prove that f s a fnte morphsm. Let x be any closed pont of X. If x s a smooth pont, then Zarsk-Nagata s purty theorem mples f s s étale over x. If x s a sngular pont of X s, then Zarsk-Nagata s purty theorem and [T, Lemma 2.1 ()] mply that f 1 (x) s a set of sngular ponts of Y s. Thus, f s a fnte morphsm. Second, we prove that f s s an admssble coverng. If y s a smooth pont, then f(y) X s a smooth pont too (cf. [R3, Lemme 6.3.5] or [Y1, Lemma 2.1]). Then Zarsk-Nagata s purty theorem mples that the morphsm f y s étale. If y s a sngular pont of Y s, then f(y) X s a sngular pont of X s too (cf. [R3, Lemme 6.3.5] or [Y1, Lemma 2.1]). Then Zarsk-Nagata s purty theorem and [T, Lemma 2.1 ()] also mply that the morphsm of local rngs ÔX s,f(y) ÔY s,y nduced by f y satsfes the condton (v) of Defnton 1.1. Thus, we have f s s a Galos admssble coverng over s f and only f f s s genercally étale. Defnton 1.6. Let Y be a stable curve over S and f : Y X a G-sem-stable coverng over S. Suppose that the morphsm f s : Y s X s on specal fbers nduced by f s not fnte. A closed pont x X s called a vertcal pont assocated to f, or for smplcty, a vertcal pont when there s no fear of confuson, f dm(f 1 (x)) = 1. The nverse mage f 1 (x) s called the vertcal fber assocated to x. Remark Suppose that R has mxed characterstc, and k s an algebrac closure of a fnte feld. Moreover, suppose that X s a stable curve over R. Then A. Tamagawa prove that, for any closed pont x, after replacng S by a fnte extenson of S, there exsts a fnte group G and a G-stable coverng f : Y X over S such that x s a vertcal pont assocated to f (cf. [T, Theorem 0.2 (v)]). Next, we recall some results concernng the p-ranks of vertcal fbers. Frst, n the case of smooth ponts, the followng result was proved by Raynaud (cf. [R1, Théorème 1]). Proposton 1.7. Let G be a fnte p-group, Y a stable curve over S, f : Y X a G-sem-stable coverng over S, and x a vertcal pont assocated to f. Suppose that x s a smooth pont of X s. Then the p-rank of each connected component of the vertcal fber f 1 (x) assocated to x s equal to 0. 5

6 In the remander of ths secton, let Y be a stable curve over S, f : Y X a Z/pZstable coverng over S and x a vertcal pont assocated to f; moreover, we suppose that x s a sngular pont of X s. Then there are two rreducble components X 1 and X 2 (whch may be equal) of X s such that x X 1 X 2. Wrte Y 1 (resp. Y 2 ) for an rreducble component of Y s such that f s (Y 1 ) = X 1 (resp. f s (Y 2 ) = X 2 ). Snce Y s a stable curve over S, the acton of Z/pZ on the generc fber Y η nduces an acton of Z/pZ on the specal fber Y s. Wrte I 1 (resp. I 2 ) for the nerta group of Y 1 (resp. Y 2 ) (note that I 1 (resp. I 2 ) does not depend on the choces of Y 1 (resp. Y 2 )). Wrte Y for the normalzaton of X n the functon feld K(Y ) nduced by f and f : Y X for the normalzaton morphsm. Let y Y be the closed pont such that f (y ) = x. Snce x s a vertcal pont assocated to f, the closed pont y s not a node of the specal fber Y s of Y. We consder the morphsm Spec O Y,y Spec O X,x nduced by f. Snce Z/pZ s a p-group, the Zarsk-Nagata s purty theorem and [T, Lemma 2.1 ()] mply that, f I 1 = I 2 = {1}, the morphsm Spec O Y,y Spec O X,x s étale. Ths means that y s a node. Thus, ether I 1 = Z/pZ or I2 = Z/pZ holds. Wthout loss of generalty, we may assume that I 1 = Z/pZ. Note that f 1 (x) s connected. For the p-rank of f 1 (x), we have the followng lemma. Lemma 1.8. Wrte Γ x for the dual graph of the sem-stable curve f 1 (x) red Y s over s, where ( ) red denotes the reduced nduced closed subscheme of ( ). (a) If I 1 = Z/pZ, and I2 s trval, then σ(f 1 (x)) = 0. (b) If I 1 = I 2 = Z/pZ, then one of the followng condtons holds: () σ(f 1 (x)) s equal to 0; () σ(f 1 (x)) = rank(h 1 (Γ x, Z)) = p 1; () σ(f 1 (x)) = p 1 and Γ x s a tree. Proof. The lemma follows mmedately from [S1, Proposton 1] or [Y2, Theorem 4.8 and Corollary 4.10] when G = Z/pZ. Remark In fact, Saïd obtaned a p-rank formula for vertcal fbers n the case where G s a cyclc p-group (cf. [S1, Proposton 1]). Moreover, the author generalzes the p-rank formula to the case where G s an arbtrary p-group (cf. [Y2, Theorem 4.8 and Corollary 4.10]). Remark We can construct some Z/pZ-stable coverngs whch satsfy the condtons of Lemma 1.8 (a) and Lemma 1.8 (b)-(). However, the author does not know that how to construct a Z/pZ-stable coverng whch satsfes the condtons of Lemma 1.8 (b)-() or Lemma 1.8 (b)-(). Remark Y. Hosh obtaned an anabelan pro-p good reducton crteron for a smooth proper ordnary hyperbolc curve (.e., the reducton s an ordnary stable curve) over a p-adc feld (cf. [H]). It s very nterestng for the author to know whether or not the pro-p good reducton crteron of Hosh can be extended to arbtrary proper hyperbolc curves. One of the man techncal dffcultes s how to construct a p-coverng of a gven proper hyperbolc curve such that there exst two rreducble components whose p-ranks are postve. We have the followng queston: Queston: Suppose that dm Fp (H 1 ét(x η, F p )) σ(x s ) > 0 (note that, f char(k) = 0, the nequalty always holds). After replacng S by a fnte extenson of S, does there 6

7 exst a Z/pZ-stable coverng over S such that, for some vertcal pont x, the vertcal fber assocated to x satsfes the condtons of Lemma 1.8 (b)-()? Proposton 1.9. Suppose that the sem-stable curve f 1 (x) red over s s ordnary. I 1 = I 2 = Z/pZ, then σ(f 1 (x)) = p 1. If Proof. We mantan the notatons ntroduced n the proof of Lemma 1.8. If σ(f 1 (x)) = 0, then for each 1 n, I P = Z/pZ. Ths means that V Y s s a proectve lne for each 1 n. Snce Y s s a stable curve over s, we have V h 1 (B) red for each 1 n. Thus, h 1 (B) red. On the other hand, snce Y s s a stable curve over s, Proposton 1.7 mples that h 1 (B) red s not ordnary. Ths s a contradcton. Then the proposton follows from Lemma 1.8 (b). 2 Ordnarness of stable coverngs In ths secton, we prove our man theorem of the present paper. Defnton 2.1. Let C 1 and C 2 be two sem-stable curves over an algebracally closed feld l of characterstc p > 0, ψ : C 2 C 1 a fnte surectve morphsm over l, and G Aut(C 2 /C 1 ) a fnte p-group. We shall call ψ a Galos coverng wth Galos group G f G acts genercally freely on C 2, G acts freely at the nodes of C 2, and ψ s equal to the quotent morphsm C 2 C 2 /G. Lemma 2.2. Let G be a p-group, C 1 and C 2 two sem-stable curves over an algebracally closed feld l of characterstc p > 0, and ψ : C 2 C 1 a Galos coverng wth Galos group G. Then we have σ(c 2 ) 1 = (#G)(σ(C 1 ) 1) + (e c2 1), c 2 C cl 2 where C cl 2 denotes the set of closed ponts of C 2, and e c2 denotes the ramfcaton ndex of ψ at c 2. Proof. There exst many proofs of the lemma. For example, t s easy to see that the proof of the Deurng-Shafarevch formula gven n [B, Theorem 3.1] can be extended to the case where ψ s a Galos coverng of sem-stable curves. Remark Lemma 2.2 extends the Deurng-Shafarevch formula to Galos coverngs of sem-stable curves. Moreover, the author also extended the Deurng-Shafarevch formula to a more general case by usng the theory of sem-graphs wth p-rank (cf. [Y2, Theorem 4.5]). Defnton 2.3. Let Γ be a fnte graph. We use the notaton v(γ) to denote the set of vertces of Γ and e(γ) to denote the set of edges of Γ. For an edge e e(γ), we use the notaton v(e) to denote the set of vertces whch are abutted by e. We defne an equvalence relaton on e(γ) as follows: e 1 e 2 f v(e 1 ) = v(e 2 ). Then we obtan a new fnte graph Γ nd := Γ/. We shall call Γ nd the nduced graph of Γ. Note that v(γ nd ) = v(γ) and e(γ nd ) = e(γ)/. 7

8 Defnton 2.4. Let Y be a stable curve over S and f : Y X a Z/pZ-stable coverng over S. For each rreducble component Y v of the specal fber Y s of Y, wrte X v for f(y v ). We shall call f s p-new-ordnary f, for each rreducble component Y v Y s, one of the followng condtons holds: () f f s Yv s a constant morphsm (.e., f(y v ) s a pont), then Y v s ordnary; () f the restrcton morphsm f s Yv s genercally étale, then f s Yv : Ỹv X v nduced by f s Yv s p-new-ordnary (cf. Defnton 1.4), where ( ) denotes the normalzaton of ( ). Remark Note that, f X s s ordnary, then f s s p-new-ordnary f and only f Y s s ordnary. Defnton 2.5. Let Z be a stable curve over an algebracally closed feld. We shall call Z sturdy f the genus of the normalzaton of each rreducble component of Z s 2. Now, let us prove the man theorem. Theorem 2.6. Let f : Y X be a Z/pZ-stable coverng over S. Suppose that X s s sturdy, and the morphsm f s : Y s X s over s nduced by f s p-new-ordnary. Then f s s an admssble coverng. If, moreover, we suppose that the p-rank of the normalzaton of each rreducble component of X s s 2, then f s s an admssble coverng f and only f σ(y s ) = p(σ(x s ) 1) + 1. Proof. Wrte {Xét } I (resp. {X n } J ) for the set of stable subcurves of X s such that the followng condtons hold: () for each I (resp. J), f s s genercally étale over Xét (resp. purely nseparable over X n ); () for each I (resp. J) and each rreducble and X v X n ), component X v X s, f X v Xét and X v Xét (resp. X v X n then f s s purely nseparable (resp. f s s genercally étale) over X v. Then we have For each I (resp. J), we wrte Γ X ét X n ) and g(xét ) (resp. g(x n X s = ( I Xét ) ( J X n ). (resp. Γ X n ) for the dual graph of Xét (resp. X n ). (resp. )) for the genus of Xét Wrte V for the set of vertcal ponts assocated to f. For each vertcal pont x V, wrte E x for the vertcal fber assocated to x (note that E x s connected) and g(e x ) for the genus of E x. If V contans a smooth pont of X s, then Proposton 1.7 and Defnton 2.4 mply that f s s not p-new-ordnary. Thus, V s contaned n the sngular locus of X s. For each sngular pont x of X s, Remark mples that f s s étale over x. Thus, we \ I Xét have V J X n. Ths means that, for each x V, we have ether x J X n or x ( J X n ) ( I Xét ). In order to prove the theorem. we wll calculate the p-rank of Y s by usng the Deurng- Shafarevch formula. By applyng Lemma 2.2, we may assume that Xét s rreducble for each I. Let L := e(γ X n ) e(γ Xs ) (cf. Defnton 2.3). We have the followng clam: Clam 1: We may deform the stable curve X s along L to obtan a new stable curve over η := Spec K such that the set of edges of the dual graph of the new stable curve may be naturally dentfed wth e(γ Xs ) \ L. 8

9 Let us prove Clam 1. Suppose that ϕ s : s M g(x),s := M g(x) Spec Z S s the classfyng morphsm determned by X s s. Thus the completon of the local rng of the modul stack at ϕ s s somorphc to R t 1,..., t 3g(X) 3, where the t 1,..., t 3g(X) 3 are ndetermnates. Furthermore, the ndetermnates t 1,..., t m may be chosen so as to correspond to the deformatons of the nodes of X s. Suppose that {t 1,..., t d } s the subset of {t 1,..., t m } correspondng to the subset L e(γ Xs ). Now fx a morphsm S Spec R t 1,..., t 3g(X) 3 such that t d+1,..., t m 0 R, but t 1,..., t d map to nonzero elements of R. Then the composte morphsm ϕ : S Spec R t 1,..., t 3g(X) 3 M g(x),s determnes a stable curve X over S. Moreover, the specal fber of X s naturally somorphc to X s over s. Wrte Xs for the geometrc generc fber X η η over η and Γ X s for the dual graph of Xs. It follows from the constructon of Xs that we have two natural maps v(γ Xs ) v(γ X s ), e(γ Xs ) \ L e(γ X s ) (the latter of whch s a becton). Ths completes the proof of Clam 1. Note that #v(γ X s ) = #I + #J. Wrte n for #(Xét ( J X n )), r Xs for rank(h 1 (Γ Xs, Z)), rx nd for rank(h 1 (Γ nd s X, Z)), s r X n for rank(h 1 (Γ X n, Z)), and rx n s for J r X n, respectvely, where Γnd X denotes to the s nduced graph of Γ X s (cf. Defnton 2.3). Then we have r Xs = r nd X s + rn X s + I n #e(γ nd X s ). For each I (resp. J), wrte Y ét \ I Xét of Y s (resp. {fs 1 ) (resp. g(y n g(y ét where F ét étale over Xét (X n (resp. Y n ) for the closed subscheme fs 1(Xét ) red ) red} of Y s, where { } denotes the closure of { }), and )) for the genus of Y ét Y ét (resp. Y n = F ét = F n (resp. Y n ( x V X ét E x) ). Then we have ( x X n (V\Xét ) E x )), (resp. F n ) denotes the closed subscheme of Y ét (resp. purely nseparable over X n (resp. Y n ) whch s genercally ). Next, we start to prove the theorem. Step 1: For any I (resp. J), let us calculate g(y ét ) and σ(y ét ) (resp. g(y n ) and σ(y n )) under the assumpton that f s s p-new-ordnary, respectvely. If F ét s rreducble, by the Remann-Hurwtz formula and Lemma 1.8 (a), we have g(y ét ) = p(g(xét ) 1) deg(r ) (p 1)#(V Xét ), where R denotes the ramfcaton dvsor of f s F ét : F ét Xét. Note that we have #Supp(R ) + #(V Xét ) = n. 9

10 Moreover, snce we assume that f s s p-new-ordnary, Remark and Defnton 2.4 mply that deg(r ) = 2#Supp(R )(p 1). Thus, we obtan For the p-rank of Y ét, we have g(y ét ) = p(g(xét ) 1) + n (p 1) + 1. σ(y ét ) = p(σ(xét ) 1)+(p 1)(#deg(R )+#(V Xét ))+1 = p(σ(xét ) 1)+n (p 1)+1. If F ét s dsconnected, then we have V Xét = Xét ( X n ). Snce we assume that f s s p-new-ordnary, Lemma 1.8 (a) and Defnton 2.4 mply that F ét = X ét, and for any x V Xét, all the rreducble components of E x are somorphc to P 1. Note that rank(h 1 (Γ Y ét, Z)) s equal to (n 1)(p 1). Thus, we have and g(y ét ) = pg(xét ) + (n 1)(p 1) = p(g(xét ) 1) + n (p 1) + 1 σ(y ét ) = pσ(xét ) + (n 1)(p 1) = p(σ(xét ) 1) + n (p 1) + 1. On the other hand, snce we assume that f s s p-new-ordnary, by Proposton 1.9, for each x X n (V \ Xét ), we have σ(e x ) = g(e x ) = p 1. Then we obtan g(y n ) = g(f n ) + g(e x ) = g(x n ) + (p 1)#(X n (V \ Xét )) ) x X n (V\ Xét and σ(y n ) = σ(f n ) + x X n (V\ Xét σ(e x ) = σ(x n ) + (p 1)#(X n ) (V \ Xét )) where g(f n ) denotes the genus of F n. Step 2: Let us prove the frst part of the theorem (.e., f s s an admssble coverng under the assumpton that f s s p-new-ordnary.). The dea of the proof of the frst part of the theorem s by comparng the genus of generc fber Y η wth the genus of specal fber Y s. We wll compute the genus of generc fber of Y η by applyng Remann-Hurwtz formula, and compute the genus of specal fber Y s by applyng the propertes of p-new-ordnary and the results obtaned n Step 1. Wrte m for #(X n (V \ Xét )). Then we have g(y s ) = g(y ét ) + g(y n ) + r Xs rx n s = (p(g(xét ) 1) + n (p 1) + 1) + (g(x n ) + m (p 1)) + r Xs r n X s. 10

11 On the other hand, by applyng the Remann-Hurwtz formula to f η : Y η X η, we obtan that the genus g(y η ) of the generc fber Y η s equal to p(( g(xét ) + g(x n ) + r Xs r n X s ) 1) + 1. Snce g(y η ) s equal to g(y s ), we obtan (1 p)( (g(x n ) m ) 1 + r Xs r n X s (n 1)) = 0. Then we have 0 = (g(x n ) m ) 1 + r Xs r n X s (n 1) = (g(x n ) m ) 1 + rx nd + s n #e(γ nd X s ) (n 1) = (g(x n ) m ) 1 + rx nd s #e(γnd X ) + #I s By applyng Euler-Poncaré characterstc formula for the graph Γ nd X, we obtan s Then we have r nd X s #e(γnd X ) + #I 1 = s #v(γnd X ) + #I = #J. s 0 = (g(x n ) m ) #J = (g(x n ) m 1). On the other hand, by the assumptons that X s s sturdy, we have g(x n ) = g( X v ) + r X n 2 #v(γ X n ) + r X n v v(γ X n ) = #v(γ X n ) + #e(γ X n ) + 1, where X v denotes the genus of the normalzaton of X v, and g( X v ) denotes the genus of X v. If {X n } J s not empty, snce #e(γ X n ) m, we have (g(xn ) m 1) > 0. Then we obtan a contradcton. Thus, {X n } J s empty. Ths means that f s s genercally étale. Then by Remark 1.5.2, we have f s s an admssble coverng. Step 3: Let us prove the moreover part of the theorem. The dea of the proof of the moreover part s by comparng the p-rank of Y s wth the p-rank of Y s when f s s p-new-ordnary. We wll compute the p-rank of Y s by applyng Deurng-Shafarevch formula, the propertes of p-new ordnary, and the results obtaned n Step 1. 11

12 If f s s an admssble coverng, then the moreover part follows from Lemma 2.2. Thus, we suppose that σ(y s ) = p(σ(x s ) 1) + 1. Then we have σ(y s ) = p(σ(x s ) 1) + 1 = p(( σ(xét ) + σ(x n ) + r Xs r n X s ) 1) + 1. Wrte m for #(X n (V \ Xét )). On the other hand, σ(y s ) attans ts maxmum f and only f f s s p-new-ordnary. Moreover, f f s s p-new-ordnary, the p-rank of Y s s σ(y ét ) + σ(y n ) + r Xs rx n s = (p(σ(xét ) 1) + n (p 1) + 1) + (σ(x n ) + m (p 1)) + r Xs r n X s. Thus, we have σ(y s ) = p(( σ(xét ) + σ(x n ) + r Xs r n X s ) 1) + 1 (p(σ(xét ) 1) + n (p 1) + 1) + (σ(x n ) + m (p 1)) + r Xs r n X s. Smlar arguments to the arguments gven n the proof above mply that (σ(x n ) m 1) 0. On the other hand, snce σ( X v ) 2 for each v v(γ X n σ(x n ) = 2 #v(γ X n ) + r X n v v(γ X n ) σ( X v ) + r X n ), we have = #v(γ X n ) + #e(γ X n ) + 1. If {X n } J s not empty, snce #e(γ X n ) m, we have (σ(xn ) m 1) > 0. Then we obtan a contradcton. Thus, {X n } J s empty. Ths means that f s s genercally étale. Then by Remark 1.5.2, we have f s s an admssble coverng. We complete the proof of the theorem. By applyng Theorem 2.6, we generalze the man result of [R3] as follows. Moreover, we obtan a numercal crteron for the admssblty of G-stable coverngs f G s a p-group. Corollary 2.7. Let G be a fnte solvable group, Y a stable curve over S, and f : Y X a G-stable coverng over S. Suppose that X s s sturdy, and that Y s s ordnary (.e., σ(y s ) = g(y s ) = (#G)(g(X s ) 1) + 1). Then the morphsm f s : Y s X s over s nduced by f s an admssble coverng. Moreover, suppose that the p-rank of the normalzaton of each rreducble component of X s s 2, and that G s a p-group. Then the morphsm f s : Y s X s over s nduced by f s an admssble coverng f and only f σ(y s ) 1 = (#G)(σ(X s ) 1). 12

13 Proof. If X s s not ordnary, then Y s s not ordnary. Thus, we may assume that X s s ordnary. Snce G s a fnte solvable group, we have a seres of subgroups {1} =: G m+1 G m G m 1... G 0 := G such that G /G +1, = 0,..., m, s a cyclc group of prme order. Note that Y := Y/G m+1, = 0,..., m, s a sem-stable curve over S. Then the seres of subgroups of G nduces a sequence of morphsms of sem-stable curves f 0 f 1 f m 1 f m Y =: Y 0 Y1... Ym X. such that f m... f 0 = f. Suppose that f s s not an admssble coverng. Then there exsts 0 w m such that (f ) s s an admssble coverng for each w + 1 and (f w ) s s not an admssble coverng. Note that snce an admssble coverng of a sturdy stable curve s sturdy, Y w+1 s sturdy. Moreover, Y, w, s a stable curve over S, and f, w s a G /G +1 -stable coverng over S. If (Y w+1 ) s s not ordnary, then Y s s not ordnary. Thus, we may assume (Y w+1 ) s s ordnary. Snce (f w ) s s not an admssble coverng, G w /G w+1 s somorphc to Z/pZ. Then the corollary follows from Theorem 2.6. The moreover part follows mmedately from the moreover part of Theorem 2.6 and Lemma 2.2. Acknowledgements I would lke to thank the referee for carefully readng the manuscrpt and for gvng many comments whch substantally helped mprovng the qualty of the paper. Ths research was supported by JSPS KAKENHI Grant Number 16J

14 References [B] I. Bouw, The p-rank of curves and covers of curves. Courbes sem-stables et groupe fondamental en géométre algébrque (Lumny, 1998), , Progr. Math., 187, Brkhäuser, Basel, [C] R. Crew, Étale p-covers n characterstc p. Composto Math. 52 (1984), [H] [LL] [M1] [M2] Y. Hosh, On the pro-p absolute anabelan geometry of proper hyperbolc curves, RIMS Preprnt Q. Lu, D. Lorenzn, Models of curves and fnte covers. Composto Math. 118 (1999), S. Mochzuk, The geometry of the compactfcaton of the Hurwtz scheme, Publ. Res. Inst. Math. Sc. 31 (1995), S. Mochzuk, The profnte Grothendeck conecture for closed hyperbolc curves over number felds. J. Math. Sc. Unv. Tokyo 3 (1996), [R1] M. Raynaud, p-groupes et réducton sem-stable des courbes. The Grothendeck Festschrft, Vol. III, , Progr. Math., 88, Brkhäuser Boston, Boston, MA, [R2] [R3] [S1] M. Raynaud, Mauvase réducton des courbes et p-rang. C. R. Acad. Sc. Pars Sér. I Math. 319 (1994), M. Raynaud, Revêtements de la drote affne en caractérstque p > 0 et conecture d Abhyankar, Invent. Math. 116 (1994), M. Saïd, p-rank and sem-stable reducton of curves. C. R. Acad. Sc. Pars Sér. I Math. 326 (1998), [S2] M. Saïd, p-rank and sem-stable reducton of curves II. Math. Ann. 312 (1998), [T] A. Tamagawa, Resoluton of nonsngulartes of famles of curves. Publ. Res. Inst. Math. Sc. 40 (2004), [Y1] Y. Yang, On the exstence of non-fnte coverngs of stable curves over complete dscrete valuaton rngs, to appear n Math. J. Okayama Unv. See also yuyang/papersandpreprnts/wrnfa.pdf [Y2] Y. Yang, Formulas for local and global p-ranks of coverngs of curves, RIMS Preprnt See also yuyang/papersandpreprnts/flg.pdf. 14

15 [Y3] Y. Yang, On the boundedness and graph-theoretcty of p-ranks of coverngs of curves, RIMS Preprnt See also yuyang/papersandpreprnts/bg.pdf Yu Yang Address: Research Insttute for Mathematcal Scences, Kyoto Unversty Kyoto , Japan E-mal: 15