Degenerations without Triple Points

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Threefold Birational Morphisms and

Degenerations without Triple Points

Annelies JASPERS

Supervisor: Prof. dr. A. H¨oring (Universit´e Paris 6)

Co-supervisor: Prof. dr. J. Nicaise (KU Leuven)

Thesis presented in fulfillment of the requirements for the degree of Master of Science in Mathematics

Academic year 2012-2013

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Acknowledgement

I would like to express my deep gratitude to professor H¨oring for his encouragement, his patience and his useful suggestions. Our weekly discussions lifted this master’s thesis to a higher level. Thank you for all the time you spend explaining and reading to improve my knowledge and insights. I could not have expected a better supervisor.

I also wish to thank professor Nicaise for proposing this interesting subject, which will be a great preparation for my Ph.D. He also suggested professor H¨oring would be a great supervisor. His support and care are highly appreciated.

I would like to extend my thanks to the people of the research unit of Algebraic Geom- etry of the University of Leuven, and in particular to Arne Smeets, for listening to my presentation and for their good feedback.

Pieter Segaert helped me a lot with some of my L^ATEX-issues and Daan Michiels installed the software to make the figures in this thesis and showed me how to work with it. Thank you!

Of course I am particularly grateful to my parents for their support and understanding when I was stressed and especially for their effective pep talks. Finally, I would like to thank Val´erie, Daan, Pieter, Tia, Lena, Anna and all my other friends in Leuven and Paris, because taking a nice break from time to time is really necessary.

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Introduction

The aim of this master’s thesis is to understand [Cra83] and [CM83]. In these papers, we consider a smooth variety X overC, such that

(i) Either X is the domain of a proper birational morphism f : X → Y, not an isomorphism, where Y is a smooth, affine variety andf :X\f⁻¹(P)→Y \ {P}is an isomorphism for some point P ∈Y.

(ii) OrXis the total space off :X →∆, a proper, surjective morphism, where ∆ =A¹. The general fiber X_t is smooth, irreducible and projective, and mK_X_t ∼0 for some m ≥1.

It was Crauder’s insight that these two situations are highly analogous. More concretely, in the first case, we can write the scheme-theoretical fiber as a divisor

X_P =X

i∈I

s_iV_i,

where s_i ≥1 and all V_i irreducible and reduced. The following formula holds K_X −f^∗K_Y ≡X

i∈I

r_iV_i,

where KX, KY are the canonical divisors of X, Y respectively, and ri ∈N for all i∈I.

In the second situation, the special fiber can be written as X0 =X

i∈I

siVi,

with V_i the reduced irreducible components of X₀ and s_i ≥1 the multiplicity. Moreover, the following formula holds:

K_X ≡X

i∈I

r_iV_i. with ri ∈Q^≥0 and KX the canonical divisor ofX.

In both situations, we define

ρi = ri+ 1 s_i ,

for alli∈I. It is this quantity which plays a central role in the proofs of our main results.

In Chapter 2, we develop the analogy and obtain formulae valid in both cases (i) and (ii).

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In this master’s thesis, we apply this similarity for X of dimension 2 and 3. In the case of dimension 3, we make the extra hypothesis that P

i∈IV_i is a simple normal crossings divisor without triple points (i.e. for i, j, k distinct, we have V_i ∩V_j ∩V_k = ∅). As we consider two possible dimensions, there are four occurrences:

Birational Morphisms Degenerations

dimX = 2 Zariski’s Theorem on factorization of birational morphisms (Theorem 3.11)

Kodaira’s Classification of singular fibers of an elliptic surface (Theorem 3.14)

dimX = 3 Analogue of Zariski’s theorem on factorization of birational morphisms (Theorem 5.1)

Crauder-Morrison Classification of the special fiber of a triple- point-free degeneration of surfaces with Kodaira dimension 0 (Theorem 6.2)

The beauty of considering cases (i) and (ii) together is that the proofs of all four theorems rely on the same principle.

In Chapter 3, we will prove the following Theorems:

Theorem 3.11. Let f : X → Y be a proper birational morphism of smooth surfaces.

Then f can be written as a finite composition of blow-downs.

Theorem 3.14. Let X be an elliptic surface. This means X is a smooth surface and there is a proper surjective morphism π :X →∆ where the general fiber X_t is an elliptic curve for t 6= 0. Suppose moreover X contains no exceptional curves. Then X₀ is a multiple of the possibilities listed in the Table on pages 27-28.

These results were already well-known in the time of [Cra83], but Crauder provided new proofs which show the strengths of considering (i) and (ii) at the same time. Moreover, the proofs of these results illustrate the methods used for dimension 3. The proofs of both theorems depend on Propositions 3.6 and 3.8. In these propositions we assume that there is no contraction X →X⁰ of a Vi, such that X⁰ is still of case (i) or (ii). Propositions 3.6 and 3.8 give lists of the possible dual graphs near every V_j by comparing ρ_j to the ρ_i’s of the curves V_i meeting V_j.

The proof of Theorem 3.11 works by contradiction. So we assume that there is no contraction X → X⁰ of a V_i, such that X⁰ is still of case (i) or (ii). Hence we can apply Propositions 3.6 and 3.8. We find a contradiction quite easily. This implies there is a blow-down and moreover, it factors through f. An induction argument shows thatf can be written as the composition of blow-downs.

For the proof of Theorem 3.14, combinatorial arguments depending on 3.6 and 3.8, provide a list of the possibilities of the singular fiber of an elliptic surface as in the Table pages 27-28.

The case when X is a threefold, is treated in Chapters 4, 5 and 6. The proofs of these theorems follow the same lines as in the case of X being a surface. We also assume that

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there is no contraction µ : X → X⁰ of a V_j, such that X⁰ is still of case (i) or (ii) and that P

i6=jµ(V_i) is still a simple normal crossings divisor without triple points. Under this condition we have three propositions: 4.15, 4.19 and 4.23. They provide a list of the possible dual graphs near a surfaceV_j under certain conditions, by comparingρ_j with the ρ’s of surfaces meeting V_j. These propositions are developed in Chapter 4.

In Chapter 5, we prove Theorem 5.1.

Theorem 5.1. Let f : X → Y be a proper, birational morphism of smooth threefolds such that

i) f :X\S

i∈IVi →Y \ {P} is an isomorphism for a point P ∈Y, ii) P

i∈IV_i is a simple normal crossings divisor without triple points.

Then f may be written as a composition of blow-downs.

The proof of this Theorem is similar to the proof of 3.11. We first assume that there is no contraction X → X⁰ of a V_j, such that X⁰ is still of case (i) or (ii) and that P

i6=jµ(V_i) is still a smooth normal crossings divisor without triple points. Then we can apply Propositions 4.15, 4.19 and 4.23, from which a contradiction follows. So there is a blow-down and it factors throughf. An induction argument shows thatf can be written as the composition of blow-downs.

Contrary to the proof of 3.11, it is now quite difficult to obtain a contradiction. The idea of how we get to the contradiction is summarized in de flowchart on page vi.

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Fact: For allj: the surface V_j has a P¹-fibration or is isomorphic to P²

Assumption: There is no smooth contraction µ : X → X⁰ of Vj such that P

i6=jµ(V_i) is a simple normal crossings divisor without triple points

Construction of a chain

V₁ ^//V₂ ^//· · · ^//V_n−1 ^//V_n, where V1 'P². The structure of V2, . . . , Vn−1

is given by Proposition 4.19 and the structure of V_n is given by Proposition 4.15(e).

The surface V1 is isomorphic to the exceptional divisor of the blow-up of Y atP.

V₂, . . . , V_n are ruled.

s_n < sn−1 < · · · < s₁

s1 = 1 V₂, . . . , V_n are

not contractible

Contradiction, since s_i ≥ 1 for all i.

Plan of the proof of Theorem 5.1

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In Chapter 6, we will prove Theorem 6.2. This theorem is discussed in [CM83].

Theorem 6.2. Let X be a smooth threefold which is the total space of f : X → ∆, a proper, surjective morphism and ∆ =A¹. Suppose that the general fiber X_t is a smooth, projective surface of Kodaira dimension 0. We write the central fiber X₀ = P

i∈Is_iV_i. Moreover, we assume P

i∈IV_i to be a simple normal crossings divisor. We also assume that there are no triple points, i.e. V_i∩V_j∩V_k =∅ for i, j, k distinct.

Define the subgraphs Γ_min = span{V_i|ρ_i minimal} and Γ₀ = span{V_i ∈Γ_min|s_i minimal}

of the dual graph Γ.

Then there is a birational model Xe →∆ of X with the following properties:

(6.2.1) Γ_min is a connected subgraph ofΓ. It is either a single surface, a cycle or a chain.

(6.2.1) Each connected component of Γ\Γ_min is chain F₀—F₁—· · ·—F_l, with only F_l meeting Γ_min, and it meets a unique surface of Γ_min. The surfaces F₁, . . . , F_l are ruled and F₀ is either ruled or is isomorphic to P². The possibilities for F₀—F₁—· · ·—F_l can be found in Tables 6.2, 6.3 and 6.4.

(6.2.1) IfΓ_min is a single surfaceP, then P is isomorphic toP², it has a P¹-fibration or it has Kodaira dimension 0. If P has aP¹-fibration over a non-rational curve, then there is a map X₀ → E where E is an elliptic curve and all fibers are Kodaira elliptic curves of (constant) type II, III, IV, I^∗₀, II^∗, III^∗ or IV^∗ as described in Theorem 3.14.

(6.2.1) If Γ_min is a cycle W₁—W₂—· · ·—W_k, then Γ = Γ_min and X₀ =s(Pk

i=1W_i). All W_i are ruled over an elliptic curve.

(6.2.1) IfΓmin is a chainW0—W1—· · ·—Wk—Wk+1, then the surfacesW1, . . . , Wk have a P¹-fibration over an elliptic curve. The subgraph Γ₀ is a chain W_α—· · ·—W_β, with0≤α≤β ≤k+1. The surfacesW₁, . . . , Wα−1 andW_β+1, . . . , W_k don’t meet Γ\Γmin. The possible forms of the chainsW0—· · ·—Wα−1 andWβ+1—· · ·—Wk+1

can be found in Table 6.7.

Note that in Theorem 6.2 we do not assume the special fiber to be reduced. So in general, such degenerations are not semistable. A classification of semistable degenerations was already given in [Kul77], [Kul81], [Mor81], [PP81], [Tsu79]. Crauder and Morrison relaxed the hypotheses of reducedness, but imposed the extra condition of no triple points and made a classification of such degenerations.

To prove Theorem 6.2, we first take a minimal modelX⁰ofX, hence there is no contraction X⁰ →X⁰⁰of aV_j, such thatX⁰⁰is still of case (i) or (ii) and thatP

i6=jµ(V_i) is still a smooth normal crossings divisor without triple points. So we can apply Proposition 4.15 and 4.23 to X⁰. From these propositions a rough classification of the dual graph Γ easily follows (summarized in (6.2.1) and (6.2.2)). We will prove that there is a birational modification Xe of X⁰, which satisfies (6.2.3), (6.2.4) and (6.2.5). The rest of the proof is essentially combinatorial, heavily relying on Propositions 4.15 and 4.23.

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Chapter 1 Preliminaries

1.1 Conventions

By a variety, we mean an integral, separated scheme of finite type over an algebraically closed fieldk. In fact, in this work all varieties will be overC, except for Chapter 1. IfX is a smooth variety over C, then we can also consider X as a complex manifold with its analytic topology. See for example [Har77, Appendix B].

We use the wordcurve to mean a proper variety of dimension 1. So in particular, a curve is always irreducible. A smooth curve is projective [Har77, Prop II.6.7]. By a curve on a variety X we mean the image of a non-constant morphism from a curve to X. A 1-cycle on a varietyX is a formal sumPs

i=1n_iC_i, where then_i are integers and theC_i are curves onX.

A surface (respectively a threefold) is a proper variety of dimension 2 (respectively 3). A smooth surface over C is projective [Har77, Thm B.3.4].

We use the word point to mean a closed point, unless we specify the generic point.

If X is a smooth variety, we can talk about divisors, without making the distinction between Cartier and Weil divisors [Har77, Prop II.6.11]. This is true because there is a bijection between the Weil divisors and the Cartier divisors, which preserves linear equivalence. Furthermore, there is a bijection between the divisors modulo linear equivalence and the invertible sheaves modulo isomorphism, since X is integral [Har77, Prop II.6.15].

The invertible sheaf associated to a divisor D on X will be denoted by O_X(D). If the divisorsD1 andD2 are linearly equivalent, we writeD1 ∼D2. We will also use the notion of Q-divisor. This is a finite formal sumPs

i=1n_iD_i, where theD_i are closed integral sub- schemes of codimension 1 and all the n_i ∈ Q. We will say that a Q-divisor is Q-Cartier if some multiple is Cartier.

Let X be a variety. To a divisor D =P

i∈In_iD_i, we can associate a graph Γ, called the dual graph, in the following manner. The graph Γ contains #I vertices {v_i}_i∈I and if D_i∩D_j 6= ∅, then there is an edge connecting the vertices v_i and v_j. It is important to note that sometimes different definitions for a dual graph associated to a divisor are used.

We see that ∪_i∈ID_i is connected if and only if Γ is connected.

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1.2. INTERSECTION THEORY

Let{1,2,· · ·, k−1, k} ⊂I be a subset ofI. We say that n₁D₁+n₂D₂+· · ·+n_kD_k is a chain in D, if the dual graph associated to span{v₁,· · ·, v_k} is of the following form:

v₁ v₂ vk−1 v_k

We will often implicitly confuse vertices of the dual graph Γ and components of the divisor D.

1.2 Intersection Theory

We follow mainly [Deb01, Section 1.2].

Definition 1.1. Let X be a proper scheme over C of dimension r. We define the intersection number of r Cartier divisors D₁, . . . , D_r as the coefficient of m₁· · ·m_r in the polynomial

(m₁, . . . , m_r)→χ(X,O_X(m₁D₁+· · ·+m_rD_r)), where χ is the Euler-characteristic.

In [Deb01, Thm 1.5], it is shown that this is indeed a polynomial. Moreover, this map is multilinear, symmetric and takes integral values [Deb01, Prop 1.8]. Hence we obtain a morphism

Div(X)^r → Z

(D1, . . . , Dr) 7→ D1· · ·Dr.

It is clear from the definition that the intersection number is invariant under linear equivalence. Indeed, if D₁ is a principal Cartier divisor, then O_X(m₁D₁ +· · ·+ m_rD_r) = OX(m2D2+· · ·+mrDr), sinceOX(D1) = OX. So the coefficient ofm1· · ·mrinχ(X,OX(m1D1+

· · ·+m_rD_r)) is 0.

Definition 1.1 is not very intuitive, but it has a geometrical interpretation. If D₁, . . . , D_r are effective and meet properly in a finite number of points, then the intersection number is the number of points in the intersection, counted with multiplicity [Kol96, Thm VI.2.8].

If Y is a subscheme of X of dimension s and if D₁, . . . , D_s are Cartier divisors, then we define

(D₁· · ·D_s)·Y =D₁|_Y · · ·D_s|_Y.

WhenY is contained in the support ofDi, thenDi|Y is defined to be the Cartier divisor on Y associated to the invertible sheaf O_X(D_i)⊗O_Xi∗O_Y, where i:Y →X is the inclusion.

We will often abbreviate O_X(D_i)⊗O_X i∗O_Y toO_X(D_i)⊗ O_Y. In [Deb01, Prop 1.8] it is proven that for Dn effective with associated subscheme Y, we have

D₁· · ·D_n = (D₁· · ·Dn−1)·Y.

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1.3. CANONICAL SHEAF AND DUALIZING SHEAF

When Dis an effective Cartier divisor with associated subscheme Z and C is a curve, we can define

D·C.

When Z does not contain C, the intersection number counts the number of points (with multiplicity) of the intersection of Z and C. This can be proven by the Riemann-Roch Theorem [Har77, Thm IV.1.3]. We can extend this definition D·C to Q-divisors D and 1-cycles C, by linearity.

Let π :X →Y be a proper morphism between varieties and let C be a curve on X. We define the 1-cycle π∗C onY as follows: If C is contracted to a point byπ, thenπ∗C = 0.

Otherwise, π(C) is a curve on Y and we set π∗C = dπ(C) where d is the degree of the morphism C →π(C) induced byπ [Har77, Def p.137].

If Dis a Cartier divisor on X, then we have the projection formula [Deb01, 1.9]

π^∗D·C =D·π∗C. (1.1)

Definition 1.2. LetX be a proper scheme. Two Cartier divisorsD₁, D₂ arenumerically equivalent, written D₁ ≡D₂, if for every curve C, we have

D₁ ·C=D₂·C.

It is clear that two linearly equivalent divisors are also numerically equivalent. We will often write D ≡ E, even when D and E are linearly equivalent, since we will mostly use only the properties of the numerical equivalence. We can extend the definition of numerical equivalence to Q-divisors. Note that if D ≡ E, where D is a Q-Cartier Q- divisor and E is a Cartier divisor, then D·C ∈ Z for every curve C. We will use this property very often, without mentioning.

Definition 1.3. The Neron-Severi group of X is defined as N¹(X) = Div(X)/≡.

1.3 Canonical Sheaf and Dualizing Sheaf

Definition 1.4. [Har77, p. 241]

Let X be a proper scheme over a field k of dimension n. A dualizing sheaf for X is a coherent sheaf ω_X^◦ on X, together with a trace morphism t :Hⁿ(X, ω_X^◦) → k, such that for all coherent sheaves F onX, the natural pairing

Hom(F, ω_X^◦ )×Hⁿ(X,F)→Hⁿ(X, ω^◦_X) followed by t gives an isomorphism

Hom(F, ω_X^◦)→Hⁿ(X,F)^∨. Theorem 1.5. [Har77, Prop III.7.2 and Prop III.7.5]

Let X be a projective scheme, then X has a dualizing sheaf. Moreover, it is unique up to unique isomorphism.

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1.3. CANONICAL SHEAF AND DUALIZING SHEAF

Definition 1.6. [Har77, p. 180]

Let X be a smooth variety over a field k of dimension n. The canonical sheaf of X is defined as

ωX =

n

^Ω_X/k, the n-th exterior power of the sheaf of differentials.

This is an invertible sheaf

Theorem 1.7. [Har77, Cor III.7.12]

Let X be a projective smooth variety. Then the dualizing sheaf ω_X^◦ is isomorphic to the canonical sheaf ω_X.

We will denote for the rest of this text by ω_X the dualizing sheaf when X is projective.

When X is smooth, we denote by ω_X the canonical sheaf. IfX is an integral variety such that ω_X is an invertible sheaf (for example when X is smooth), then we can associate to ω_X a Cartier divisor K_X, which is defined up to linear equivalence. We call K_X the canonical divisor.

We now state the adjunction formula, which we will use very often.

Theorem 1.8. [Vak13, Prop 30.4.8]

Let X be a smooth, proper variety of dimension n. Let Y be an effective Cartier divisor on X. Then

ω_Y = (ω_X ⊗ O_X(Y))⊗ O_Y. In particular, ω_Y is an invertible sheaf on Y.

Proposition 1.9. Let X be a projective curve, possibly singular. Then degK_X = 2g−2,

where g is the genus of X.

Proof. For a divisor DonX with support in the set of regular points ofX, we denote by l(D) the dimension of Γ(X,O_X(D)). By the Riemann-Roch Theorem [Har77, Exercise IV.1.9], we have

l(K_X)−l(0) = degK_X + 1−g.

From l(KX) =g [Har77, Prop IV.1.1] and l(0) = 1, we obtain the degree of KX.

Let X be a smooth, proper surface and let Y be a projective curve on X. Then, using Proposition 1.9, the adjunction formula can be rewritten as

2g(Y)−2 =KX ·Y +Y². (1.2)

We call this formula the genus-formula.

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1.4. RULED SURFACES AND P¹-FIBRATIONS

1.4 Ruled Surfaces and P¹-fibrations

Definition 1.10. A ruled surface is a smooth projective surface X together with a surjective morphism π : X → C to a smooth curve C, such that the fiber X_y is isomorphic toP¹ for every point y ∈C.

Definition 1.11. A smooth projective surfaceX has aP¹-fibration if there is a surjective morphism π : X → C to a smooth curve C, such that all fibers are connected and that for all but finitely many points y of C, the fiberX_y is isomorphic to P¹.

Note that every ruled surface has a P¹-fibration. All fibers of π:X →C are numerically equivalent, since two points on C are linearly equivalent. If a smooth curve S is not contained in a fiber ofπ, then for every fiber F, we haveF ·S >0. IfF ·S = 1, we callS asection of π. If F ·S >1, then S is called amulti-section of π. Note that every section S is isomorphic to C via the restriction of π toS.

Proposition 1.12. Let X be a smooth projective surface with a P¹-fibration π:X →C.

Let F be any fiber of π. Then we have that F² = 0 and the arithmetic genus pa(F) = 0.

Proof. Since all fibers ofπ are numerically equivalent and don’t meet each other, we have F² = 0. By [Har77, Prop III.9.7], the morphism π : X → C is flat. Hence the genus is constant over all fibers of X [Har77, Cor III.9.13]. Since the general fiber is isomorphic toP¹, we conclude thatp_a(F) = 0.

Proposition 1.13. Let X be a smooth, projective surface with a P¹-fibration π:X →C.

LetDbe an irreducible component of a reducible fiberF ofπ. ThenD² <0andg(D) = 0.

Proof. Since all fibers of π are numerically equivalent, and D is disjoint from a general fiber, we have D·F = 0. Write F =mD+P

miDi as a divisor, with m, mi >0. Then we have

mD²+X

m_iD·D_i = 0.

SinceF is reducible and connected, there is at least one indexj such thatD·D_j >0. As D·D_i ≥0 for alli, we have P

m_iD·D_i >0. We conclude D² <0.

Let i :D → F be the closed immersion of D in F. Then we have the exact sequence of sheaves

0→ I_D → O_F →i∗O_D →0,

where I_D is the sheaf of ideals associated to D. We obtain the long exact sequence:

· · · →H¹(F,O_F)→H¹(F, i∗O_D)→H²(F,I_D)→ · · ·

By Grothendieck’s Vanishing Theorem [Har77, Thm III.2.7], one has H²(F,I_D) = 0.

Moreover we have H¹(F, i∗O_D) = H¹(D,O_D) [Har77, Lemma III.2.10]. So we have a surjective morphism

H¹(F,O_F)→H¹(D,O_D).

Since p_a(F) = 0, the dimension of H¹(F,O_F) is 0. This means that the dimension of H¹(D,O_D) must be 0 too. We concludeg(D) = 0.

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Corollary 1.14. Let X be a smooth projective surface with a P¹-fibration π : X → C.

Let D be an irreducible component of a reducible fiber F of π. Then K_X ·D≥ −1.

Proof. By the genus-formula (1.2), we get

−2 = 2g(D)−2 = D²+K_X ·D.

Since D² ≤ −1, we have K_X ·D≥ −1.

Corollary 1.15. Let X be a smooth, projective surface with a P¹-fibration π : X → C.

Let F be an arbitrary fiber of π. For every effective divisor H on X, we have H·F ≥0.

Proof. Since all fibers are numerically equivalent, we can change the fiber to an irreducible fiber that is not an irreducible component of H.

Proposition 1.16. [Har77, Prop V.2.3]

Let π :X →C be a ruled surface, let C₀ ⊂X be a section of π and let F be a fiber of π.

Then the Neron-Severi group of X is generated by C₀ and F, i.e.

N¹(X)'Z⊕Z.

Theorem 1.17. A smooth projective surface X has a P¹-fibration if and only if there exists a composition of blow-downs µ:X →X⁰ with X⁰ a ruled surface.

Proof. LetX⁰ be a ruled surface with ruling π:X⁰ →C. If there exists a composition of blow-downs µ: X → X⁰, then π◦µ : X → C is a surjective morphism with connected fibers and for all but finitely many points y of C the fibers are still isomorphic to P¹. So X has a P¹-fibration.

Conversely, let X be a smooth projective surface with a P¹-fibration π : X → C. Let n be the number of irreducible components in reducible fibers of π. Note that n = 1 is not possible because a reducible fiber has at least two irreducible components. So if n < 2, then n = 0. We will prove the theorem by induction on n. If n = 0, then X is already ruled, so we can take for µthe identity. Ifn > 0, then we will find an irreducible component L of a reducible fiberF with L² =−1. The contraction of L will be the first blow-down in the composition of µ.

Let F =Pk

i=1m_iD_i be any reducible fiber of π, with D_i the irreducible components and mi >0. By the genus-formula (1.2) and Proposition 1.12 we have

−2 = 2g(F)−2 = K_X ·F +F² =K_X ·F.

Suppose that D_i² 6= −1 for i = 2, . . . , k. We show D₁² = −1. By Proposition 1.13, we have D²_i ≤ −2 for i = 2, . . . , k. Hence we have K_X ·D_i = 2g(D_i)−2−D_i² ≥ 0 by the genus-formula (1.2). It follows from

−2 = K_X ·F =m₁K_X ·D₁+

k

X

i=2

m_iK_X ·D_i,

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that K_X ·D₁ <0. Since −2 =K_X ·D₁ +D₁² and D²₁ <0 by Proposition 1.13, this gives D²₁ =−1.

Since all components of fibers have genus 0, they are all isomorphic to P¹. So by Castel- nuovo’s criterion [Har77, Thm V.5.7], we can contract L =D1 to a point P, so we have µ₁ :X→X. Sincee µ₁ has connected fibers and sinceπ contracts each fiber ofµ₁, we can apply the rigidity lemma [Deb01, Lemma 1.15]. Hence the fibration π : X → C factors through µ₁, such that π =π₁ ◦µ₁. So π₁ is a P¹-fibration on X, ande µ₁(F) is a fiber of π1. Because X\L' Xe \ {P}, the fiber µ1(F) has one irreducible component less than F. By induction we obtain a composition of blow-downs X →X⁰, with X⁰ ruled.

Theorem 1.18. Let X be a smooth, projective surface with non-trivial canonical divisor KX. Suppose there exists an s ∈N\0 such that| −sKX| 6=∅. Then X is isomorphic to P² or X has a P¹-fibration.

Proof. First we state the Enriques-Kodaira Classification, the proof can be found in [BPVdV84, Chapter VI]:

Let X be a proper, smooth, algebraic surface. Then there exists a composition of blow-downs µ:X →X⁰ with X⁰ one of the following:

1. K_X⁰ nef and κ(X⁰) = κ(X) ≥ 0, where κ denotes the Kodaira dimension,

2. X⁰ 'P²,

3. X⁰ is a ruled surface.

First we show that case 1 cannot happen in the situation of the theorem. So suppose there exists a composition of blow-downs µ : X → X⁰ with K_X⁰ nef and κ(X⁰) = κ(X) ≥ 0.

Because | −sKX| 6= ∅, there is an effective divisor D ∼ −sKX. And hence for every multiple −snK_X of −sK_X with n >0, we have

Γ(X,O_X(−snK_X))6= 0. (1.3)

On the other hand, since tr.deg ⊕n≥0 Γ(X,OX(nKX))

−1 = κ(X) ≥ 0, there is an integer m≥1 with Γ(X,O_X(mK_X))6= 0. Hence for every positive integern:

Γ(X,O_X(mnK_X))6= 0. (1.4) Set t =sm. Then by (1.3) and (1.4):

Γ(X,OX(−tKX))6= 0 and Γ(X,OX(tKX))6= 0.

Hence −tK_X and tK_X are both linearly equivalent to effective divisors [Har77, Prop II.7.7]. So we can write

−tK_X = X

i

a_iC_i with a_i >0, tK_X = X

j

b_jD_j with b_j >0.

Degenerations without Triple Points

Threefold Birational Morphisms and

Degenerations without Triple Points

Introduction

Contents

Chapter 1

Preliminaries