p -Rank and semi-stable reduction of curves II

Math. Ann. 312, 625–639 (1998) MathematischeAnnalenc© Springer-Verlag 1998

p-Rank and semi-stable reduction of curves II

Mohamed Saıdi

Mathematisches Institut der Universitat Bonn, Beringstraße 6, D-53115 Bonn, Germany(e-mail: [email protected])

Received: 30 March 1998

Mathematics Subject Classification (1991):11G20, 14H30, 14G40

LetR be a discrete complete valuation ring, with algebraically closed residuefield of characteristicp > 0, and field of fractionsK. Let X be a smoothand properR-curve, with geometrically connected generic fibreXK :=X×RK. LetYK → XK be a Galois covering of degreepwithYK connected.In this paper we study thep-rank of the special fibre of a semi-stable modelof YK .

0. Introduction

This paper is a continuation of the previous paper [Sa]. LetR be a com-plete discrete valuation ring, with fractions fieldK, and algebraically closedresidue fieldk of characteristicp > 0. Consider agermX := SpecOX,x of aflat proper and normalR-curveX at a closed pointx which is either asmoothpoint or anordinary doublepoint. Letf : Y → X be a finite Galois coveringof groupG = Z/pZ, with Y normal. Assume that there exists a proper andbirational morphismf : Y → Y such that the special fibreY ×Spec R Spec k

of Y is semi-stable. For a closed pointy of Y, we give all the possible valuesof thep-rankof the fibref−1(y) of y in Y. In particular we give a bound forthep-rank of f−1(y), which depends only on the cardinality of the branchlocus in the morphismfK : Y ×Spec R SpecK → X ×Spec R SpecK be-tween the generic fibres, and the ramification at the generic points in themorphismfk : Y ×Spec R Spec k → X ×Spec R Spec k between the specialfibres, cf. theorem 1 for the case wherex is smooth, and theorem 2 for thecase wherex is a double point.

626 M. Saıdi

As an application, letX be a proper and semi-stableR-curve with smoothand geometrically connected generic fibre. Consider a finite Galois coveringf : Y → X of groupG = Z/pZ, with Y normal and connected. Assumethat there exists a proper and birational morphismf : Y → Y such thatY is semi-stable (such a morphism exists after eventually a finite extensionof R, by the theorem of semi-stable reduction for curves cf. [De-Mu]).We give bounds for thep-rank of the special fibreY ×Spec R Spec k ofY , in terms of thep-rank ofX ×Spec R Spec k, of the branch locus in themorphismfK : Y ×Spec R SpecK → X ×Spec R SpecK between thegeneric fibres, and the ramification at the generic points in the morphismfk : Y ×Spec R Spec k → X ×Spec R Spec k between the special fibres, cf.theorem 3 and theorem 4. Thep-rank of Y ×Spec R Spec k is known to bebounded by the genus ofY ×Spec R SpecK if char(K) = 0, and by thep-rank ofY ×Spec R SpecK if char(K) = p. The bounds we give are in somecases better than these bounds, in other words we can affirm in some casesthatY ×Spec R Spec k can not beordinary, i.e itsp-rank can not be maximalequal to the above bounds. For example we can prove the following:

Theorem.LetX be a proper and smoothR-curve with geometrically con-nected fibre. Assume that the genus ofX ×Spec R SpecK is greater or equalto 2. Let f : Y → X be a finite Galois covering of groupG = Z/pZ,with Y normal and connected. Assume that there exists a semi-stable modelf : Y → Y of Y over R. If the morphismfk : Y ×Spec R Spec k →X ×Spec R Spec k between the special fibres is inseparable, then the specialfibre Y ×Spec R Spec k can not be ordinary.

For more precise results see theorem 3, theorem 4, and remark 1. FinallyI would mention that I think the above results could be generalized to thecase of a covering with Galois group ap-group which is cyclic (and moregenerally anyp-group). I would like to come back to this question in asubsequent work.

1. Notations and definitions

For a finite setG, we will denote by|G| the cardinality ofG.

In all this work we consider a complete discrete valuation ringR, withfield of fractionsK, and algebraically closed residue fieldk of characteristicp > 0. For anR-schemeX, we denote byXK := X ×Spec R SpecK thegenericfibre of X, andXk := X ×Spec R Spec k its specialfibre. In whatfollows by a germX of an R-curve we mean thatX := SpecOX,x isthe spectrum of the local ring of a proper flat and normalR-schemeX, ofrelative dimension1, at a closed pointx.

p-Rank and semi-stable reduction of curves II 627

Let X be a germ of anR-curve. Then it follows from the theorem ofsemi-stable reduction for propre curves (cf. [De-Mu]) thatX admits afterpossibly a finite extension ofR a semi-stablemodel, by which we meanthat there exists a birational and proper morphismf : X → X such thatXK ' XK , and the following conditions hold

(i) The special fibreXk of X is reduced.

(ii) Xk has only ordinary double points as singularities.

We recall from [Sa] the definition ofp-rank for germ of curves.

Definition 1. LetX := SpecOX,x be agerm of anR-curveX at a closedpoint x, which has a semi-stable modelf : X → X overR. Thep-rankrx of X at x is defined as thep-rank of the proper and semi-stablek-curvef−1(x), which equals todimZ/pZ H1

et(f−1(x), Z/pZ). In particular, if X

is semi-stable at x, we haverx = 0.

Note that thep-rankrx is uniquely defined, and does not depend on thechoice of the semi-stable model ofX . One can also define thegenusgx ofX atx as the genus off−1(x) which equals todimk H1(f−1(x),Of−1(x)),and one hasrx ≤ gx (cf. [Ra-1]). LetOx be the local ring ofXk at x. Letδx := dimk Ox/Ox whereOx is the normalisation ofOx, and letmx bethe number of maximal ideals inOx. Thengx = δx − mx + 1, in particulargx = 0 if x is a smooth or an ordinary double point.

For a germX of anR-curve as above letµx := 2δx −mx +1. Considera finite morphismf : Y = Spec OY,y → X := SpecOX,x between germsof R-curves. Let(xj)j∈J be the points of the normalisationXk of Xk abovex, and for a fixedj let (yi,j)i∈Ij be the points of the normalisationYk

of Yk which are abovexj . Assume that the morphismfk : Yk → Xk isgenerically separable. Under this assumption we have the following “localRiemann-Hurwitz formula” which is due to Kato (cf. [Ka], and [Ma-Yo]) :

µy − 1 = n(µx − 1) + D −∑

i,j

dwi,j

wheren is the degree of the morphismf : Y → X , D is the degree ofthe divisor of ramification in the morphismfK : YK → XK , anddw

i,j :=vxj (δyi,j ,xj )− e+1, whereδyi,j ,xj is the discriminant ideal of the extension

OXk,xj→ OYk,yi,j

of complete discrete valuation ring, ande its ramificationindex.

628 M. Saıdi

2. Galois covers of degreep and localp-rank

In this section we will consider a germX of a semi-stableR-curve at asmooth or at an ordinary double point, and Galois coversf : Y → Xof degreep. We will study the localp-rank of Y at a closed point in thecase where the morphismfK : YK → XK is ramified. The case wherefK : YK → XK is etale has been considered in [Ra] and [Sa]. Our mainresults are the following:

Theorem 1. Let X := SpecOX,x be a germ of anR-curve at a smoothpoint x (i.e. the completionOX,x is isomorphic toR[[s]]). Letf : Y → Xbe a finite Galois covering with groupG = Z/pZ, and withY normal. LetB = {x1, ..., xm} ⊂ XK be the branch locus of the morphismfK : YK →XK , which we assume non empty of cardinalitym ≥ 1. Letη be the genericpoint ofXk, and letI(η) be the inertia subgroup at a generic point ofYk

aboveη. Assume thatY has a semi-stable modelf : Y → Y overR, andlet y be the closed point ofY (Y is necessarily local). The following holds:

Case 1. The morphismfk : Yk → Xk is generically separable, i.e.I(η) = 1is trivial.

(i) If char(K) = 0, then necessarilym ≥ 2, andry = α(p − 1) for aninteger0 ≤ α ≤ (m − 2)/2. In particular ry ≤ (m − 2)(p − 1)/2, andry ≤ (m − 3)(p − 1)/2 if m is odd.

(ii) If char(K) = p, thenry = β(p − 1) for an integerβ ≤ (m − 1). Inparticular ry ≤ (m − 1)(p − 1).

Case 2. The morphismfk : Yk → Xk is purely inseparable, i.e.I(η) = G.

(i) If char(K) = 0, thenry = α(p − 1) for an integer0 ≤ α ≤ m/2. Inparticular ry ≤ m(p − 1)/2, andry ≤ (m − 1)(p − 1)/2 if m is odd.

(ii) If char(K) = p, thenry = β(p − 1) for an integer0 ≤ β ≤ m. Inparticular ry ≤ m(p − 1).

Theorem 2. LetX := SpecOX,x be a germ of anR-curve at an ordinarydouble pointx (i.e. the completionOX,x is isomorphic toR[[s, t]]/(st−πe)for a certain integere ≥ 1 whereπ is a uniformizing parameter ofR). Letf : Y → X be a finite Galois covering with groupG = Z/pZ, and withYnormal. LetB = {x1, ..., xm} ⊂ XK be the branch locus of the morphismfK : YK → XK , which we assume non empty of cardinalitym ≥ 1. Letη1andη2 be the generic points ofXk, and letI(ηi) be the inertia subgroup ata generic point ofYk aboveηi for i = 1, 2. Assume thatY has a semi-stablemodelf : Y → Y overR, and lety be the closed point ofY. The followingholds:


Case 1.The morphismfk : Yk → Xk is generically separable, i.e.I(η1) =I(η2) = 1.

(i) If char(K) = 0, then necessarilym ≥ 2 andry = α(p − 1) for aninteger0 ≤ α ≤ (m − 2)/2. In particular ry ≤ (m − 2)(p − 1)/2, andry ≤ (m − 3)(p − 1)/2 if m is odd.

(ii) If char(K) = p, thenry = β(p − 1) for an integerβ ≤ (m − 1). Inparticular ry ≤ (m − 1)(p − 1).

Case 2. I(η1) and I(η2) are not both trivial. We can assume for examplethat I(η1) = G.

(i) If char(K) = 0, thenry = α(p − 1) − (p − min(|I(η1)|, |I(η2)|))for an integer0 ≤ α ≤ m. In particular ry ≤ (m − 1)(p − 1) if I(η2) = 1,andry ≤ m(p − 1) if I(η2) = G.

(ii) If char(K) = p, thenry = β(p − 1) − (p − min(|I(η1)|, |I(η2)|))for an integer0 ≤ β ≤ m + 1. In particular ry ≤ m(p − 1) if I(η2) = 1,andry ≤ (m + 1)(p − 1) if I(η2) = G.

The proofs for different characteristics cases are similar to each others.We will give complete proofs in the characteristic0 case and indicate brieflyhow this works in the characteristicp case.

Proof of Theorem 1

Case 1.The morphismfk : Yk → Xk is generically separable.

Assumechar(K) = 0. We apply the Kato formula (cf. 1) to the mor-phismY → X . In this caseµx = 0, andD = m(p − 1), which implythatµy = (m − 1)(p − 1) − dw, wheredw :=

∑i,j dw

i,j . Note that eithermy = 1 in which casedw ≥ p − 1, or my = p in which casedw = 0. Inboth cases we deduce easily thatgy = δy − my + 1 ≤ (m − 2)(p − 1)/2,hencery ≤ gy ≤ (m − 2)(p − 1)/2.

Assumechar(K) = p. Matignon and Youssefi constructed in [Ma-Yo]a compactification of the morphismSpec OY,y → Spec OX,x between thecompletions ofY andX at the pointsy andx. More precisely they constructa finite Galois morphismY ′ → X ′ = P

1R of degreep between proper and

normalR-curves, with a closed pointy ∈ Y ′ and its imagex ∈ X ′, thismorphism gives rise to the above morphismSpec OY,y → Spec OX,x whencompleted aty andx, and this morphism is unramified outside the points{xi}i=1,m. Moreover one hasry ≤ rY ′

k≤ rY ′

KwhererY ′

k(resp.rY ′

K) is

the p-rank of Y ′k (resp.Y ′

K) (cf. [Ra-1]), the later can be computed using

630 M. Saıdi

the Deuring-Shafarevich formula (cf. [Sa], 1) and is in this case equal to(m − 1)(p − 1).

Case 2.The morphismfk : Yk → Xk is purely inseparable.

Note that in this caseYk is homeomorphic toXk, in particular there isa unique pointy of Yk abovex. Let f : Y → Y be a minimal semi-stablemodel ofY. The action ofG = Z/pZ onY induces an action ofG onY, andthe quotientX := Y/G is a semi-stable model ofX (cf. [Ra] Appendice).Moreover one can chooseY andX such that the points{xi}i=1,m of XK

specialize in distinct smooth points{xi}i=1,m of Xk. We have the followingdiagram :

Y f−−−→ Xf

x g

x

Y g−−−→ XWe will proceed by induction on the cardinalitym of the branch locus

B, and also on the length of the graphΓ associated tof−1(y). The specialfibre Xk of X is a tree-like. LetX1 be the strict transform ofXk in X , itmeets a unique irreducible componentX2 of g−1(x) at a unique doublepoint x′. The strict transformY1 of Yk in Y is its normalisation, it meetsa unique irreducible componentY2 of Yk at a unique double point. Let uscontract inX all the irreducible components ofXk other thanX1 andX2(cf. [Bo-Lu-Ra] 6.6). We obtain a semi-stable modelX ′ of X whereX1 andX ′

2 : the image ofX2, meet atx′. Let Y ′ be the normalisation ofX ′ in YK .We have the following diagram:

Y f−−−→ Xx

x

Y ′ h−−−→ X ′x

x

Y g−−−→ XThe points{xi}i=1,m specialize now in non necessarily distinct smooth

points{x′j}j=1,n of X ′

k, for n ≤ m, lying on X ′2. Assumechar(K) = 0.

We first consider the casem = 1. In this case letx be the unique point ofXK which ramify inYK → XK , and which specializes in the smooth pointx of X2. Let Y ′

2 be the image ofY2 in Y ′. We deduce from case 1 that themorphismY ′

2 → X ′2 is necessarily purely inseparable (otherwise there must


be at least 2 ramified distinct points which specialize inx). In particular thereis a unique pointy ∈ Y ′

2 abovex, and thep-rank of the normalisation ofY ′2

equals0. Note thatry =∑

z∈Y ′2rz. If z ∈ Y ′

2 is a closed point other thanyand the double point ofY ′

k, then by a result of Raynaud (cf. [Sa], 2)rz = 0.This imply thatry = ry. By localizing aty and its imagex we are under thehypothesis of theorem 1, case 2, but the length of the graph of the fibre ofyin Yk, which is a semi-stable model ofSpecOY ′,y, is strictly less than theone ofΓ , hence we deduce by the induction hypothesis thatry = 0 whichimply thatry = 0.

Remark that as above ifchar(K) = p then the morphismY ′2 → X ′

2 canbe separable in which casery = rY ′

2+ ry, rY ′

2= (p − 1) andry = 0 by

case 1, hencery = p − 1 in this case.

Assume now thatm > 1. We make the induction hypothesis that theassertions of theorem 1, case 2, are true in the case of a covering with abranch locus on the generic fibre of cardinality strictly less thanm. With thesame notations as above two cases can occur :

(i) The morphismY ′2 → X ′

2 is separable.

(ii) The morphismY ′2 → X ′

2 is inseparable.

(i) For each1 ≤ j ≤ n, let mj be the number of ramified points whichspecialize inx′

j (∑n

j=1 mj = m), and lety′j be the point ofY ′

2 abovex′j .

Let n1 be the number of points among the{y′j}n

j=1 that are unibranche(these are the points which ramify in the morphismY2 → X2 between thenormalisations). Thenry = rY2 +

∑nj=1 ry′

j+ (n − n1)(p − 1), thep-rank

of Y2 is rY2 = n1(p − 1), andry′j

≤ (mj − 2)(p − 1)/2 by induction

hypothesis. This imply thatry ≤ ∑j mj(p − 1)/2 = m(p − 1)/2.

(ii) In this caseY ′2 is homeomorphic toX ′

2, hence thep-rank of its nor-malisation equals0. thep-rankrz at a closed pointz of Y ′

2 other than they′

j is equal to0 by Raynaud’s result (cf. [Sa], 2), hencery =∑n

j=1 ry′j

≤∑j mj(p − 1)/2 = m(p − 1)/2, wheremj is as above.

Finally it is easily seen by induction on the cardinalitym of the branchlocusB that in both cases 1 and 2 thep-rankry is a multiple ofp − 1.

Proof of theorem 2

1. The case where the morphismfk : Yk → Xk is generically separable.

Assumechar(K) = 0. We apply the Kato formula (cf. 1) to the mor-phismf : Y → X . In this caseµx = 0, andD = m(p−1), which imply that

632 M. Saıdi

µy = (m−1)(p−1)−dw, wheredw :=∑

i,j dwi,j . Note that eithermy = 2

in which casedw ≥ 2(p−1), ormy = p+1 in which casedw ≥ (p−1), orfinally my = 2p in which casedw = 0. In both cases we deduce easily thatgy = δy −my +1 ≤ (m−2)(p−1)/2, hencery ≤ gy ≤ (m−2)(p−1)/2.

Assumechar(K) = p. We use the same argument as in the proof oftheorem 1, case 1. One constructs a compactification of the morphismSpec OY,y → Spec OX,x between the completions ofY andX at the pointsy andx. This means a finite Galois morphismY ′ → X ′ of degreep be-tween proper curves, a closed pointy ∈ Y ′ and its imagex ∈ X ′, and thismorphism gives rise to the above morphismSpec OY,y → Spec OX,x whencompleted aty andx, and is unramified outside the points{xi}i=1,m (thegeneric fibre ofX ′ has genus 0). Moreover one hasry ≤ rY ′

KwhererY ′

K

is thep-rank ofY ′K , which can be computed using the Deuring-Shafarevich

formula (cf. [Sa], 1) and is in this case equal to(m − 1)(p − 1).

2. The case whereI(ηi) are not both trivial fori = 1, 2. We assume forexample thatI(η1) = G.

As in the case 2 of theorem 1 one considers a minimal semi-stable modelf : Y → Y of Y, and the quotientX := Y/G such that the points{xi}i=1,m

of XK specialize in distinct smooth points{xi}i=1,m of Xk. We have thefollowing diagram:

Y f−−−→ Xf

x g

x

Y g−−−→ XDenote byX1 andX2 the irreducible components ofXk. Let X1 (resp.

X2) be the strict transform ofX1 (resp.X2) in X . The graph associatedto thek-curveg−1(x) is a tree, and each irreducible component ofg−1(x)is a projective line. In this tree we will consider the “geodesic”Γx linkingthe strict transformsX1 andX2. This geodesic corresponds ing−1(x) toa chain of projective lines linkingX1 andX2. Moreover after eventuallycontracting inX all the irreducible components ofXk other thanX1, X2,and the components ofΓx, one can assume thatXk consists only ofX1,X2, which are linked byΓx. But now the points{xi}i=1,m specialize in nonnecessarily distinct smooth points ofΓx. Let C1 (resp.C2) be the first irre-ducible component ofΓx that we enconter moving fromX1 in the directionof X2 (resp. moving fromX2 in the direction ofX1), and such that one ofthe points{xi}i=1,m specializes inC1 (resp. specializes inC2). Let us nowcontract inX all the irreducible components ofXk other thanX1, X2, C1andC2. We obtain a semi-stable modelX ′ of XK , whereC1 meetsX1 at


a double pointx1, C2 meetsX2 at a double pointx2, andC1 meetsC2 ata double pointx′. Let Y ′ be the normalisation ofX ′ in YK . We have thefollowing diagram:

Y f−−−→ Xx

x

Y ′ h−−−→ X ′x

x

Y g−−−→ X

Let {zi}i∈I (resp.{zj}j∈J ) be the points ofC1 (resp.C2) on whichspecialize the points of the branch locus offK : YK → XK . For eachi ∈ I(resp. eachj ∈ J) Let mi (resp.mj) be the number of those ramified pointswhich specialize inzi (resp. inzj), also letm′ be the number of those ramifiedpoints which specialize inx′. One has

∑i mi +

∑j mj + m′ = m. We

will proceed by induction on the cardinality of the branch locus{xi}i=1,m.Assumechar(K) = 0. We first consider the casem = 1, and the branchlocus consists of one pointx. In this caseC1 = C2 =: C, andx specialize ina smooth pointx of C. It follows that the inertia subgroup at a generic pointof Y ′

k above the generic point ofC equalsG, otherwise there must be at leasttwo ramified point which specialize inx (cf. theorem 1). In particular thereis a unique irreducible componentC ′ of Y ′

k aboveC, and the morphismC ′ → C is a homeomorphism. In particular thep-rank of the normalisationof C ′ equals0. Let y1 (resp.y2 and y) be the point ofY ′

k abovex1 (resp.abovex2 andx). If z ∈ C ′ is a closed point other thany1, y2, andy, thenby Raynaud’s result (cf. [Sa], 2)rz = 0. Hencery = ry1 + ry2 + ry. Bytheorem 1 one hasry = 0. Also by [Sa], proposition 1, we havery1 ≤ (p−1),andry2 ≤ p − p/|I(η2)|. In particular ifI(η2) = 1, thenry2 = 0. Alsoif I(η2) = 1, then necessarilyry1 = 0. Otherwise there would be in thegeodesic linking the componentsC andX1 in Xk an irreducible componentC1 such that the morphismg : Y → X is separable aboveC1. If then wecontract all the irreducible components inXk other thanX1, C1 andX2we will contradict the conclusion of theorem 2 , case 1, thatm ≥ 2 in thecharacteristic0 case (the pointx specializes then at a double point linkingC1 andX2). Also for the same reason as above, ifI(η) = G, thenry1 andry2

can not be both equal top−1. Hencery ≤ p−1−(p−min(|I(η1)|, |I(η2)|)as claimed.

Next we assumem > 1. Different cases can occur, we will treat onlythe following case :C1 6= C2, andI(η′

1) = G (resp.I(η′2) = G) where

634 M. Saıdi

I(η′1) (resp.I(η′

2)) is the inertia subgroup at a generic pointη′1 (resp.η′

2) ofY ′

k above the generic point ofC1 (resp. above the generic point ofC2). Theother cases are similarily treated and are left to the reader. In particular inthe above case there is a unique irreducible componentsC ′

1 (resp.C ′2) of Y ′

kaboveC1 (resp. aboveC2). The morphismC ′

i → Ci is a homeomorphism,in particular thep-rank of the normalisation ofC ′

i equals0 for i = 1, 2.Let {yi}i∈I (resp.{yj}j∈J ) be the points ofC ′

1 (resp. ofC ′2) above{zi}i∈I

(resp. above{zj}j∈J ), and lety be the point ofY ′k abovex. Thep-rankry at

y equals∑

i∈I ryi +∑

j∈J ryj +ry +ry1 +ry2 . By theorem 1 one hasryi ≤mi(p−1)/2, ryj ≤ mj(p−1)/2, for eachi ∈ I, and eachj ∈ J , and by theinduction hypothesis (note thatm′ < m in the above case)ry ≤ m′(p− 1).Also by [Sa], proposition 1, one hasry1 ≤ p−1, andry2 ≤ p−p/|I(η2)|. Inparticular ifI(η2) = 1, thenry2 = 0. Also if I(η2) = 1 andry1 6= 0, then bythe same argument used in the conclusion for the casem = 1, one deduceseasily from case 1, thatry ≤ (m−2)(p−1)/2+(p−1). Hence ifI(η2) = 1we can assume thatry1 = 0. In this casery =

∑i∈I ryi +

∑j∈J ryj + ry,

which imply thatry ≤ (m−m′)(p−1)/2+m′(p−1), and asm′ ≤ m−2we deduce thatry ≤ (m − 1)(p − 1). Assume now thatI(η2) = G. Then ifry1 = ry2 = p−1, with the same argument as above, and using the result oftheorem 2 case 1, one concludes thatry ≤ 2(p − 1) + (m − 2)(p − 1)/2 ≤m(p − 1). So if I(η2) = G we can assume for example thatry1 = 0, hencery =

∑i∈I ryi +

∑j∈J ryj + ry + ry2 ≤ ∑

i mi(p − 1)/2 +∑

j mj(p −1)/2 + m′(p − 1) + (p − 1) ≤ (m + m′ + 2)/2(p − 1) ≤ m(p − 1), sincem′ ≤ m − 2. And this concludes the proof of the case 2 of theorem 2 in theabove considered case.

Example 1.We will give examples for the different values of thep-rankry

that can occur in theorem 1, case 2, and in the case wherechar(K) = 0. Fixan integerm ≥ 2, and an integerα ≤ m/2. Letk be an algebraically closedfield of characteristicp > 0. Considerα+1 points{xi}α+1

i=1 on the projectiveline P

1k, and integers{mi}α

i=1, with mi ≥ 2, and such that∑

i mi = m.

There exists a finite Galois coveringY1 → X1 = P1k of groupG =

Z/pZ, which is etale outside the points{xi}α+1i=1 , and which is ramified

above each pointxi with conductormi, for i = 1, α. The p-rank rY1 ofY1 equalsα(p − 1). By a result of Sekiguchi, Oort and Suwa ([Se-Oo-Su],cf. also [Ma-Gr]), the above covering can be lifted to a Galois coveringY ′′ → P

1R of proper and smooth curves over a discrete complete valuation

ring R which is finite over the ring of Witt vectorW (k). At each pointxi, for i = 1, α, specialize exactlymi points among the ramified points inY ′′

K → P1K , whereK := FrR. Consider a blowing-upX ′ of P

1R at the closed

point xα+1, such that the ramified points inY ′′K → P

1R which specialize in


xα+1 specialize in smooth points ofX ′k. The special fibre ofX ′ consist of

two componentsX ′1 andX ′

2 which intersect at a double pointx′. Let Y ′be the normalisation ofX ′ in Y ′′

K . It follows from the theorem ofp-purityfor double points (cf. [Sa], 1) that the inertia subgroup at a generic pointof an irreducible component ofY ′

k aboveX ′2 equalsG. In particular there

is a unique irreducible componentY ′2 of Y ′

k aboveX ′2, and the morphism

Y ′2 → X ′

2 is purely inseparable. Let us now contract inX ′ the componentX ′1

to a closed smooth pointx lying onX ′2. We obtain theR curveX = P

1R. Let

Y be the normalisation ofX in Y ′′R . We have a canonical morphismY → X

which is Galois of groupG = Z/pZ. A localisation of this morphism abovethe pointx provides an example of the case 2 of theorem 1, in the case wherechar(K) = 0, with ry = α(p − 1).

1. Global results

In this section we will deduce from the last section some global resultsconcerning thep-rank of the semi-stable reduction of coverings of degreepbetween proper curves.

Theorem 3. Let X be a smooth and properR-curve, whose generic fibreXK is geometrically connected. LetfK : YK → XK be a finite Galois cov-ering of groupG = Z/pZ, withYK connected, and letB = {x1, ..., xm} ⊂XK be its branch locus, which we assume non empty of cardinalitym.Let {xj}n

j=1, for n ≤ m, be the set of points ofXk where specialize thepoints ofB, and for eachj let mj be the number of those points whichspecialize inxj (m =

∑j mj). LetJodd := {1 ≤ j ≤ n, mj odd} (resp.

Jeven := {1 ≤ j ≤ n, mj even}) be the set of thosej for whichmj is odd(resp. the set of thosej for whichmj is even).

LetY be the normalisation ofX in YK , and letf : Y → X be the canon-ical morphism. Assume that there exists a proper and birational morphismf : Y → Y such thatY is semi-stable. The following hold:


(i) If char(K) = 0, then thep-rank rYkof Yk satisfy the inequalityrYk

≤prXk

+ (m − |Jodd| − 2)(p − 1)/2. In particular if |Jodd| ≥ 1, then thespecial fibreYk of Y can not be ordinary.

(ii) If char(K) = p, thenrYk≤ rXk

+ (m − 1)(p − 1). In particular if thep-rank rYk

is maximal equal torYK, then necessarilyrXK

= rXk.


636 M. Saıdi

(i) If char(K) = 0, thenrYk≤ prXk

+(m−|Jodd|)(p−1)/2. In particularif 2gXK

− 2 + |Jodd| > 0, wheregXKis the genus ofXK , then the special

fibre Yk of Y can not be ordinary.

(ii) If char(K) = p, thenrYk≤ rXk

+m(p− 1). In particular if thep-rank

rXKof XK satisfyrXK

≥ 1, thenYk can not be ordinary.

The case where the branch locusB is empty, and under the same hy-pothesis as in theorem 3, has been treated by Raynaud (cf. [Ra] and [Ra-1]).

Proof


This morphism is necessarily ramified above eachxj , 1 ≤ j ≤ n. Letyj be the unique closed point ofYk abovexj . The morphismfk : Yk → Xk

is etale outside the points{yj}j (cf. theorem of Purity [SGA-1], expose X,3.1, p. 275), in particularYk is smooth outside this points. One hasrYk

=rY ′

k+

∑nj=1 ryj +k′(p−1), whererY ′

kis thep-rank of the normalisationY ′

k

of Yk, which is the strict transform ofYk in Yk, andk′ is the number of pointsamong the{xj}n

j=1 which does not ramify in the morphismY ′k → Xk (each

point yj above anxj which does not ramify inY ′k hasp branches which

pass through it and this introducesp − 1 cycles in the configuration ofYk

which contributes byp−1 to thep-rank ofYk). Thep-rankrY ′k

of Y ′k equals

prXk+ (n − k′ + 1)(p − 1).

Assumechar(K) = 0. By theorem 1, case 1,ryj ≤ (mj − 2)(p − 1)/2if mj is even, andryj ≤ (mj − 3)(p − 1)/2 if mj is odd. This imply thatrYk

≤ prXk+(n−k′+1)(p−1)+k′(p−1)+

∑j∈Jodd

(mj −3)(p−1)/2+∑j∈Jeven

(mj −2)(p−1)/2, hencerYk≤ prXk

+(m−|Jodd|−2)(p−1)/2.

Assume now thatYk is ordinary, i.e.rYk= gYK

the genus ofYK , whichby the Riemann-Hurwitz formula equalspgXK

+ (m − 2)(p − 1)/2. Theabove inequality imply in this case thatrYk

= pgXK+(m−2)(p−1)/2 ≤

prXk+ (m − |Jodd| − 2)(p − 1)/2. On the other handrXk

≤ gXk= gXK

,hencepgXK

+ (m − 2)(p − 1)/2 ≤ pgXK+ (m − |Jodd| − 2)(p − 1)/2,

and this is impossible if|Jodd| ≥ 1.

Assume now thatchar(K) = p. By theorem 1, case 1,ryj ≤ (mj −1)(p−1), for 1 ≤ j ≤ n. HencerYk

≤ prXk+(n−k′ +1)(p−1)+k′(p−

1)+∑

j(mj −1)(p−1), which imply thatrYk≤ prXk

+(m−1)(p−1)/2.One hasrYk

≤ rYK(cf. for example [Ra-1]). Assume now that thep-rank


rYkis maximal equal torYK

which is equal toprXK+(m−1)(p−1), then

the above inequality imply thatrXK≤ rXk

. On the other handrXk≤ rXK

,hence necessarilyrXk

= rXK.


In this caseYk is homeomorphic toXk, and the normalisationY ′k of Yk,

which is the strict transform ofYk in Yk, is isomorphic toXk. For each1 ≤j ≤ n let yj be the unique point ofYk abovexj . one hasrYk

= rY ′k+

∑j ryj

(if y is a closed point ofYk other than the{yj}j thenry = 0 by a result ofRaynaud (cf. [Sa])), andrY ′

k= rXk

.

Assumechar(K) = 0. By theorem 1, case 1,ryj ≤ mj(p − 1)/2 ifmj is even andryj ≤ (mj − 1)(p − 1)/2 if mj is odd. This imply thatrYk

≤ rXk+

∑j∈Jodd

(mj − 1)(p − 1)/2 +∑

j∈Jevenmj(p − 1)/2, hence

rYk≤ rXk

+ (m − |Jodd|)(p − 1)/2. Assume now thatYk is ordinary,i.e. rYk

= gYK= gXK

+ (m − 2)(p − 1)/2. The above inequality implythen thatpgXK

+ (m − 2)(p − 1)/2 ≤ rXk+ (m − |Jodd|)(p − 1)/2 ≤

gXK+(m−|Jodd|)(p−1)/2, hencegXK

+(m−2)/2 ≤ (m−|Jodd|)/2),which imply that necessarily2gXK

− 2 + |Jodd| ≤ 0.

Assume now thatchar(K) = p. By theorem 1, case 1,ryj ≤ mj(p−1),for 1 ≤ j ≤ n, which imply thatrYk

≤ rXk+

∑j mj(p − 1) = rXk

+m(p − 1). In particular assume now thatYk is ordinary, i.e.rYk

= rYK=

prXK+ (m − 1)(p − 1). Then the above inequality imply in this case that

prXK+ (m − 1)(p − 1) ≤ rXk

+ m(p − 1) ≤ rXK+ m(p − 1), hence

rXK(p − 1) + (m − 1)(p − 1) ≤ m(p − 1). Which imply thatrXK

≤ 1.

The above result can be generalized to the case whereX is semi-stable,in the sens that one can give bounds for thep-rank of the special fibreYk of asemi-stable model ofYK . We will state our result only in the case whereXis semi-stable, and its special fibre consists of two irreducible componentswhich intersect at a double point. The proof (as well as the statement of theresult forX semi-stable in general) is an easy consequence of theorem 1,theorem 2, and theorem 1 in [Sa], and is left to the reader.

Theorem 4. Let X be a proper and semi-stableR-curve, whose genericfibre XK is geometrically connected, and whose special fibre consists oftwo irreducible componentsX1 andX2 of strictly positif genus, and whichmeet at an ordinary double pointx. LetfK : YK → XK be a finite Galoiscovering of groupG = Z/pZ, with YK connected. LetB ⊂ XK be thebranch locus offK . In caseB = {x1, ..., xm} is non empty of cardinalitym, let {xi}i∈I (resp.{xj}j∈J ) be the set of points ofX1 (resp. set of pointsof X2) other than the pointx and on which specialize the points ofB. For

638 M. Saıdi

i ∈ I (resp.j ∈ J) let mi (resp.mj) be the number of those points ofBwhich specialize inxj (resp. specialize inxj). Letm′ be the number of thosepoints ofB which specialize inx. One has|I|+ |J |+m′ = m. LetY be thenormalisation ofX in YK , and letf : Y → X be the canonical morphism.Assume that there exists a proper and birational morphismf : Y → Y suchthat Y is semi-stable. The following holds:

The case whereB is empty.

(i) If fk : Yk → Xk is generically separable, then it isetale andY issemi-stable, moreoverrYk

= prX1 − (p − 1) + prX2 .

(ii) If fk : Yk → Xk is generically separable aboveX1 and purely insepa-rable aboveX2, thenrYk

= prX1 + rX2 .

(iii) If fk : Yk → Xk is purely inseparable aboveX1 andX2, thenrYk≤

rX1 + rX2 + (p − 1).

The case whereB = {x1, ..., xm} is non empty.

(i) If fk : Yk → Xk is generically separable, thenrYk≤ prXk

+ (m −|Iodd| − |Jodd| − 2)(p − 1)/2 if char(K) = 0, andrYk

≤ prXk+ (m −

1)(p − 1) if char(K) = p.

(ii) If fk : Yk → Xk is generically separable aboveX1 and purely insepa-rable aboveX2, thenrYk

≤ prX1 + rX2 +(m+2m′ −|Iodd|−2)(p−1)/2if char(K) = 0, andrYk

≤ prX1 + rX2 + m(p − 1).

(iii) If fk : Yk → Xk is purely inseparable aboveX1 andX2, thenrYk≤

rX1 + rX2 + (m − |Iodd| − |Jodd|)(p − 1) if char(K) = 0, andrYK≤

rX1 + rX2 + (m + 1)(p − 1) if char(K) = p.

Remark 1.One can easily deduce from the above result, as in theorem 3for the case whereX is smooth, a necessary condition such that the specialfibre Yk of Y is ordinary, i.e. has maximalp-rank equal to the genus ofYK

if char(K) = 0, and equal to thep-rank ofYK if char(K) = p.

References

[Bo-Lu-Ra] S. Bosch, W. Lutkebohmert, M. Raynaud (1990): Neron Models, Springer-Verlag,21.

[De-Mu] P. Deligne, D. Mumford (1969): The irreducibility of the space of curves ofgiven genus, Pub. I.H.E.S.36, 75–109.

[SGA-1] A. Grothendieck (1971): Revetementsetales et groupe fondamental, Lect.Notes in Math,224, Springer-Verlag.

[Ka] K. Kato (1987): Vanishing cycles, ramification of valuations, and class fieldtheory, Duke Math. J55, 629–659.


[Ma-Gr] M. Matignon, B. Green: On lifting of Galois covers of smooth curves, to appearin Compositio Math.

[Ma-Yo] M. Matignon, T. Youssefi (1992): Prolongement de morphismes de fibresformelles et cycles evanescents, preprint, Universite Bordeaux I.

[Ra] M. Raynaud (1990): p-Groupes et reduction semi-stable des courbes, TheGrothendieck Festschrift Vol.3, Birkhauser, Boston, 179–197.

[Ra-1] M. Raynaud (1994): Mauvaise reduction des courbes etp-rang, C. R. Acad.Sci. Paris, t.316, Serie I, 1279–1282.

[Sa] M. Saıdi (1998):p-Rank and semi-stable reduction of curves, C. R. Acad. Sci.Paris, t.326, Serie I, 63–68.

[Se-Oo-Su] T. Sekiguchi, F. Oort, N. Suwa (1989): On the deformation of Artin-Schreierto Kummer, Ann. Scient.Ec. Norm. Sup.,4e serie, t.22, 345–375.

p -Rank and semi-stable reduction of curves II

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