Semistable Extension of Families of Curves

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Semistable Extension of Families of Curves

DISSERTATION ZUR ERLANGUNG DES DOKTORGRADES DER NATURWISSENSCHAFTEN (Dr. rer. nat.)

AN DER FAKULTÄT FÜR MATHEMATIK DER UNIVERSITÄT REGENSBURG

vorgelegt von Mohammad Dashtpeyma

aus Ahvaz, Iran

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Promotionsgesuch eingereicht am: 03.02.2014

Die Arbeit wurde angeleitet von: Prof. Dr. Uwe Jannsen Prüfungsausschuss:

Vorsitzender: Prof. Dr. Harald Garcke Erst-Gutachter: Prof. Dr. Uwe Jannsen Zweit-Gutachter: Prof. Dr. Moritz Kerz weiterer Prüfer: Prof. Dr. Klaus Künnemann

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Introduction

The stable reduction theorem of Deligne-Mumford says that for any smooth projective curve C over the function fieldK of a discrete valuation ring R, there exists a finite separable extensionLof K such that the curveC⊗_K Lhas stable reduction over the integral closure ofRinL. For the reason why one can choose the field extension to be separable, see, for example [23], Exercise 10.4.2.

One can prove the properness of the moduli space of stablen−pointed genusgcurves using the above theorem.

Deligne generalised the result above (See [9] or Theorem 3.12 [2]) as follows: for any proper stable curveCof genusg≥2 over an open dense subscheme of a quasi-compact quasi-separated integral schemeS there exists a proper surjective morphismS⁰ → S such thatC×_S S⁰can be extended to a proper stable curve overS⁰. The theorem is so-called Stable Extension Theorem.

Remark:The curveCneed not be smooth.

A. de Jong extended Deligne’s result in [5] (See also [7]) showing that for any proper curveCover an integral quasi-compact excellent schemeS, there exists an alteration S⁰ → S and a modificationC⁰ →C×_S S⁰such thatC⁰is a proper semi-stable curve overS⁰. For the precise statement see Chapter 5, Theorem 5.3.1.

Our Main Concern:Roughly speaking, we would like to give an explicit construction describing “the best possible model” for a given semi-stable curve (Definition 4.2.1) which is defined over a discrete valuation field.

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We proceed in two different ways;

I)Using properties of moduli space of stable curves: since the moduli space of stable curves of genusgis proper, we can use the ‘extended’ version of valuation criterion for properness of morphisms of algebraic stacks given by Deligne and Mumford to show that the reduction remains stable after possibly a finite extension of the base field (See 5.2).

II)Using the semi-stable reduction theorem on every component of the given semistable curveCand then gluing the suitable models in a certain way (See 5.3).

In the last chapter we introduce some monodromy conditions which ensure that families of curves, defined over an open dense subset of the base scheme, can be extended to the whole base in a stable manner (i.e., the family extends to a “stable family” over the whole scheme).

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Chapter 2

Normality and gluing along sections

2.1 Product

In this chapter we collect the prerequisite results which are needed for the next chapters.

They can be found in several books written on algebraic geometry. Our main source for this chapter is Algebraic geometry and arithmetic curves, Q. Liu [23].

2.1.1 Product along arbitrary base scheme

Definition and convention 2.1.1. IfX is a scheme and x ∈ X is a point, thenk(x) denotes the residue field of the local ringO_X,x. Sometimes we consider a point as a scheme x = Speck(x). A scheme S is integral if it is irreducible and reduced. Its function field is denotedR(S).

Definition 2.1.2.LetS be a scheme, and letXandYbe two schemes overS (S−schemes).

The fibre product ofX andY overS is defined to be anS−schemeX×_S Y, together with two morphisms ofS−schemesp:X×_SY →X,q:X×_SY →Y(the projections), verifying the following universal property:

Let f : Z → X,g : Z → Y be two morphisms ofS−schemes. Then there exists a unique morphism ofS−schemes a

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(f,g) :Z→X×_S Ywhich makes the following diagram commutative:

X

Z

f

66

(f,g) //

g

((

X×_S Y

p

OO

q

Y

It is well known that the fibered product (X×_SY,p,q) exists and is unique up to unique isomorphism.

The following facts easily follow from the universal property of the fibered product.

Theorem 2.1.3. LetS be a scheme andX,Y andZ areS−schemes. The following properties hold.

• X×_S S 'X,Y×_S X'X×_S Y,(X×_S Y)×_S Z'X×_S (Y×_S Z)

• LetZbe aY−scheme, considered as anS−scheme viaZ → Y → S.Then we have a canonical isomorphism ofS−schemes (X×_S Y)×_YZ 'X×_S Z. Where X×_S Yis endowed with the structure of aY−scheme via the second projection.

• Let f :X→X⁰,g:Y →Y⁰be morphisms ofS−schemes. There exists a unique morphism ofS−schemesf×g:X×_SY →X⁰×_SY⁰which makes the following diagram commutative:

X ^f //X⁰

X×_S Y ^f×g //

p

OO

q

X⁰×_S Y⁰

p⁰

OO

q⁰

Y

q

OO

g //Y⁰

• leti:U→X,j:V→Ybe open subschemes. Then the morphismi×jincludes an isomorphismU×_S V 'p⁻¹(U)∩q⁻¹(V)⊆X×_S Y.

Definition 2.1.4. We say that the morphism of schemesX→S is projective if it factors into a closed immersionX→Pⁿ_S followed by the canonical morphismPⁿ_S →S.

2.1.2 Fibres of morphisms

Definition 2.1.5. Letf :X→Ybe a morphism of schemes. For anyy∈Y X_y=X×_YSpeck(y).

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This is the fibre of f overy. The second projectionXy → Speck(y) makesXyinto a scheme overk(y).In caseY is irreducible of generic pointξ; we callXξ the generic fibre off.

It can be seen that the first projection

p:Xy=X×_YSpeck(y)→X induces a homeomorphism fromXyontof⁻¹(y).

Example 2.1.6. (See [23]) Letmbe a non-zero integer. Let f :X=SpecZ[Y,Z]/(YZ²−m)→SpecZ

be the canonical morphism. For any prime numberp, we letXpis the fibre of f over the pointpZ∈SpecZ.Then the generic fibre ofXis

SpecQ[Y,Z]/(YZ²−m)=SpecQ[Z,1/Z].

And the fibreX_pis equal to SpecFp[Y,Z]/(YZ²−m).Therefore if the primepdoes not dividemthe fibre X_pis integral, being isomorphic to SpecFp[Y,1/Y]. Otherwise (if p|m) it has two irreducible components. Note that the schemeXitself is integral. So, we can ‘cut’Xinto slices, most of these slicesXp(forp-m) staying integral, but some become reducible. This phenomenon is called ‘degeneration’.

2.2 Base Change

Definition 2.2.1. A morphism of schemes f : X → Y is of finite type if f is quasi- compact (inverse image of any affine open subset of codomain can be covered by a finite number of open subsets of domain) and if for every affine subsetVofY, and for every affine open subsetUof f⁻¹(V), the canonical homomorphismO_Y(V)→ O_X(U) makesO_X(U) into a finitely generatedO_Y(V)−algebra.

Definition 2.2.2.Letkbe a field. A scheme of finite type over Speckis called algebraic variety overk. (We sometimes restrict the definition by imposing additional conditions such as irreducibility.)

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Theorem 2.2.3. Let X be an algebraic variety over k, and let K/k be an algebraic extension. Then the following properties are true.

1. We have dimXK =dimX.

2. IfXis reduced andK/kis separable, thenXKis reduced.

Proof. Since we may assume thatXis affine, sayX=SpecA.

(1) is obvious becauseA→ A_K =A⊗_kKis injective and integral therefore dimA = dimA_K. In general for ifφ:A→ Bis an injective morphism of rings andq∈SpecB andp=φ⁻¹(q)∈SpecAthen we have dimV(p)=dimV(q).

(2) Since the ringAcan be embedded in⊕_iA/piforp1, . . .pnthe minimal prime ideals of A, one can assumeA is an integral ring. Now AK is a subring of Frac(A)⊗_k K, therefore it is enough to show that the ringF⊗_kKis reduced for any fieldFcontaining k. Every element ofF⊗_kKis containedF⊗_kK⁰, withK⁰finite separable overk, and we therefore we can assume thatKis finite overk. It follows thatK'k[T]/(P(T)),where P(T) ∈ k[T] is a separable polynomial butP(T) is still separable in F[T],therefore

F⊗_kK'F[T]/(P) is reduced.

2.2.1 Extending varieties to algebraically closed field

Definition 2.2.4. LetXbe an algebraic variety overk. Let ¯kbe the algebraic closure ofk. We say thatXis geometrically reduced (resp. geometrically irreducible, geometrically integral, geometrically connected) ifXk¯ is reduced (resp. irreducible, integral, connected).

Example 2.2.5. Letkbe a field of characteristic different from 2, anda∈kwhich is not a square. Consider the projective variety

X=Projk[x,y,z]/(x²−ay²)

Letα ∈ k¯ be a square root ofaandK =k[α]. Then we see that Xis integral while XK =ProjK[x,y,z]/(x−αy)(x+αy) is not.

Definition 2.2.6. LetXbe an algebraic variety over a fieldk, and let K/kbe a field extension andX(K) denote the set of morphisms ofK−schemes from Spec(K)→ X.

The elements ofX(K) are calledK−valued points ofX. The instant remark is thatX(K) is not in general the set of pointsx∈Xsuch thatk(x)⊆K.

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Theorem 2.2.7. IfX is an algebraic variety overk, and K/kis a field extension, the following properties hold.

1. There is a canonical bijectionX(K)→XK(K)

2. The data consisting of a pointx∈Xand a homomorphism ofk−algebrask(x)→ Kuniquely determine an element ofX(K)

3. For any extensionK⁰/K, we have a natural inclusionX(K)⊆X(K⁰).

Proof. (1) If we denote the set of morphisms ofS−schemes fromXtoYby Mor_S(X,Y) then the projectionspandqinduce the maps

Mor_S(Z,X×_S Y)→Mor_S(Z,X) MorS(Z,X×_S Y)→MorS(Z,Y) whereZis anS−scheme. This gives a map

MorS(Z,X×_S Y)→MorS(Z,X)×MorS(Z,Y),

By the universal property of product it is bijective. Now by takingZ =S we have a canonical bijection of sections

(X×_S Y)(S)'X(S)×Y(S).

IfY =Zand we have

Mor_S(Y,X×_S Y)'Mor_S(Y,X),

in whichX×_SYis endowed with the structure of aY−scheme via the second projection.

(2) Take s ∈ X(K) and let x ∈ X be the image of s : SpecK → X. We have s^#_x:O_X,x→Kinduces a homomorphismk(x)→KConversely, ifx∈Xandk(x)→K a given homomorphism, define a morphism ofk−schemes SpecK→Speck(x). Com- pose this with the canonical morphism Speck(x)→ Xwe obtain an element ofX(K) and these two processes are inverses.

(3) It is obvious because the composition with SpecK⁰ → SpecK induces a map X(K)→X(K⁰) in (2) we showed that the map is injective.

Theorem 2.2.8. LetXbe an integral algebraic variety overk. ThenXis geometrically reduced if and only ifK(X) is a finite separable extension of a purely transcendental extensionk(T₁, . . . ,T_d).

Proof. It is obvious ifK(X) is a finite separable extension of the fieldL:=k(T1, . . . ,T_d), thenK(X) can be written asL[S]/P(S) for an irreducible separable polynomialP(S)∈ L[S]. ThereforeK(X)⊗_kk¯=L⁰[S]/P(S),whereL⁰=L⊗_kk¯=¯k(T1, . . . ,Td).ButP(S)

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remains separable in L⁰[S].Therefore K(X)⊗_kk¯ is reduced. So X is geometrically reduced.

Conversely, If X is geometrically reduced and K(X) is a finite extension of L := k(T1, . . . ,Td), then ifK(X)/Lis separable, there is nothing to prove. So assume that K(X) , LswhereLsis the separable closure of LinK(X) and f ∈ K(X)\Lsis such that f^p ∈Ls. p=Char(k). We show in the following thatLs[f] is finite and separable over a purely transcendental extension. SinceK(X)/Lis finite extension, this process decomposes the extensionK(X)/L_s into a sequence of purely inseparable extensions which proves the theorem.

Letp(S)=S^r+Qr−1S^r−1+· · ·+Q₀ ∈L[S] be the minimal polynomial of f^poverL.

We show that at least oneQ_i<k(T₁^p, . . . ,T_d^p). If not,P(S^p)=H(S)^pfor a polynomial H(S)∈k[S¯ ].ThereforeLs[f]⊗_kk¯=(Ls⊗_kk)[S¯ ]/P(S^p) is not reduced. This is in contradiction with the fact thatK(X)⊗_kk¯is reduced andLs[f]⊗kk¯⊆K(X)⊗kk¯as algebras.

So, a power of say,T1prime topappears in one of theQi. HenceT1is algebraic and separable overk(f,T2, . . . ,Td).SinceLs[f] is finite separable overk(f,T1,T2, . . . ,Td), we haveLs[f] finite and separable overk(f,T2, . . . ,Td). This extension is purely transcendental because its transcendental degree overkis equal to that ofLs[f],which is

d.

2.2.2 Rational points of geometrically reduced varieties

Theorem 2.2.9. LetXbe a geometrically reduced algebraic variety over a fieldk. Let k^sbe the separable closure ofk. ThenX(k^s),∅.

Proof. (See [23], Proposition 3.2.20): We may assume thatk=k^sreplacingXbyX_k^s (X_k^s is geometrically reduced). Now we would like to show thatX(k),∅.Again by replacingXby an irreducible affine open sebset, we may assume that Xis affine and integral. Since nowXis integral algebraic variety which is geometrically reduced, by theorem above its fraction field K(X) is finite separable overk(T1, . . . ,Td) therefore K(X)=k(T1, . . . ,Td)[f]. Assume thatP(S)∈ Frac(A)[S] is the minimal polynomial of f andA=k[T1, . . . ,Td],B=O_X(X). We may assume thatA[f]⊆Bafter localising Bif necessary. Since Frac(B)=Frac(A)[f], there exists ag∈ Asuch thatB⊆ Ag[f] andP(S)∈ Ag[S]. It follows thatBg =Ag[f]= Ag[S]/P(S).AsP(S) is a separable polynomial, its resultanth:=Res(P(S),P(S⁰))∈Agis non-zero. The fieldkis infinite because it is separably closed. Therefore there exists at ∈ k^d such thatg(t), 0 and h(t),0. Lety∈SpecAgbe the point corresponding tot. Thenk(y)=kand

Bg⊗_A_gk(y)=k[S]/P(S˜ ),

in which ˜P(S) is the image of P(S) ∈ k(y)[S]. The resultant of ˜P(S) ish(t) , 0.

Therefore ˜P(S) is separable andB_g⊗_A_gk(y) is a direct sum ofk’s. Hence the points of SpecBgoveryare rational overkwhich meansX(k),∅.

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2.3 Some global properties

2.3.1 Separatedness

Definition 2.3.1. LetXbe a topological space. Let∆: X → X×Xbe the diagonal map sendingxto (x,x) in whichX×X is endowed with the product topology. From topology we know thatXis separated if and only if the image of diagonal map∆(X) is closed inX×X.

Although the underlying topological space of a scheme is almost never separated, we would like to define the separatedness of schemes in accordance to the above equiva- lence.

Definition 2.3.2. Let f : X → Y be a morphism of schemes. The morphism∆ := (IdX,Idx) :X→ X×_YXis called the diagonal morphism off. We sayXis separated overY if ∆is a closed immersion of schemes. This is a local property onY. The schemeXis separated if it is separated overZ.

It is obvious that any morphism of affine schemesX → Y is separated because in this case X×_Y X and∆are respectively B⊗_A B andρ : B⊗_A B → B defined by ρ(b₁⊗b₂)=b₁b₂.Sinceρis surjective,∆is a closed immersion.

From this we conclude that if f :X→Yis a morphism of schemes such that∆(X) is a closed subset ofX×_YX. Then f is separated.

Theorem 2.3.3. IfXis scheme thenXis separated if and only if there exists a covering ofX by affine open subsetsUi such that for alli,j,Ui∩Uj is affine andO_X(Ui)⊗_Z O_X(Uj)→ O_X(Ui∩Uj) is surjective.

Proof. First of all we have the inverse image ofU×_ZVunder the diagonal morphism

∆⁻¹(U×V)=U∩Vand also∆:U∩V →U×_ZVcorresponds toO(U)⊗_ZO(V)→ O(U∩V). Now ifX is separated then∆is a closed immersion andU×V is affine thereforeU∩Vis affine as well.

Conversely, since∆:∆⁻¹(Ui×_ZUj)→Ui×Ujis a closed immersion foriand jand

Ui×_ZUjcoverX×_ZX,∆is a closed immersion.

Application of the theorem above. 2.3.4. The projective space Pⁿ_Z is separated because we can cover it with the affine open subsetsD₊(Ti) for 0 ≤ i ≤ n and the condition of the theorem above is satisfied.

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It is trivial from the definition of separatedness that open and closed immersions are separated morphisms and that the composition of two separated morphisms is a separated morphism. Therefore any projective morphism is separated.

2.3.2 Properness

Definition 2.3.5. We say that a morphism f :X → Y is closed if f maps any closed subset ofX onto a closed subset ofY. We say that f is universally closed if for any base changeY⁰→Y,morphismX×_YY⁰→Y⁰stays a closed morphism.

We say that a morphism of schemesf :X→Yis proper if it is of finite type, separated and universally closed. We say that aY−scheme is proper if the structure morphism is proper. Clearly, properness is a local property onY.

Example 2.3.6. Closed immersions are proper. It is also well-known that if the morphism SpecB → SpecAis proper then Bis finite over A. Moreover if X is proper over SpecAthenO_X(X) is integral overA. From this one can deduce ifXis a reduced algebraic variety which is proper over a fieldkthenO_X(X) is ak−vector space of finite dimension. Even more generally, ifXis a scheme (not necessarily reduced) over an arbitrary Noetherian ringA, one can show thatO_X(X) is finite overA, using the finiteness theorem of coherent sheaves.

Definition 2.3.7. LetK be a field. A valuation ofKis a mapνfromK^∗to a totally ordered Abelian groupΓ, verifying the following properties:

1. ν(αβ)=ν(α)+ν(β), (i.e.,νis a group homomorphism);

2. ν(α+β)≥minν(α), ν(β).

Convention:ν(0) = +∞. The setO_ν ={α∈ K|ν(α)≥0}is called the valuation ring ofν(or the valuation ring ofK). In general, a ring is called a valuation ring if it is the valuation ring of a field for a valuation. The valuation ring is a local ring with the maximal idealmν={α∈K|ν(α)>0}.

Lemma 2.3.8. LetO_K be a valuation ring, K = Frac(OK), andAa local subring of K which dominates O_K, meaningO_K ⊆ A and the morphism O_K → A is a local homomorphism of local rings. ThenA=O_K.

Proof. If there exists ana ∈ A\O_K, thenν(1/a) > 0.Hence 1/a ∈ mν ⊆ mA. This

implies that 1=a.(1/a)∈m_Awhich is impossible.

2.3.3 Extending of rational points

The following result characterizes the properness (See [16], Theorem II.4.7).

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Theorem (Valuation criterion for properness) 2.3.9. A morphism of finite type f : X→Yis proper if and only if for any valuation ringO_KoverY, with fraction fieldK, the canonical map

X(OK)→XK(K) is bijective.

Proof. See [16], Theorem II.4.7.

Remark 2.3.10. Concerning the property of properness, it follows from the above criterion that if the morphism of schemes f : X → Y is proper then the fibre X_y → Speck(y) is proper. One can ask whether the converse is true. If we do not impose any additional condition, then it is trivially false; as an counter-example we can takeY to be a Noetherian scheme and f :X→Yan open immersion. Nevertheless under some additional assumptions we can see that the converse holds. The most important case for us is as follows;

LetO_Kbe a discrete valuation ring,Xan irreducible scheme,X→SpecO_Ka surjective morphism and of finite type. If the fibresX→ SpecO_K are geometrically connected, and if the special fibreXs→Speck(s) is proper thenX→SpecO_Kis proper. See [15], IV 15.7.10 (the main reference) and [23], Remark 3.3.28

In what follows, we mention an important theorem from Nagata without proof.

2.3.4 Compactifications of schemes and Chow’s lemma

Theorem (Nagata) 2.3.11. LetXbe a separated scheme of finite type over a Noethe- rian schemeY. There exists a proper scheme ˆX overY such thatXis embedded in ˆX through an open immersion which scheme-theoretically has a dense image; i.e., one has the following commutative diagram

X ^ι //'' Xˆ

π

Y whereιis a dense open immersion and whereπis proper.

Proof. For a proof of the theorem see [31].

Definition 2.3.12. A proper algebraic variety over a field is called a complete variety.

The most common class of proper morphisms is that of projective morphisms. Projec- tive morphisms enjoy the same good properties as proper morphisms, such as stability under base change and composition. Projective morphisms and proper morphisms are

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connected with one another by Chow’s lemma.

Theorem 2.3.13. (Chow’s lemma) LetY be a Noetherian scheme. For any proper morphismX→Y, there exists a commutative diagram

X⁰ ^f //

g

''

X

Y

with f,gprojective, and f⁻¹(U)→ Uis an isomorphism for some everywhere dense open subsetU⊆X.

Definition 2.3.14. The morphism f : X → Y is quasi-projective if it can be decom- posed into an open immersion of finite typeX→Zand a projective morphismZ→Y.

2.4 Normality

Definition 2.4.1. A normal Noetherian integral domain of dimension 0 or 1 is called a Dedekind domain. LetX be a scheme. We say x ∈ X is a normal point of X if the local ringO_X,x is normal (i.e., it is integral and integrally closed in its fraction field). A schemeX is normal if it is normal at every pointx ∈ X. A normal locally Noetherian scheme of dimension 0 or 1 is called a Dedekind scheme. We have included dimension 0 to the definition so that the property of being a Dedekind domain is stable by localisation, which in turns means an open subscheme of a Dedekind scheme is – by our definition – a Dedekind scheme as well.

Theorem 2.4.2. LetXbe a normal locally Noetherian scheme andFbe a closed subset ofXof codimension≥2. Then the restriction

O_X(X)→ O_X(X\F)

is an isomorphism. This means that every regular function onX\Fextends uniquely to a regular function onX.

Proof. By assumingX =SpecAis affine, every prime ideal p ⊂Aof height 1 is in X\F. Therefore the theorem immediately follows from the fact thatA= T

p∈SpecA,htp=1

Ap.

for normal rings of dimension≥1.

Lemma 2.4.3. LetXbe a reducedS−scheme,Ya separatedS−scheme and also consider two morphisms ofS−schemes f,g: X →Y. If f|_U =g|_U for some everywhere dense open subsetU⊆X, then f =g.

Proof. Set∆ = ∆Y/S andh = (f,g) : X → Y ×_S Y.We have∆◦ f = (f,f) due to the universal property of the morphism (f,f). Therefore∆◦ f andhcoincide onU.

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Consequently,U ⊆ h⁻¹(∆(Y)).Since∆(Y) is closed, we haveX = h⁻¹(∆(Y)).Hence f(x)=g(x) for everyx∈X.

Now for showing f = g we can assume that X = SpecA andY = SpecBand let ϕ, ψbe ring homomorphisms corresponding to f andg respectively. Forb ∈ B set a =ϕ(b)−ψ(b).Thena|U =0. It follows thatU ⊆ V(a),and henceV(a) =SpecA sinceUis dense. This implies thatais nilpotent and sinceAis reduced, we havea=0

Thereforeϕ=ψ,and f =g.

2.4.1 Extending morphism to points of codimension 1

Theorem 2.4.4. LetY →S be a proper morphism over a locally Noetherian scheme.

LetXbe a normalS−scheme of finite type and consider a morphism ofS−schemes f :U→Y defined on a non-empty open subsetUofX. Then f extends uniquely to a morphismV→Y, whereVis an open subset ofXcontaining all points of codimension 1.

Proof. (See [23], Proposition 4.1.16) SinceY → S is separated and Xreduced from the lemma above the uniqueness is obvious. Letξbe the generic point ofX f induces a morphism f_ξ : SpecK(X) →Y. Letx∈ Xbe a point of codimension 1. ThenO_X,x is a discrete valuation ring with the field of fractionsK(X). From Corollary 2.3.10 f_ξ extends to a morphism fx: SpecO_X,x→Y. SinceY is of finite type overS, fxcan be extended tog:Ux→YwhereUxis an open neighbourhood ofx.

SetWto be an affine open neighbourhood ofg(x). Consider the restriction offandgto ξ∈U⁰:= f⁻¹(W)∩g⁻¹(W). SinceO_X(U⁰)⊆K(X), the ring homomorphismsO_Y(W)→ O_X(U⁰) corresponding to f|_U⁰ andg|U⁰ are identical. Therefore their corresponding morphisms of schemes are identical as well; f|_U⁰ =g|U⁰by the Theorem 2.3.9. Now using the lemma 2.4.3 above f andgcoincide onU∩Ux. If we take another point of codimension 1, sayx⁰∈Xthe same reasoning shows thatg⁰:Ux⁰ →Y coincide with f andgrespectively onU∩Ux⁰andUx∩Ux⁰Therefore f can be extended to an open

subsetV⊆Xcontaining all points of codimension 1.

With the same assumptions as theorem 2.4.4, if dimX = 1 then f extends uniquely to a morphismX → Y. In what follows, we are going to explain a useful lemma for normality.

2.4.2 A criterion for normality

Theorem 2.4.5. ( [23], Lemma 4.1.18) LetO_K be a discrete valuation ring, with field of fractionsKand residue fieldk. LetXbe anO_K−scheme such thatO_X(U) is flat over O_K for every affine open subsetUof X. We suppose thatXK is normal and thatXkis reduced. ThenXis normal.

Proof. As always, we may assume thatX =SpecAis affine. SinceAis flat overO_K thenA → A⊗_O_K Kis injective. ThereforeAis integral domain. Iftis a uniformizer

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forO_K takeα ∈Frac(A) so thatαis integral overA. SinceA⊗_O_K Kis normal, there exista∈A, r∈Zsuch thatα=t^−ra. Now ifa<tAwe are going to show thatr≤0.

We haveαⁿ+an−1αⁿ⁻¹+· · ·+a0 =0 an integral equation forαoverA. Ifr>0 by multiplying this equation byt^rnwe see thatαis nilpotent inA/tA. Hencea∈tAwhich

is a contradiction. Thereforer≤0 andα∈A.

There are several other criteria for normality such as Serre’sRkandSkconditions.

Definition 2.4.6. LetX be a locally Noetherian scheme andk≥0 be an integer. We say thatXhas the property (R_k) isXis regular at all of its points of codimension≤k.

We say thatXverifies property (S_k) if for anyx∈Xwe have depthO_X,x≥inf{k,dimO_X,x}.

Then one can show that a locally Noetherian connected schemeXis normal if and only if it verifiesR1andS2.

Definition 2.4.7. LetX be an integral scheme. A morphismπ: X⁰ → Xis called a normalisation morphism ifX⁰is normal and if every dominant morphism f :Y → X withYnormal factors uniquely throughπ:

Y ^f //

X

X⁰

π

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We can extend the definition of normalisation to reducible schemes.

Extended Definition 2.4.8. LetXbe a scheme having only finite number of irreducible componentsX₁, . . . ,X_n (endowed with the reduced closed subscheme structure). The disjoint unionX⁰ = `

1≤i≤n

X_i⁰whereX_i⁰ is the normalisation of the integral schemeX_i (defined in 1.4.6 above) is called the normalisation ofX. By construction, X⁰is endowed with a surjective integral morphismπ: X⁰ →X. IfXred is the reduced scheme associated toX, thenX⁰_red=X⁰.

Definition 2.4.9. Assume X is an integral scheme and L is an algebraic extension of its function fieldK(X). We define the normalisation of X in Lto be an integral morphismπ : X⁰ → X with X⁰ normal, K(X⁰) = L, and such that π extends the canonical morphism SpecL→X.

Remarks 2.4.10. 1. Ifπ : X⁰ → X is a normalisation of X then for any open subschemeUofX, the restriction morphismπ⁻¹(U)→ Uis a normalisation of U.

2. IfAis an integral domain andA⁰is the integral closure ofAin Frac(A) then the morphism SpecA⁰ → SpecAinduced by the canonical injection A → A⁰ is a normalisation morphism.

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Theorem 2.4.11. The normalisation of an integral schemeXexists and it is unique up to isomorphism. Moreover, a morphism f :Y → Xis the normalisation morphism if and only ifYis normal, and f is birational and integral.

Proof. According to the universal property of normality, the uniqueness is immediate.

For the existence it is enough to coverX with affine open subsets Ui and apply the remark above to get normalisation morphismsU⁰_i →Ui. Now gluing morphisms along with their intersections gives us the desired morphism. The rest of statement is obvious

because of the remark above.

In the same manner, one can show that the normalisationXinL:=K(X⁰) exists and is unique. Moreover, for any affine open subsetU ⊆X,π⁻¹(U) is affine andO⁰_X(π⁻¹(U)) is integral closure ofO_X(U) inL.

Theorem 2.4.12. (See [23], Proposition 4.1.25) LetXbe a normal Noetherian scheme andLbe a finite separable extension ofK(X). Then the normalisationX⁰→ XofXin Lis a finite morphism.

Proof. We may assume thatX=SpecA. IfBis the integral closure ofAinL, we want to show thatBis finite as anA−module. SinceAis Noetherian, one can extendL(By definition,L=FracB) in such a way that it is Galois overK:=K(X).

Now consider the trace form TrL/K:L×L→Ksending (x,y)7→TrL/K(xy).From linear algebra, the form is non-degenerate and bilinear becauseL/Kis separable extension.

Take{e1, . . . ,en}as a base ofL/Kwith theei∈B. There exists a basis{e^∗₁, . . . ,e^∗_n} ⊂L dual to the basis above, meaning Tr_L/K(e_ie^∗_j)=δ_{i j}. If we takeb∈Bwe can represent it asb=P

jλ_je^∗_jwithλ_j∈K. But then we haveλ_j=Tr_L/K(be_j)∈B∩K=A.Therefore Bis a sub-A−module ofP

jAe^∗_jand finite overA.

Since the integral closure ofk[x1, . . . ,xn] inL(finite extension ofk(x1, . . .xn)) is finite overk[x1, . . . ,xn], the normalisation inLof an integral algebraic variety over the field kis finite. In particular, the normalisation of an integral algebraic variety over a fieldk is again an algebraic variety over the same field.

2.4.3 Integral closure of Dedekind rings

Theorem 2.4.13. LetAbe a Dedekind ring with field of fractions K. If Lis a finite extension ofK, andBis the integral closure ofAinL, thenBis a Dedekind ring and the canonical morphismf : SpecB→SpecAhas finite fibres.

Proof. Consider the field extensionL/Kwe can decompose it into a separable extension and a purely inseparable extension. Due to Theorem 2.4.12, we only have to deal with the inseparable part. So, we can assume that the extensionL/K is purely inseparable. Therefore there exists a powerpêof the characteristicp =Char(K) such that L^pê ⊆ K. Hence we haveB^pê ⊆ A. Let p ∈ SpecAthen √

pBis the unique prime ideal ofBlying above p. This shows that f is bijective and sinceBis integral overA

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dimB=dimA=1.

Now takeq∈SpecBand setp:=q∩Ato be a maximal ideal ofA. SinceApis a discrete valuation ring, assume thatν:K → Zis a discrete valuation associated toApDefine νL(β)=ν(β^p^e). We see thatνLis also a discrete valuation ofLwith the valuation ring Bq. ThereforeBqis a discrete valuation ring for allq∈ SpecB. All we have to show now is thatBis Noetherian. LetIbe a non-zero ideal ofB. We show thatIis finitely generated. Take 0 , b ∈ I. Consider the ring B/bB. It is integral over A/bB∩A.

We haveb^p^e ∈ bB∩A, therefore it is non-zero and dimB/bB =dimA/bB∩A =0.

Since f is bijective,V(b) is a finite setq₁, . . . ,q_n. On the other hand we haveB/bB'

⊕_1≤i≤nB_q_i/(b) is Noetherian (note that all B_q_i are Noetherian local rings). Therefore I/(f) is finitely generated ideal and consequentlyIis finitely generated.

Definition 2.4.14. Assume thatXis a scheme andx∈X. Letmxbe the maximal ideal ofO_X,xandk(x) = O_X,x/mx be the residue field. Then mx/m²_x = mx⊗_O_X,x k(x) as a k(x)−vector space. Its dual (mx/m²_x)^∨is called the (Zariski) tangent space toXatx. We denote it byTX,x. If f : X → Y is a morphism of schemes and x∈ X andy= f(x), thenf_x^#:O_Y,y→ O_X,xinduces ak(x)−linear mapTf,x:TX,x→TY,y⊗_k(x)k(x), which is called tangent map of f atx.

Let (A,m) be a Noetherian local ring with residue fieldk=A/m. We know that always dim_km/m²≥dimA. The Noetherian local ringAis called regular if the equality holds which means ifmis generated by dimAelements.

Now, letX be a locally Noetherian scheme, and x ∈ X be a point. We say that Xis regular atx∈ XifO_X,xis a regular local ring, which means if dimO_X,x=dimk(x)TX,x

ifx∈Xis not regular, we call it a singular point ofX.

2.4.4 The relation between regularity and normality

Theorem 2.4.15. IfX is a Noetherian scheme then X is regular if and only if it is regular at its closed points. Moreover, ifXis regular then any connected component of Xis normal.

Proof. We may assume that X = SpecAis affine. Since we have (A_m)_pA_m = Ap for p∈SpecAandmmaximal ideal ofAsuch thatp⊂m, the first part of the theorem is trivial (recall that a ring is regular if its localisations are regular local rings). The rest of theorem follows from the fact that local regular rings are integrally closed in their

field of fractions.

Example 2.4.16. The affine spaceAⁿkand the projective spacePⁿkare regular. The affine space is regular because its local rings at its closed points are regular. The projective space is regular because it is a union of open subschemes isomorphic toAⁿ_k.

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2.5 Smoothness

In the study of a family of varieties parameterised by a base scheme, the flatness is a crucial notion which somehow shows the continuity of fibres.

Definition 2.5.1. Let f :X→Ybe a morphism of scheme. We say that f is flat atx∈ Xis the corresponding homomorphism f_x^#:O_Y,f_(x) → O_X,xis flat. From commutative algebra, the property of flatness is stable under base change, composition and fibre product.

Example 2.5.2. Since any algebra over a fieldkis flat, the morphismX →Speckof an algebraic variety overkis flat. As another class of examples, one can mention open immersions as well, while closed immersions are not in general flat. One can go further and see that a closed immersion is flat if it is an open immersion.

2.5.1 A criterion for flatness

As an important consequence for flatness, we have the following theorem.

Theorem 2.5.3. ( [23], Lemma 4.3.7) Let f : X → Y be a flat morphism with Y is irreducible. Then every non-empty open subsetU ofX dominatesY (i.e., f(U) is dense inY). IfXhas only a finite number of irreducible components, then every one of them dominatesY.

Proof. We may assume that Y = SpecA and U = SpecB are affine. Since open immersions are flat, we haveU→ Yis flat. Letηbe the generic point ofYandNthe nilradical ofA. By flatness ofBas anA−module we have

B/N B=B⊗_A(A/N)⊆B⊗_AFrac(A/N)=B⊗_Ak(η)=O(U_η)

If the fibreUη = ∅thenB = N Bwhich means the ring Bis nilpotent and therefore U=∅, a contradiction. Therefore the fibreUη,∅andf(U) is dense inY.

IfX has only finite number of irreducible components, then every component has a non-empty open subset. Hence every one of them dominatesY. Theorem 2.5.4. ( [23], Proposition 4.3.9) LetYbe a Dedekind scheme. Let f :X→Y be a morphism withXreduced. Then fis flat if and only if every irreducible component ofXdominatesY.

Proof. First we suppose every component ofXdominatesY. Letx∈ Xandy= f(x).

Ifyis the generic point ofY, thenO_X,xis anO_Y,y =K(Y)−module therefore it is flat.

So, let us suppose thaty∈ Y is a closed point andπ ∈ O_Y,yis a uniformizer. We are going to show thatπis not a zero divisor inO_X,x. By definition, this shows thatO_Y,y is flat overO_X,x. By assumptionπis not contained in any of minimal prime ideals of O_X,x. SinceXis reduced this implies thatπis not a zero divisor inO_X,x. The converse

is deduced from theorem 2.5.3.

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Corollary 2.5.5. Let Y be a Dedekind scheme, and f : X → Y be a non-constant morphism withXintegral. Then f is flat.

Proof. SinceY is irreducible and of dimension 1, f(X) is dense inY (because f is not

constant). Therefore by theorem 2.5.4 f is flat.

2.5.2 Dimension of fibres

Theorem 2.5.6. (See [23], Theorem 4.3.12 and [16], III 9.5) If f : X → Y is a morphism of Noetherian schemes and ifx∈Xandy=f(x) then

dimO_X_y_,x≥dimO_X,x−dimO_Y,y. Moreover if f is flat then the equality holds.

Proof. We may assume thatY is affine and the spectrum of a Noetherian local ring.

Because simply one can change the base via SpecO_Y,y→Y. So,yis a closed point of Y.

We proceed by induction on dimY. If dimY=0 then we haveXred=(Xy)red,therefore the equality holds.

If dimY ≥1,we may assume thatYis reduced via the base changeYred→Y. We have X×_YY_red //

Y_red

X ^f //Y

This does not change the dimensions ofY andX_y at x. And since flatness is stable under base change, therefore we may assume thatY is reduced. Taket ∈ Awhich is neither a zero divisor nor invertible. We have

dim(A/tA)=dimA−1, dim(B/tB)≥dimB−1

in whichB:=O_X,xBis a flatA−module. Therefore tensoring the injective homomor- phismA→−^t AbyBkeeps the injectivity and thereforet∈B(we show the image oftin Bby the same lettert) is a non-zero divisor inB. Hence dim(B/tB)=dimB−1.

SetY⁰=Spec(A/tA) andX⁰=X×_YY⁰. Then by the induction hypothesis we have dimO_X⁰_y_,x≥dimO_X⁰_,x−dimO_Y⁰_,y

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and the equality holds if f is flat, because if so,X⁰→Y⁰is also flat.

But nowX⁰_y=Xytherefore we have

dimO_X_y_,x≥(dimO_X,x−1)−(dimO_Y,y−1)=dimO_X,x−dimO_Y,y,

where the equality holds if f is a flat morphism.

Corollary 2.5.7. Let f : X → Y be a flat and surjective morphism of algebraic varieties. IfY is irreducible andXis equidimensional (which means that all of its irreducible components have the same dimension), then for everyy ∈ Y, the fibreXyis equidimensional, and we have

dimXy=dimX−dimY .

Proof. Ifx∈Xyis a closed point, then for every irreducible componentXiofXpassing throughxwe have

dimO_X

i,x=dimXi−dim{x} therefore

dimO_X,x=dimX−dim{x}

Sincexis the generic point of{x}and the latter is an algebraic variety we have dim{x}=trdeg_kk(x)

On the other hand,x∈Xyis a closed point thereforek(y)/k(x) is an algebraic extension.

Hence

dim{x}=trdeg_kk(x)=trdeg_kk(y)=dim{y}=dimY−dimO_Y_,y. By the above equalities and theorem 2.5.6 we finally have

dimX_y=dimO_X,x−dimO_Y,y=dimX−dimY.

2.5.3 Étale morphisms

Definition 2.5.8. Let f : X → Y be a morphism of finite type of locally Noetherian schemes. Letx∈ X andy= f(x). We say that f is unramified atxif the homomor- phismO_Y_,y → O_X,x for which we havemyO_X,x = mx(meaningO_X,x/m_yO_X,x =k(x)), and if the (finite) extension of residue fieldsk(y)→k(x) is separable. We say that f is étale atxis it is uniramified and flat atx.

A local homomorphism of Noetherian local ringsA→ Bis called étale if it is flat and unramified morphism such thatBis a localisation of a finitely generatedA−algebra.

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Example 2.5.9. IfL/Kis a finite field extension then SpecL→SpecKis unramified (and therefore étale) if and only if the extensionL/Kis separable.

Definition 2.5.10. AssumeXis an algebraic variety over a fieldkand ¯kis the algebraic closure ofk. We say thatX is smooth atx ∈ X if the points ofXk¯ lying abovexare regular points ofXk¯. As the simplest examples,Aⁿ_kandPⁿ_kare smooth varieties.

2.5.4 Relation between regularity and smoothness

In the following theorem we show that the Jacobian criterion is a criterion for smoothness.

Theorem 2.5.11. ( [23], Proposition 4.3.30) LetXis an algebraic variety over a field k, and letx∈Xbe a closed point. IfXis smooth atx, then it is regular atx. Moreover, the converse is true ifk(x) is separable overk.

Proof. We may assume thatXis affine. Letx⁰∈Xk¯ lying abovex. The fibers of mor- phismXk¯ → Xare of dimension 0. By theorem 2.5.6, we have dimO_X_¯

k,x⁰ =dimO_X,x Let us takeX =V(I) ⊆Aⁿ_k andI =(f1, . . . ,fr) andJxdenote the Jacobian ofXatx.

Of course,Jx = Jx⁰ as matrices in ¯k. SetDxP : k[X1, . . . ,Xn]→ (kⁿ)^∨(dual ofkⁿ as k−vector space), defined by

DxP: (x1, . . . ,xn)7→ X

1≤i≤n

∂P

∂X_i(y)xi. Then we have a surjective mapI/I∩m²→DxI. Therefore we have

dimk(x)TX,x=n−dimk(x)((I/I∩m²))≤n−rankDxI=n−rankJx.

Note that so far we have not used the assumption of smoothness. IfX is smooth atx then we have

dimk(x)TX,x≤n−rankJ⁰_x=dimO_X_¯

k,x⁰=dimO_X,x

HenceXis regular at x. For the second part, ifk(x)=k(i.e., ifxis rational) then the we have

dim_k(I/I∩m²)=rankD_xI.

Therefore it is obvious by the Jacobian criterion (applied toX¯k) thatX¯kis regular atx⁰. If not, consider the mapXk¯ → X. Since Speck⁰→Speckis finite and étale according to the assumption, we haveXk¯ → Xétale and finite as well. Now all the points ofX⁰_k lying abovexare rational overk⁰Therefore we return to the first case. SoXis smooth atx.

In general if f :X→Yis of finite type and étale atx∈Xthen we haveXis regular at xif and only ifY is regular at f(x) because in this case we have dimO_X,x=dimO_Y,f(x) and alsoTX,x'TY,f(x)⊗_k(y)k(x)

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Corollary 2.5.12. IfXis an algebraic variety over a fieldkandx∈ Xis an arbitrary point. IfXis smooth atx, thenXis regular atx.

Proof. Takex⁰∈Xk¯ as a point abovex∈X. SinceXis smooth atxthenx⁰is a regular point. Now consider{x⁰}. It is a reduced algebraic variety over an algebraically closed field then it is well-known that it contains a closed regular pointy⁰∈X¯k. Lety∈Xbe the image ofy⁰. Then from Theorem 2.5.11yis regular. Sinceyis a specialisation ofx

thereforexis regular as well.

Smooth morphisms are a generalisation of étale morphisms.

Definition 2.5.13. LetYbe a locally Noetherian scheme andf :X→Ybe a morphism of finite type. We sayf is smooth at a pointx∈Xif f is flat atxand ifX_y→Speck(y) (in whichy= f(x)) is smooth atx∈X_y

We say thatf is smooth of relative dimensionnif it is smooth at allx∈Xand all of its non-empty fibres are equidimensional of dimensionn. As an example, étale morphisms of finite type are smooth of relative dimension 0.

By the same process as the corollary 2.5.12, one can show that ifYis a locally Noethe- rian regular scheme and iff :X→Yis smooth morphism thenXis regular too.

2.6 Divisors

2.6.1 Cartier divisors

Definition 2.6.1. LetXbe a scheme. The sheaf of algebras associated to the presheaf K_X⁰ (defined by K_X⁰(U) = Frac(OX(U)) := (OX(U)\Z(OX(U)))⁻¹O_X(U), in which Z(OX(U)) is the set of zero divisors of the ring) is denoted by K_X. We call it the sheaf of stalks of meromorphic functions onX.

The subsheaf of invertible elements ofK_X is denoted byK_X^∗. An element ofK_X(X) is called a meromorphic element onX.

Definition 2.6.2. LetXbe a scheme. Denote the groupH⁰(X,K_X^∗/O^∗_X) by Div(X).The elements of Div(X) are called divisors onX. If f ∈H⁰(X,K_X^∗); its image in Div(X) is called a principal Cartier divisor and denoted by div(f).

Convention: We show the group law in Div(X) additively.

We say two divisorsD1andD2are linearly equivalent (D1∼D2) ifD1−D2is principal.

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A Cartier divisorXis called effective if it is in the image of H⁰(X,O_X∩ K_X^∗)→H⁰(X,K_X^∗/O^∗_X) and we denote it byD≥0.

We can represent a Cartier divisorDby{(Ui,fi)i}where theUiare open subsets ofX and they coverX. the fi∈ K_X^∗(Ui) the quotient of two regular elements ofO_X(Ui), and

fi|_U_i_∩U_j ∈ fj|_U_i_∩U_jO_X(Ui∩Uj)^∗for everyi,j.

Two systems{(Ui,fi)i}and{(Vj,gj)j}represent the same divisor if onUi∩Vj, fiand gjdiffer by a multiplicative factor inO_X(Ui∩Vj)^∗.

2.6.2 Weil divisors

For more details see for example [23].

Definition 2.6.3. LetXbe a Noetherian scheme. A prime cycle onXis an irreducible closed subset ofX. A cycle onX is an element ofZ^(X) and can be represented in a unique way by

Z=X

x∈X

nx{x}

If all ofnx=0, Z =0 Thenxis called the multiplicity ofZatx. If al of multx(Z)≥0 for everyx∈X, we say thatZis positive.

The finite union of{x}for whichnx,0 is called the support ofZ. It is a closed subset ofX. The support of divisor 0 is set to be the empty set by convention. If all of irreducible components of Supp(Z) are of codimension 1, we say thatZis of codimension 1. Note that{x}if and only if dimO_X,x=1.

IfXis a Noetherian integral scheme, a cycle of codimension 1 onXis called a Weil divisor onX. They form an abelian group by component-wise addition. IfXis a normal Noetherian scheme andf ∈K(X) be a non-zero divisor, then forx∈Xof codimension 1 we haveO_X,x normal local ring of dimension 1 (by definition, a discrete valuation ring). Therefore we can define

mult_x:K(X)→Z∪ {∞}

to be the normalised valuation ofK(X) Set (f) := X

x∈X,dimO_X,x=1

multx(f)[{x}].

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Such a divisor is called a principal Weil divisor.

The quotient of the group of cycles of codimension 1 onXby the subgroup of principal divisors is denoted by Cl(X). Two Weil divisorsZ1andZ2are equivalent (Z1 ∼Z2) if Z1−Z2is a principal Weil divisor.

2.6.3 Relation between Cartier divisors and Weil divisors

Definition 2.6.4. Let Abe a Noetherian local ring of dimension 1. We know that if f ∈Ais a regular element then length_A(A/f A) is a finite integer and it is additive over a short exact sequence. Therefore the map

f 7→length_A(A/f A) extends to a group homomorphism

Frac(A)^∗→Z.

Since the invertible elements ofAare contained in the kernel we obtain a homomorphism

multA : Frac(A)^∗/A^∗→Z.

Now letXbe a Noetherian scheme andD∈Div(X) a Cartier divisor. For anyx∈Xof codimension 1. We have

(K_X^∗

O^∗_X)x= FracO^∗_X,x O^∗_X,x . Define

multx(D) :=multO_X,x(Dx) .

We can assign to a Cartier divisor a Weil divisor as follows;

IfDis a Cartier divisor, we set

[D]= X

x∈X, dimOX,x=1

multx(D)[{x}].

Therefore [D] is a cycle of codimension 1 such that mult_x([D]) = mult_x(D) at every point of codimension 1.

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One can show that ifXis a regular Noetherian (hence it is normal) integral scheme.

Then the canonical homomorphisms

Div(X)→Z¹(X), CaCl(X)→Cl(X)

are isomorphism. (CaCl(X) is the quotient group of Cartier divisors onXmodulo the principal Cartier divisors and Z¹(X) is the group of cycles of codimension 1 onX.)

2.6.4 Meeting transversally

Definition 2.6.5. Let (A,m) be ann−dimensional Noetherian local ring. It is well known (See for instance [25], Theorem 13.4) that there exists an m−primary ideal generated bynelements but not by any fewer number of elements. Ifa₁, . . . ,a_n ∈m generate anm−primary ideal, then{a₁, . . . ,a_n}is called a system of parameters ofA. A system of parameters which generates the maximal idealmis called a regular system of parameters. Of course in the latter case, (A,m) by definition is a regular local ring.

Definition 2.6.6. ( [23], Definition 9.1.6) LetY be a regular Noetherian scheme, and Dbe an effective Cartier divisor onY. We say thatDhas normal crossings at a point y ∈ Y if there exist a regular system of parameters f1, . . . ,fn of Y at y, an integer 0≤m≤n, and integersr1, . . . ,rm≥1 such that the idealO_Y(−D)y∈ O_Y_,yis generated by f₁^r¹. . .f_m^r^m. The divisorDhas normal crossings if it has normal crossings at every pointy∈Y. We say that the prime divisorsD1, . . . ,Dlmeets transversally aty∈Y if they are pairwise distinct and if the divisorD1+· · ·+Dlhas normal crossings aty.

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2.6.5 Intersection with horizontal divisors

Theorem 2.6.7. ( [23], Proposition 9.1.30) Supposeπ:X → S be an arithmetic surface (i.e.,Xis regular, integral, projective, flat scheme of dimension 2 over a Dedekind schemeS of dimension 1). Letηbe the generic point andsbe a closed point. Then for any closed pointP∈Xη, we have

{P} ·Xs=[K(P) :K(S)] :=deg_k(s)O_X({P})|Xs,

in which{P}is the Zariski closure of{P}inX, endowed with the reduced closed subscheme structure.

Proof. Let denote the horizontal divisor{P}byD. Ifi:D→Xis the canonical closed immersion andh:D→S the finite surjective morphismπ◦i, then we haveX_s=π^∗s (becauseX→S is regular fibered surface) and we haveXs|_D=i^∗(π^∗s)=h^∗s.Now

h∗[Xs|_D]=h∗[h^∗s]=d[s], where d=[K(D) :K(S)], and

h^∗: Div(S)→Div(D), h∗: Div(D)→Div(S).

are the canonical morphisms of abelian groups of Weil divisors. On the other hand we have

h∗[Xs|_D]=X

s∈S

is(V,D)[s].

whereis(V,D) :=deg_k(s)O_X(D)|V.ThereforeD·Xs=dbecauseK(D)=K(P).

2.6.6 Exceptional divisors and van der Waerden’s purity theorem

We conclude this chapter by stating without proof an important theorem by Van dar Waerden which is the so-called the purity theorem. It shows that the notion of divisors is connected to the exceptional locus of a separated birational morphisms of finite type.

It is crucial for constructing minimal regular models in the next chapters.

Assume X andY are Noetherian integral schemes and f : X → Y is a separated birational morphism of finite type. There exists an open setW ⊆Yfor which we have x ∈ f⁻¹(W) if and only ifO_Y,f_(x) → O_X,xis an isomorphism (See, for instance, [23], Corollary 4.4.3 (b)). We call the subsetE := X\f⁻¹(W) the exceptional locus of f. The definition is local in the sense that ifU⊆Xis open then the exceptional locus of

f|_U:U→YisU∩E.

Theorem 2.6.8. ( [23], Theorem 7.2.22. van der Waerden’s purity theorem) LetX,Y be Noetherian integral schemes, and let f :X→Y be a separated birational morphism of finite type. Suppose thatYis regular. Then the exceptional locusEof f is empty or of pure codimension 1 inX.

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There is a nice criterion distinguishing exceptional divisors among vertical divisors.

Theorem 2.6.9. (Castelnuovo’s criterion) IfXis a regular, integral, projective and flat S−curve andE⊂Xsis a vertical prime divisor thenEis an exceptional divisor if and only ifE'P¹_k⁰andE²=−[k⁰:k(s)] in whichk⁰:=O_E(E).

Proof. See Theorem 9.3.8 [23].

In the following theorem, we see that the projective lineP¹_kis the unique (up to isomorphism) geometrically integral, projective curve of genusg≤0 which has ank−rational point.

Theorem 2.6.10. (Curves of small genus) LetXbe a geometrically integral projective curve of arithmetic genusp_a≤0 over a fieldk. We haveX'P¹kif and only ifX(k),∅.

Proof. The curve X is smooth overkbecause if X⁰ is the normalisation ofX¯k, then pa(X⁰)≥0 (H⁰(X⁰,O_X⁰)=k) and hence¯ X⁰=Xk¯.Now takey∈X(k) and consider the k−vector space

L(y) :={f ∈K(X)^∗| multy(f)+1≥0} ∪ {0}.

From Riemann-Roch Theorem, dimkL(y)=2.ThereforeX'P¹_k(In general, a normal projective curveXis isomorphic to the projective line if and only if there exists a Cartier

divisorDsuch that degD=1 andl(D)≥2).

2.7 Abelian varieties

In this section we state the preliminary definitions and facts about the extensive subject of abelian varieties. We will use the following results in the last chapter, where we take advantage of the Jacobian of families of curves so as to extract properties of the given families (See for instance Chapter 6. Theorem 6.3.1). For more details and proves in this subsection, refer to [30] or [28], or [40].

2.7.1 Group schemes

Definitions and basic properties 2.7.1. A schemeGoverS is a group scheme if it is endowed with the following morphisms overS,

m:G×_S G→G :S →G inv :G→G with the following properties;

Semistable Extension of Families of Curves