SIMULTANEOUS ℚ-FACTORIALIZATION OF DIVISORS AND APPLICATIONS TO 3-FOLD

DIVISORIAL CONTRACTIONS THAT CONTRACT A

DIVISOR ONTO A CURVE THROUGH A cD_{5} SINGULARITY

NIKOLAOS TZIOLAS

Abstract. Let Γ ⊂ X be a smooth curve on a 3-fold. In this paper we classify terminal divisorial contractions E ⊂ Y ⟶ Γ ⊂ X contracting an irreducible surface E onto Γ, in the case when the general section of X containing Γ has D_{5} type singularities. In doing so we consider the problem of existence of MMP in families of canonical 3-folds, and we get results about existence of simultaneous ℚ-factorialisation of a family of divisors.

0. Introduction

In the birational classification of algebraic varieties, one wants to identify in each birational class some distinguished “simple” members which are called minimal models and then study the structure of birational maps between them. In dimension two, satisfactory results were known for over one hundred years. To treat the problem in higher dimensions, the minimal model program (MMP) was developed. After contributions of Kawamata, Kollár, Reid, Shokurov and others, the program was completed in dimension 3 by Mori in 1988. According to Mori’s theorem, every ℚ-factorial terminal projective threefold X, is birational to a threefold X′ with either K_{X′} nef (in which case it is called a minimal model), or X′ is a Fano fiber space. It is known that any birational map between minimal models is a composition of flops [Kaw88]. Terminal flops have been classified by the work of Kollár [Ko91]. However, the structure of birational maps between Fano fiber spaces is one of the most important open problems of threefold birational geometry. The Sarkisov program was developed by Corti, Reid and Sarkisov in order to factorize every birational map between Fano fiber spaces, as a composition of “elementary links”. Therefore, in order to better understand the geometry of the Fano fiber spaces, it is important to understand these links which consist of flips, flops and divisorial contractions. Flips have been classified by the work of Kollár and Mori [Ko-Mo92]. The structure of divisorial contractions is still an open problem.

1991 Mathematics Subject Classification. Primary 14E30, 14E35.

Key words and phrases. Algebraic geometry.

let E ⊂ Y ⟶^{f} Γ ⊂ X be a divisorial contraction. The most results concerning the case that Γ is a point are due to Corti [Cor-Rei00] and Kawakita [Ka02], [Ka01]. The case that Γ is a curve has been studied by Kawamata [Kaw94], and the author [Tzi02]. This paper can be considered as a continuation of [Tzi02]. In that paper, divisorial contractions when Γ was a smooth curve and X had a DuVal section of type D_{2n} through Γ, were classified. It was also explained that the case when the general section of X through Γ is D_{2n+1}, is very different than the D_{2n} one. In this paper we completely classify divisorial contractions E ⊂ Y ⟶^{f} Γ ⊂ X, in the case that the general section of X through Γ has D_{5} type singularities and Y is terminal. This is done in theorem 2.3 and corollaries 2.3 and 2.5. In doing so we follow the method presented in [Tzi02]. However, we need to consider cases with respect to the form of the equation of X. We deform X to an X_{0} whose equation is much simpler and then we show that in some cases the divisorial contractions exist in families. This is not true in general as shown by example 1.6. Then we obtain information about X by using results about deformations of terminal singularities [Nak98]. In order to construct divisorial contractions in families, we must consider the problem of existence of simultaneous ℚ-factorialization of divisors. In particular, given a family of threefolds X ⟶ T, and a global divisor Z ⊂ X, does there exist a morphism Y/T ⟶ X/T, such that Y_{t} is a ℚ-factorialization of Z_{t}? If T is a point and X is a canonical threefold this was done by Kawamata [Kaw88]. The problem is of interest by itself from the point of view of the MMP in higher dimensions. The case that X_{0} is terminal is treated in theorem 1.1. The canonical case is more complicated. Theorem 1.3 gives conditions for a simultaneous ℚ-factorialization to exist. Example 1.6 shows that these conditions are necessary. Finally in order to construct divisorial contractions in families, it is essential to use MMP in families of canonical threefolds. The terminal case was done by Kollár and Mori [Ko-Mo92]. The main difficulty in the canonical case is the existence of canonical flops in families. This is also an interesting problem by itself from the higher dimensional point of view. I should mention that this is not solved by Shokurov’s latest work. Propositions 1.9 and 1.10 give cases when canonical flops exist in families. These results are sufficient for the purposes of this paper but are a very superficial approach to this problem.

ℚ-FACTORIALIZATION-DIVISORIAL CONTRACTIONS 3

I plan to return to it in a future paper. Finally I should say that I believe that the general D_{2n+1} case could be done by the methods of this paper. However my feeling is that there is a big amount of calculations needed to do and that would result into a disproportionally large to the results paper.

1. ℚ-factorialization in families of canonical 3-folds

In this section we will consider the following problem.

Problem: Let X ⟶ T be a family of 3-folds, and Z ⊂ X a divisor in X flat over T. Assume that the scheme theoretic intersection Z_{t} = Z ∩ X_{t} is a divisor. When is there a simultaneous ℚ-factorialization of Z_{t}? That is, when is there a morphism Y ⟶ X such that Y_{t} ⟶ X_{t} is a ℚ-factorialization of Z_{t} for all t?

The case of a family of terminal threefolds is particularly nice and it is possible to give a complete answer.

Theorem 1.1. Let X ⟶^{f} T be a one dimensional family of terminal 3-folds. Let Z ⊂ X be a divisor in X that dominates T. Assume that the scheme theoretic intersection Z_{t} = X_{t} ∩ Z is a divisor in X_{t}. Then there exist a crepant morphism Y ⟶ X flat over T, such that Y_{t} ⟶^{f_{t}} X_{t} is a ℚ-factorialization of Z_{t}. In particular, the birational transform Z′_{t} of Z_{t} in Y_{t} is ℚ-Cartier and f_{t}-ample.

The situation for families of canonical 3-folds is much more complicated. In general it is not true that a common ℚ-factorialization exists as shown by example 1.6. The number of crepant divisors of the members of the family is important. In this direction we have the following: Theorem 1.2. Let X ⟶ T be a family of canonical threefolds. Let e(X_{t}) be the number of crepant exceptional divisors of X_{t}. Then

e(X_{0}) ≥ e(X_{t})

for t in a small neighborhood of 0 ∈ T. Moreover, if equality holds then there exist a common ℚ-terminalization. That is, there is a morphism Y ⟶^{g} X, such that Y_{t} ⟶^{g_{t}} X_{t} is a ℚ-factorial and terminal.

Question: Is it true that with assumptions as in the problem above, a simultaneous ℚ-factorialization of Z_{t} exists iff e(X_{0}) = e(X_{t}), or better if we only consider crepant divisors with center in Z_{t}?

In the canonical case we have the following. Theorem 1.3. Let X ⟶ T be a family of index 1 canonical 3-folds, with T smooth. Let Z ⊂ X be a divisor in X that dominates T such that the scheme theoretic intersection Z_{t} = X_{t} ∩ Z is a smooth integral divisor in X_{t}. Moreover, assume that

(1) X_{t}^{sing} ∩ Z_{t} is a smooth curve L_{t}. Let S_{t} = SpecO_{X_{t},L_{t}}. This is a Duval surface singularity defined over ℂ(t). Z_{t}|_{S_{t}} is a line through the origin such that the extended dual graph of S_{t} and Z_{t}|_{S_{t}} is independent of t. This means that S_{t} are isomorphic singularities and that the position of Z_{t}|_{S_{t}} in the fundamental cycle does not change with t.

(2) X_{0} is either cDV along Z_{0}, or there is a smooth curve L ⊂ X dominant over T such that L ∩ X_{t} is a reduced point P_{t} and m_{L}X = 2, 3 or k ≥ 3. This means that P_{t} ∈ X_{t} is elliptic with constant k. Moreover, if k ≥ 3 then B_{P_{0}}X_{0} is cDV and if k = 1 or 2, then the (3, 2, 1, 1,) or (2, 1, 1, 1) weighted blow ups are cDV along the birational transform Z̃_{t}.

Then there is a birational morphism Y ⟶ X flat over T, such that Y_{t} ⟶ X_{t} is a ℚ-factorialization of Z_{t}, for all t.

We can use this result to construct divisorial contractions in families. Corollary 1.4. Let X ⟶ T be a 1-dimensional family of terminal 3fold singularities P_{t} ∈ X_{t}. Let Γ ⊂ X be a surface such that the scheme theoretic intersection Γ_{t} = Γ ∩ X_{t} is a smooth curve through P_{t}. Let H_{t} be the general hyperplane section of X_{t} through Γ_{t} Assume it is A_{n} for all t and that Γ_{t} intersects an end of the fundamental cycle of H_{t} for all t. Then there exist a morphism g: W ⟶ X flat over T, such that W_{t} ⟶ X_{t} is a terminal extremal contraction of an irreducible divisor E_{t} onto Γ_{t}. Therefore divisorial contractions of a surface to a curve exist in families for this type of singularities.

We can use the previous result to improve the result of [Tzi02, Theorem 5.1]. Corollary 1.5. Let E ⊂ W ⟶ Γ ⊂ X be a terminal 3-fold divisorial contraction contracting an irreducible divisor E onto a smooth curve Γ. Suppose that the general hyperplane S of X containing Γ is A_{n}, and that Γ intersects an end of the fundamental cycle of S. Then W has index n + 1.

Proof. By [Tzi02] it is possible to write the equation of X as

xy + z^{n+1} + tϕ(x, y, z, t) = 0,

where the curve Γ is given by x = z = t = 0. By a result of Hironaka-Rossi, the equation of X is equivalent to

xy + z^{n+1} + tϕ(x, y, z, t) + t^{m} = 0,

for m >> 1. Now we can deform X to X_{0} given by xy + z^{n+1} + t^{m} = 0, and it can be explicitely seen with the same method as in the next example, that there is a terminal contraction E_{0} ⊂ W_{0} ⟶ Γ_{0} ⊂ X_{0}, and W_{0} has index n + 1. Since the index is constant in families, the corollary follows. □

The conditions of theorem 1.3 are necessary. The next example gives a family of canonical 3-folds and a family of divisors that a simultaneous ℚ-factorialization does not exist. Example 1.6. Let X ⊂ ℂ_{x,y,z,t}^{4} × ℂ_{ϵ}^{1} be the family over ℂ_{ϵ}^{1} given by ϵz^{2} + xy + z^{3} + t^{3} = 0. Let Γ ⊂ X be the plane given by x = z = t = 0. Let f: Y = B_{Γ}X ⟶ X. Then f^{−1}(Γ) = E + F. E is not ℚ-Cartier and there is no simultaneous ℚ-factorialization of E in Y ⟶ ℂ_{ϵ}^{1}.

Proof. It is clear that Y_{ϵ} = B_{Γ_{ϵ}}X_{ϵ}. Describe Y explicitely. By making the substitution x = xt, z = zt, we see that an affine open subset of Y is given by

ϵz^{2}t + xy + z^{3}t^{2} + t^{2} = 0.

t = 0 gives that f^{−1}(Γ) = E + F, where E is given by x = t = 0 and F by y = t = 0. The other charts are checked similarly. Suppose that a simultaneous ℚ-factorialization g: Z ⟶ Y of E_{ϵ} in Y_{ϵ} exists. According to [Ko-Mo92, theorem 12.3.1], (the theorem is only for terminal threefolds but the important is the existense of the versal deformation spaces Def(Y) which exists if Y is a modification of a terminal singularity which is true in our case), the proof of [Tzi02, theorem 1.8] runs for Z and X over T and we get a morphism h: W ⟶ X flat over ℂ_{ϵ}^{1}, and W_{ϵ} ⟶ X_{ϵ} is an extremal contraction contracting an irreducible divisor E_{ϵ} onto the curve Γ_{ϵ}. The special fiber X_{0} for ϵ = 0 is given by xy + z^{3} + t^{3} = 0 and Γ_{0} by x = z = t = 0. The general hyperplane section of X_{0} through Γ_{0} is A_{2}. For ϵ ≠ 0, X_{ϵ} is given by ϵz^{2} + xy + z^{3} + t^{3} = 0. In this case, the general hyperplane section through Γ_{ϵ} is A_{1}. Therefore, W_{ϵ} has index 2 if ϵ ≠ 0 [Tzi02, theorem 5.1].

Claim: W_{0} has index 3.

Since the index is constant in families, we get a contradiction and therefore a common ℚ-factorialization does not exist.

Proof of claim. We will construct the contraction W_{0} ⟶ X_{0} explicitely. Y_{0} is given by xy + z^{3}t^{2} + t^{2} = 0. E_{0} is given by x = t = 0 and F_{0} by y = t = 0. We want to construct a ℚ-factorialization of E_{0} and then contract F_{0} to obtain W_{0}. The equation of Y_{0} can be written as xy + (z^{3} + 1)t^{2} = 0. Therefore, 2E_{0} is Cartier at 0. To obtain a ℚfactorialization all we have to do is blow up the ideal I = (x, t^{2}). So let g: Z_{0} ⟶ Y_{0} be the blow up of I. g is an isomorphism in codimension 1. I claim that the g-exceptional curves C, are not contained in the birational transform F_{0}^{Z} and therefore F_{0}^{Z} = F_{0} = ℙ^{2}. To see this we will describe Z_{0} explicitely.

Z ⊂ ℂ^{4} × ℙ_{u,v}^{1}. It is given by

xu − t^{2}v = 0
xy + z^{3}t^{2} + t^{2} = 0

In the chart v = 1, Z_{0} is given by

xu − t^{2} = 0
y + z^{3}u + u = 0

It is easy to see that the g-exceptional curves are given by x = y = t = z^{3} + 1 = 0, and that F_{0}^{Z_{0}} by y = t = u = 0. Therefore no g-exceptional curve is contained in F_{0}^{Z_{0}} and hence F_{0}^{Z_{0}} = F_{0} = ℙ^{2}. Let l ⊂ F_{0}^{Z} be a line. Then l · K_{Z_{0}} = l · K_{Y_{0}} = −1 < 0. Now let p: Z_{0} ⟶ W_{0} be the contraction of F_{0}^{Z_{0}} over X_{0}. Hence there is also a commutative diagram:

\xymatrix{ & Z_{0} \ar[dl]_{g_{0}} \ar[dr]^{p} \\ Y_{0} \ar[dr]_{f_{0}} &  & W_{0} \ar[dl]^{q} \\  & X_{0}}

We know that 2E_{0}^{Z_{0}} and 2F_{0}^{Z_{0}} are Cartier. Let a > 0 such that

(1)  K_{Z_{0}} = p^{∗}K_{W_{0}} + aF_{0}^{Z_{0}}.

Let l ⊂ F_{0}^{Z_{0}} = ℙ^{2} be a line. Then l · F_{0}^{Z_{0}} = −b/2 with b a positive integer. We want to find b.

l · (E_{0}^{Z_{0}} + F_{0}^{Z_{0}}) = l · g_{0}^{∗}(E_{0} + F_{0}) = l · (E_{0} + F_{0}) = −1.

Now, l · E_{0}^{Z_{0}} = 1/2(2E_{0}^{Z_{0}}) · l. This can easily be computed. 2E_{0}^{Z_{0}} is given by x = t^{2} = y + z^{3}u + u = 0 and F_{0}^{Z_{0}} by y = t = u = 0. We can take any line in F_{0}^{Z_{0}}, so take the one given by y = t = z = u = 0. Clearly l ∩ 2E_{0}^{Z_{0}} is a reduced point. Therefore l · 2E_{0}^{Z_{0}} = 1. Hence

l · F_{0}^{Z_{0}} = −1 − l · E_{0}^{Z_{0}} = −3/2.

Intersect (1) with l. We get that a = 2/3. Therefore the index of W_{0} is 3. □

□ 1.1. Proof of theorem 1.1. In order to prove the theorem we are going to use the MMP in dimesion 4 which is now known to work in this dimension [Sho02]. To do so we must show that X is terminal. This will follow from the next lemma. Lemma 1.7. Let f: X ⟶ T be a one dimensional family of canonical 3-folds over a smooth curve. Suppose that X has finite index. Then X is also canonical. If X_{0} is terminal, then X is also terminal.

Proof. First we reduce to the index 1 case. The index is invariant under deformation and therefore index(X) = index(X_{t}), for all t. Let

p: X̃ ⟶ X

be the index 1 cover of X. Then X̃_{t} ⟶ X_{t} is the index 1 cover of X_{t}. Since KX̃ = p^{∗}K_{X}, we only need to consider the case that X_{t} is index 1. So we assume that now. Since f^{∗}(t) = X_{t} which is Cartier in X, it follows by adjunction that

K_{X}|_{X_{t}} = K_{X_{t}}.

X_{t} has rational singularities. Therefore by [Ko-Mo98], X has also rational singularities. Since it is index 1, it must be canonical. Looking at the proof of [Ko-Mo98, theorem 5.35], we see that if X is canonical of any dimension and H a Cartier divisor that is also canonical, then the pair (X, H) is canonical too. Therefore (X, X_{t}) is canonical. This implies that there are no crepant exceptional divisors with center in X_{t}. Let g: Y ⟶ X be a resolution of X. If an exceptional divisor E is crepant, then it’s center is not in X_{t}. Therefore it does not appear in g^{∗}X_{t} and hence by adjunction, E_{t} = E ∩ X̃_{t} must be crepant and exceptional over X_{t}. But this is impossible since X_{t} is terminal near 0. Therefore X is also terminal. □

Back to the proof of Theorem 1.1. Let

g: W ⟶ X

be a resolution of X and let E_{i} be the exceptional divisors. Run a (W, 0) MMP over X. Since X is terminal, we arrive at a Y′ ⟶^{g′} X over T, such that g′ is an isomorphism in codimension 1 and Y′ is ℚ-factorial. Therefore Z′ = g′_{∗}^{−1}Z is ℚ-Cartier, and hence so is Z′_{t}. To make it g′_{t}-ample, run a (Y′_{t}, −ϵZ′_{t}) MMP over X_{t}. By [Ko-Mo92, Prop. 11.4, Theorem 11.10], and since X is terminal, this can be done in families. Therefore we obtain Y ⟶^{g} X with the required properties.

Note: It would be of interest to get a proof of 1.1 without using the MMP in dimension 4.

1.2. Proof of Theorem 1.2. To see the first part, let W ⟶^{g} X be a resolution of X. Let {F_{i}} be the set of crepant exceptional divisors. Now Let Y ⟶^{h} W be a log resolution of (W, ∑_{i} F_{i} + g_{∗}^{−1}X_{0}). In particular, h_{∗}^{−1}F_{i} is smooth, i.e., all crepant divisors are smooth and snc. Let f = g ∘ h. By generic smoothness we can assume that Y_{t} is irreducible and smooth for t ≠ 0. Let E be f-exceptional such that E_{t} is f_{t}-crepant. Write

K_{Y} = f^{∗}K_{X} + aE + {other}.

Then by adjunction it follows that

K_{Y_{t}} = K_{Y} |_{Y_{t}} = f_{t}^{∗}K_{Y_{t}} + aE_{t} + {other}|_{Y_{t}}.

Therefore a = 0 and hence E is crepant. It is possible that E_{t} has many components. Look at E ⟶ T. This is flat and surjective and E is smooth. Therefore by generic smoothness E_{t} is smooth and irreducible for general t. Since there are finitely many crepant divisors, we may assume that E_{t} is irreducible for t ≠ 0. Let X′_{0} = f_{∗}^{−1}X_{0}. Now I claim that E_{0} = E|_{X′_{0}} is also crepant. Let

f^{∗}X_{0} = X′_{0} + ∑_{i} b_{i}E_{i}.

Then by adjunction

K_{X′_{0}} = f^{∗}K_{X_{0}} + ∑_{i}(a_{i} − b_{i})E_{i}.

Since X_{0} is canonical it follows that E_{0} is also crepant. Therefore e(X_{0}) ≥ e(X_{t}).

Suppose that e(X_{0}) = e(X_{t}). Let W ⟶^{f} X be a resolution of X as before. By generic smoothness, for general t ∈ T, W_{t} is smooth and in particular irreducible. Therefore by removing a finite number of points of T, we can assume that W_{t} is smooth for all t ≠ 0. Now run a (W, 0) MMP over X. We arrive at a crepant morphism Y ⟶^{g} X such that Y_{t} is terminal for t ≠ 0. Moreover we claim that Y_{0} is irreducible. To see this, observe that X is canonical since X_{0} (we may only work with the index 1 cover of X, hence we may assume that X is index 1. Since X_{0} is rational, so is X. Now rational of index 1 implies canonical). Therefore, as in the proof of lemma 1.6, the pair (X, X_{0}) is canonical. In particular this implies that X is terminal around X_{0} and therefore there is no crepant divisor with center in X_{0}. Hence Y_{0} is irreducible. Moreover, Y_{0} ⟶ X_{0} must be a small contraction since e(X_{0}) = e(X_{t}). This concludes the proof of the theorem. □

In order to prove theorem 1.3, we will use the minimal model program in families of canonical threefolds. For families of terminal threefolds it is known to work [Ko-Mo92]. To work the canonical case we need to prove existence of canonical flops in families. Since we will be applying it to non proper families, it is desirable to obtain some results about the behaviour of ℚ-factoriality in families. The next result is similar to [Ko-Mo92, Theorem 12.1.10] but we have removed the properness condition and instead we have a stronger restriction for the singularities.

Proposition 1.8. Let X ⟶^{f} T be a 1-dimensional family of canonical threefolds such that Y_{t}^{sing} is proper, and let D be a divisor on X such that D_{0} = D|_{X_{0}} is ℚ-Cartier. Then there is a neighborhood V of 0 ∈ T such that D is ℚ-Cartier in f^{−1}(V).

Proof. Let m > 0 such that m(D|_{X_{0}}) is Cartier. Let U = {x ∈ X, s.t. mD is Cartier at x}, and Y = X − U. Since X is normal, codim(Y, X) ≥ 2. Let i, j denote the open immersions U ⟶ X and U ∩ X_{0} ⟶ X_{0}. Then by assumption, the sheaf F = O_{X}(mD)|_{U} is invertible.

Claim:

(i_{∗}F)|_{X_{0}} = j_{∗}(F |U ∩ X_{0}).

Since i_{∗}F = O_{X}(mD), and j_{∗}(F |U ∩ X_{0}) = O_{X_{0}}(m(D|_{X_{0}})), which is invertible, the claim will imply that O_{X}(mD) is invertible in a neighborhood of X_{0}. Therefore by the properness of the singular part of X, Y is over a finite number of points of T and therefore there is a O ∈ V ⊂ T such that mD|_{f^{−1}(V)} is Cartier.

To prove the claim we follow the idea of [Ko-Mo92, Theorem 12.1.10]. There is an exact sequence

0 ⟶ F ⟶ F ⟶ F |U ∩ X_{0} ⟶ 0.

This gives

0 ⟶ i_{∗}F ⟶ i_{∗}F ⟶ i_{∗}(F |U ∩ X_{0}) ⟶ (2)

⟶ R^{1}i_{∗}F ⟶ R^{1}i_{∗}F ⟶ R^{1}i_{∗}(F |U ∩ X_{0})

Let F_{0} = F |U ∩ X_{0}. We will show that R^{1}i_{∗}F_{0} = R^{1}i_{∗}F = 0. This is local, so we may take X to be affine. Then R^{1}i_{∗}F_{0} = H^{1}(X, i_{∗}F_{0}) = H^{1}(X_{0}, j_{∗}F_{0}). By using local cohomology we find that H^{1}(U∩X_{0}, F_{0}) = H_{X_{0}∩Y}^{2} (X_{0}, j_{∗}F_{0}). But j_{∗}F_{0} = O_{X_{0}}(m(D|_{X_{0}})) and hence free and in particular S_{3} and coherent. Hence by [Har67], H_{X_{0}∩Y}^{2} (X_{0}, j_{∗}F_{0}) = 0. To show that R^{1}i_{∗}F = 0, we must show that it is coherent. Then from (2) it will follow that it must be zero. By [Har67], it suffices to show that

depth_{z}F + codim(Z ∩ Y, Z) ≥ 3

for all z ∈ U. Since F is invertible, it is CM and hence we only need to check the above condition at the generic point of X. This will follow once we show that codim(Y, X) ≥ 3. Since X_{0} is canonical, X is also canonical. If dim Y = 2, then at the generic point of Y, X is a rational surface singularity. Therefore ℚ-factorial. So dim Y = 1. Therefore R^{1}i_{∗}F is coherent and hence it must be zero. This proves the claim and the proposition. □

Problem: Do canonical flops exist in families?

The answer for terminal flops is yes [Ko-Mo92]. I will treat this problem in another paper. However for the purposes of this paper, we only need existence under some very strong assumptions. Proposition 1.9. Let f_{0} : Y_{0} ⟶ X_{0} be a flopping contraction in any dimension. Suppose that X_{0} has only hypersurface double points. Let Y ⟶ T be a deformation of Y_{0}. Then there is a projective morphism f: Y ⟶ X over T, such that f_{0} : Y_{0} ⟶ X_{0}, is the given contraction. Let f′_{0} : Y′_{0} ⟶ X_{0} be the flop of f_{0}. Then there is a projective morphism f′ : Y′ ⟶ X, such that f′_{0} : Y′_{0} ⟶ X_{0} is the flop of f_{0}. In other words flops to double points exist in families.

Proof. 0 ∈ X_{0} can be written as

x^{2} + f(y_{1}, …, y_{n}) = 0.

X exists by [Ma-Ro71]. Then X can be written as

x^{2} + f(y_{1}, …, y_{n}, t_{1}, …, t_{m}) = 0.

Now we proceed as in the 3-fold terminal case. If −D is f-ample, then to prove the existence of the flop for f it suffices to show that R(X, D) = ⊕_{d}O_{X}(df_{∗}D) is finitely generated. X has an involution i. Then D + i^{∗}D = 0. Since R(X, −D) is finitely generated, then so is R(X, D). Therefore there is an Y′ ⟶ X the flop of f. But Y′ = Y and the involution is also in every fiber. Therefore the flop exists in families. □

Proposition 1.10. Let f: Y/T ⟶ X/T be a morphism between two families of canonical 3-folds over T. Suppose that f_{0} : Y_{0} ⟶ X_{0} is a flopping contraction. Suppose that there is a morphism g: Z ⟶ T, such that Z_{0} ⟶ Y_{0} is a ℚ-factorial terminalization of X_{0}. Then there is a morphism f′: Y′ ⟶ X, such that f′_{0} : Y′_{0} ⟶ X_{0}, is the flop of f_{0}. Therefore in this case flops exist in families.

Note that the conditions of this proposition are not always satisfied as shown by example 1.6.

Proof. Let f′_{0} : Y′_{0} ⟶ X_{0} be the flop of f_{0}. Let g′_{0} : Z′_{0} ⟶ Y′_{0} be a ℚ-factorial terminalization of Y′_{0}. Z_{0}, Z′_{0} are birational and nef over X_{0}. Therefore, ϕ_{0} : Z_{0} −−→ Z′_{0} is an isomorphism in codimension 1 and a composition of terminal flops [Kaw88]. Terminal flops exist in families [Ko-Mo92], therefore ϕ_{0} extends to a map ϕ: Z −−→ Z′ over T. Then by [Ma-Ro71], there is a morphism g′: Z′ ⟶ Y′ over X, extending g′_{0}. Hence f′: Y′ ⟶ X is the required flop. □

In order to prove theorem 1.3, we will need a few simple lemmas. Lemma 1.11. Let Z ⊂ X be a smooth divisor in a normal variety X which has canonical hypersurface singularities only, and that every irreducible component of X^{sing} ∩ Z is smooth. Let Y = B_{Z}X ⟶^{f} X be the blow up of X along Z. Then

(1) Y is normal and has hypersurface singularities only,

(2) K_{Y} = f^{∗}K_{X},

(3) every irreducible component E_{i} of the f-exceptional divisor E is smooth.

Proof. Let n = dim X. Then we may assume that X ⊂ ℂ^{n+1}. Let f: B_{Z}ℂ^{n+1} ⟶ ℂ^{n+1} be the blow up of ℂ^{n+1} along Z. Then B_{Z}ℂ^{n+1} is smooth and Y = B_{Z}X ⊂ B_{Z}ℂ^{n+1} is codimension 1 and hence Y has hypersurface singularities only. Let F be the f-exceptional divisor. Then by adjunction, K_{B_{Z}ℂ^{n+1}} = f^{∗}K_{ℂ^{n+1}} + F, and f^{∗}X = Y + F. Therefore

K_{Y} = (K_{B_{Z}ℂ^{n+1}} + Y)|_{Y} = f^{∗}K_{X}.

Since ℂ^{n+1} is smooth and X canonical, the proof of [Ko-Mo98, Theorem 5.34] show that the pair (ℂ^{n+1}, X) is also canonical. By adjunction we see that

K_{B_{Z}ℂ^{n+1}} + Y = f^{∗}(K_{ℂ^{n+1}} + X).

Therefore the pair (ℂ^{n+1}, Y) is also canonical. Hence by [Ko-Mo98, Proposition 5.51], Y is normal. Finally, since every irreducible component of X^{sing} ∩ Z is smooth and f^{−1}(z) = ℙ^{1}, for all z ∈ Z, it follows that every irreducible component of F ∩ Y is also smooth. □

Lemma 1.12. Let (0 ∈ X) be a canonical 3-fold singularity such that the general hyperplane section H of X through 0 is an elliptic surface singularity with invariant k = 1, 2 or 3. Let 0 ∈ Z ⊂ X be a smooth divisor. Let Y ⟶^{f} X be either the blow up of X at 0 if k = 3, or the (3, 2, 1, 1) or (2, 1, 1, 1) weighted blow up of X at 0 in the case that k = 2 or 1. Then the birational transform Z′ of Z in Y is smooth and Y^{sing} ∩ Z is a union of rational curves.

Proof. Suppose that k = 3. Then Y is just the blow up of X at 0 and therefore Z′ = B_{0}Z is smooth.

Now suppose that k = 2 or 3. Then m_{0}X = 2. Let Z be given by x = y = 0. Then X is given by an equation of the form

xf(x, y, z, t) + yg(x, y, z, t) = 0.

Since t = 0 is elliptic, we can write the equation of X as

x^{2} + xf(x, y, z, t) + yg(x, y, z, t) = 0.

By using the Weierstrass preparation theorem, the equation of X can be written as

(1)  x^{2} + 2xϕ(y, z, t) + yh(y, z, t) = 0.

Moreover, ϕ(y, z, t), h(y, z, t) ∈ (y, z, t)^{2}, because otherwise 0 ∈ X is cA_{n}. Eliminate x from (1). X becomes

(2)  F (x, y, z, t) = x^{2} + yh(y, z, t) − ϕ^{2}(y, z, t) = 0,

ans Z is given by x − ϕ(y, z, t) = y = 0. Moreover from the previous discussion, ϕ(y, z, t), h(y, z, t) ∈ (y, z, t)^{2}. Now consider cases with respect to h.

Case 1: mult_{0}h(y, z, t) ≥ 3.

In this case assign weights to x, y, z and t as follows. Let w(x) = 2 and w(y) = w(z) = w(t) = 1. Let Y = B_{0}^{w}X ⟶^{f} X be the (2, 1, 1, 1) weighted blow up of X. We want to understand the birational transform Z′ of Z in Y.

Claim: Z′ = B_{0}Z and therefore Z is smooth.

By the definition of the weighted blow up,

Z′ = Proj ⊕_{n≥0} \frac{m^{w}(n) + I_{Z}}{I_{Z}} ,

and I_{Z} = (x−ϕ(y, z, t), y). We may look at Z as given by x−ϕ_{1}(z, t) = 0 in ℂ_{x,z,t}^{3}, where ϕ_{1}(z, t) = ϕ(0, z, t). Note that ϕ_{1}(z, t) ≠ 0, since otherwise ϕ(y, z, t) = yλ(y, z, t) and hence 2Z given by x^{2} = y = 0 will be Cartier. Then mult_{0}ϕ_{1}(z, t) ≥ 2. Now

\frac{m^{w}(1) + I_{Z}}{I_{Z}} = (\overline{x}, \overline{z}, \overline{t}) = (\overline{z}, \overline{t}, ϕ_{1}(\overline{z}, \overline{t})) = (\overline{z}, \overline{t}) = \frac{m + I_{Z}}{I_{Z}}

In general,

\frac{m^{w}(n) + I_{Z}}{I_{Z}} = (\overline{x}^{i}\overline{t}^{j}\overline{z}^{k}, 2i + j + k ≥ n) =

(ϕ_{1}(\overline{z}, \overline{t})^{i}\overline{t}^{j}\overline{z}^{k}, 2i + j + k ≥ n) = (\overline{t}^{j}\overline{z}^{k}, j + k ≥ n) = \frac{m^{n} + I_{Z}}{I_{Z}}

Therefore, Z′ = B_{0}Z, and hence Z′ is smooth. Note that this is the case that k = 2.

Case 2: mult_{0}h(y, z, t) = 2, and k = 1.

Let h_{2}(y, z, t) be the degree 2 part of h(y, z, t). Since the section t = 0 is elliptic, the cubic term of F (x, y, z, 0) must be a cube because otherwise the section t = 0 is DuVal. Therefore,

h_{2}(y, z, t) = ay^{2} + tl(y, z, t),

where l(y, z, t) is linear in y, z, t. Therefore the equation of X becomes

(3)  x^{2} + ay^{3} + ytl(y, z, t) + yh_{≥4}(y, z, t) − ϕ^{2}(y, z, t) = 0.

If l(y, z, t) ≠ 0, then one of y^{2}z, yzt, yt^{2}, appears in the above equation and the section z = t is given by x^{2} + ψ(y, t) = 0, and ψ_{3}(y, t) is not a cube and hence it must be DuVal. Therefore l(y, z, t) = 0. Therefore the equation of X becomes

(4)  x^{2} + y^{3} + yh_{≥4}(y, z, t) − ϕ^{2}(y, z, t) = 0,

where m_{0}ϕ ≥ 2. Moreover, Z is given by x − ϕ(y, z, t) = y = 0, and as before, ϕ(0, z, t) ≠ 0. By using the Weierstrass preparation theorem it is possible to write

x^{2} + y^{3} + yh_{≥4}(y, z, t) − ϕ^{2}(y, z, t) =

x^{2} + u_{1}[y^{3} + α(z, t)y^{2} + β(z, t)y + γ(z, t)] =

u_{1}[(\frac{x}{\sqrt{u_{1}}} )^{2} + y^{3} + α(z, t)y^{2} + β(z, t)y + γ(z, t)]

From the above equations, it is clear that γ(z, t) = unit · ϕ_{≥2}^{2}(0, z, t) = −ψ_{≥2}^{2}(z, t). Therefore Z is given by x − ψ_{≥2}(z, t) = y = 0. Now eliminate y^{2}. The equation of X can be written

x^{2} + y^{3} + 3α(z, t)y^{2} + β(z, t)y − ψ_{≥2}^{2}(z, t) = 0.

The change of variables y ↦ y − α(z, t) makes the equation of X

(5)  x^{2} + y^{3} + (y − α(z, t))δ(z, t) − α^{3}(z, t) − ψ^{2}(z, t) = 0,

where δ(z, t) = β(z, t) − 3α^{2}(z, t). Z is given by x − ψ_{≥2}(z, t) = y − α(z, t) = 0. Now it is easy to see that m_{0}(α^{3}(z, t) + ψ^{2}(z, t)) ≥ 6. Therefore, m_{0}(ψ) ≥ 3 and m_{0}(α) ≥ 2. Now an argument as in case 1 shows that Z′ = B_{0}Z and therefore it is smooth. □

Lemma 1.13. Let X ⟶ T be a 1-dimensional family such that X_{0} is canonical. Assume that dim X_{0} ≥ 3 and that X has finite index. Then the pair (X, X_{0}) is also canonical.

Proof. By lemma 1.6 it follows that X is canonical. Since X has finite index it must have index 1. X_{0} is canonical and therefore as in the proof of lemma 1.10, (X, X_{0}) is also canonical since X_{0} is Cartier in X. □

Lemma 1.14. Let T be a smooth curve and Y/T ⟶^{f} X/T be a morphism over T. Assume that X_{0} is a canonical 3-fold. Suppose that K_{Y} = f^{∗}K_{X}. Then dim f^{−1}(x) ≤ 2 for any x ∈ X_{0}.

Proof. As before we see that the pair (X, X_{0}) is canonical. Therefore, for any x_{0} ∈ X_{0}, x_{0} ∈ X is a terminal singularity. If there is a divisor E ⊂ f^{−1}(x_{0}), then E is crepant over x_{0} which is impossible. Therefore dim f^{−1}(x_{0}) ≤ 2. □

Proof of Theorem 1.3.

Suppose that X_{0} is given by F (x, y, z, w) = 0 in ℂ^{4}. Then X is

given by F (x, y, z, w) + tΨ(x, y, z, w) = 0 in ℂ^{5}. Let X_{1} ⟶^{f_{1}} X be

the (1, 1, 1, 1, 1), or (2, 1, 1, 1, 1), or (3, 2, 1, 1, 1) weighted blow up of X along L. Consider the (1, 1, 1, 1, 1) blow up case first. Let ℂ̃ = B_{0}ℂ^{5}. Then

Kℂ̃ = f_{1}^{∗}K_{ℂ^{5}} + 3E
f_{1}^{∗}X = Y + 3E.

Therefore

K_{X_{1}} = f_{1}^{∗}K_{X}.

By lemma 1.13, it follows that dim f_{1}^{−1}(x) =≤ 2 for any x ∈ X_{0}. Therefore X_{1} ⟶ T is also a family of canonical 3-folds. Moreover, e(X_{1}) < e(X). By lemma 1.11, Z_{1} = (f_{1})_{∗}^{−1}Z is smooth and X_{1}^{sing} ∩ (Y_{1})_{0} is a union of smooth rational curves. Let X_{2} ⟶^{f_{2}} X_{1} be the blow up of X_{1} along Z_{1}. Then by lemma 1.10, K_{X_{2}} = f_{2}^{∗}K_{X_{1}}, X_{2} is normal and e(X_{2}) < e(X_{1}). Moreover, the conditions of part 1 of the theorem guarantee that there is a smooth f_{2}-exceptional divisor (over the generic point of L_{0} it is just the blow up of a line through a DuVal singularity). We continue this process and get a sequence

⋯ ⟶ X_{k} ⟶^{f_{k}} X_{k−1} ⟶ ⋯ ⟶^{f_{3}} X_{2} ⟶^{f_{2}} X_{1} ⟶^{f_{1}} X,

such that

e(X_{1}) > e(X_{2}) > ⋯ e(X_{k}) > ⋯

Since there are finitely many crepant divisors, there is a n > 0 such that e(X_{n}) = 0 and therefore X_{n} is terminal and g_{n}: X_{n} ⟶ X is crepant. Moreover X_{n} ⟶ T is a family of terminal 3-folds. Let E_{i} be the g_{n}-exceptional divisors. By lemma 1.10 they are all smooth. Since (X_{n})_{0} is terminal, it has only isolated hypersurface singularities. Therefore we can obtain a ℚ-factorialization of it, W, by just blowing up Z_{n} and the exceptional E_{i}. Now consider the map W/T ⟶ X/T. A ℚ-factorialization of Z_{0} can be obtained by running a (W_{0}, ϵ ∑ E_{i}^{W}) MMP over X_{0}. Since W_{0} is terminal and by propositions 1.8 and 1.9, the operations of the MMP extend in the family. Therefore we obtain a morphism Y/T ⟶^{f} X/T, such that Y_{0} ⟶^{f_{0}} X_{0} is a ℚ-factorialization of Z_{0}. Now by proposition 1.8, Y_{t} ⟶ X_{t} is a ℚ-factorialization of Z_{t} in a neighborhood of 0 ∈ T. □

Proof of Corollary 1.4. Let f: Y ⟶ X be the blow up of X along Γ. Then as in [Tzi02], there are two f-exceptional divisors E_{1} and E_{2}, which are smooth. E_{1} is over Γ. Then from [Tzi02] it follows that Y_{0} is cDV. The conditions of theorem 1.3 are satisfied and therefore there is a morphism g: Z ⟶ Y, such that Z_{0} ⟶ Y_{0} is a ℚ-factorialization of (E_{1})_{0}. Now the construction in [Tzi02] can be done in families and hence there is a morphism W ⟶ X which is a family of divisorial contractions. □

2. cD_{5} type divisorial contractions.

In this section we will get a complete classification of terminal 3-fold divisorial contractions E ⊂ W ⟶ Γ ⊂ X, in the case that Γ is smooth and the general section of X through Γ is of type D_{5}. At this moment I would like to point out that I believe that the same method that will treat the D_{5} case, can also give the general D_{2n+1} case. However, my feeling is that the amount of calculations needed is disproportionate to the value of the result. I should also mention that the D_{2n} case was completely done in [Tzi02], but as it was shown there it is very different than the D_{2n+1} case.

We begin with a definition. Definition 2.1. A divisorial contraction is a morphism f : E ⊂ W ⟶ Γ ⊂ X, such that X and W are ℚ-factorial, W − E ≅ X − Γ, E is a prime divisor, −K_{W} is f-ample, and ρ(W/X) = 1.

The contraction is called terminal if X and W have terminal singularities. Also note that when working in the analytic category, the ℚ-factoriality of X and the requirement that ρ(W/X) = 1 is lost.

We also need the following definition.

Definition 2.2 ([Tzi02]). Let 0 ∈ S be a DuVal singularity of type D_{n}, and let 0 ∈ Γ ⊂ S be a smooth curve through the singularity. Let U ⟶^{f} S be the minimal resolution and Γ′ = f_{∗}^{−1}Γ. Let

∘^{E_{n−1}} ; | ; ∘^{E_{1}} — ∘^{E_{2}} — ⋯ — ∘^{E_{n−2}} ; | ; ∘^{E_{n}} .

be the dual graph. Then Γ′ intersects either E_{1}, or E_{n−1} or E_{n}. In the first case we call Γ of type DF_{l}, and in the others of type DF_{r}.

In the D_{5} case we have the following. Theorem 2.3. Let 0 ∈ Γ ⊂ S ⊂ X. Suppose that P ∈ S is a D_{5} singularity for the general S through Γ. Then

(1) If Γ ⊂ S is of type DF_{l}, then there is no terminal contraction.

(2) If Γ ⊂ S is of type DF_{r}, then it is always possible to write the equation of Γ ⊂ S ⊂ X as

x^{2} + y^{2}z + 2xz^{2} + t[xzψ(z, t) + axt + bxt^{k} + ϕ_{≥2}(y, z, t)] = 0,

so that y^{2}, yz and z^{ν} do not appear in ϕ_{≥2}(y, z, t) for any ν. Moreover, k ≥ 1, ab = 0 and I_{Γ} = (x, y, t). Let a_{i,j,k} denote the coefficient of y^{i}z^{j}t^{k} in ϕ_{≥2}(y, z, t). Then a terminal contraction does not exist iff

ϕ_{2}(y, z, t) = 0

and (a)

a_{0,0,4} = a_{1,0,2} = 2a_{0,1,2} − aψ(0, 0) = 4a_{0,0,3} − a^{2} = 0

or (b)

a_{0,2,1}^{2} − 2aa_{0,2,1} + 4a_{0,0,3} = 0
2a_{0,1,2} − a_{0,2,1}ψ(0, 0) = 0.

If there is a terminal contraction W ⟶ X, then W has index 4. Example 2.4. Let 0 ∈ Γ ⊂ S ⊂ X be given by

X : x^{2} + y^{2}z + 2xz^{2} + tϕ_{≥4}(y, z, t) = 0,

I_{Γ} = (x, y, t). Then there is no terminal divisorial contraction of an irreducible suface to Γ.

Corollary 2.5. Let Γ ⊂ X as in the theorem. Let E ⊂ W ⟶^{f} Γ ⊂ X the canonical divisorial contraction, which by [Tzi02] always exists. Let Σ be the general section of X through 0. Then W is terminal iff Σ′ = f_{∗}^{−1}Σ is normal.

Conjecture 2.6. Let 0 ∈ Γ ⊂ X. Assume that the general section S of X through Γ is DuVal. Let Σ be the general section of X through 0. Let E ⊂ W ⟶^{f} Γ ⊂ X the canonical divisorial contraction. Then W is terminal iff Σ′ = f_{∗}^{−1}Σ is normal.

This is true for all the D_{2n} cases as well as the D_{5}.

Corollary 2.7. Let 0 ∈ Γ ⊂ X as above. Let Σ be the general hyperplane section of X through 0. Write X as deformation of Σ, X ⟶ Δ, using Γ as the parameter. There is a morphism Δ ⟶^{ϕ} Def(Σ) inducing the above deformation. Then there is a subset Z of Def(Σ), such that, a terminal divisorial contraction exists iff Im(ϕ) ⊂ Z.

Proof of Theorem 2.3. The method that we are going to use is based on the method that appears in [Tzi02] which I describe next. There is always a canonical contraction whose construction is described from the diagram

\xymatrix{Z \ar[d]_{g} \ar@{-->}[rr]^{ϕ} &  & Z′ \ar[d]^{g′} \\ Y \ar[dr]_{f} &  & W \ar[dl]^{p} \\  & X}

We start by taking the blow up Y of X along Γ. There are two fexceptional divisors, E_{1} and E_{2}. E_{1} is a ruled surface over Γ and E_{2} ≅ ℙ^{2} is over the singular point 0 ∈ X. Then take a ℚ-factorialisation Z of E_{1}. After running a suitable MMP over X, E_{2} contracts giving the required contraction. The main difficulty is to calculate Z and especially it’s singularities away from E_{2}^{Z}. In the D_{2n} case, Z was relatively straightforward to obtain. To treat the D_{5} case we will do the following to understand Z. We will first put some restrictions on the equation of X. Then Z will be obtained from the following diagram:

\xymatrix{ & Y_{2} \ar[dl]_{f_{2}} \ar[dr]^{g_{2}} \\ Y_{1} \ar[dr]_{f_{1}} &  & Z \ar[dl]^{g} \\  & Y}

Y_{1} is the blow up of Y along E_{1}. Let F be the f_{1}-exceptional divisor. Then we will get an explicit ℚ-factorialisation Y_{2} of E_{1}^{Y_{1}}. We will then show that no f_{2}-exceptional curve is contained in F^{Y_{2}} and hence we can contract F^{Y_{2}} to obtain Z. Then in order to decide whether W is terminal, all we must do is to study the singularities of Y_{2} away from F^{Y_{2}} ∪ E_{2}^{Y_{2}}. Finally in order to remove the restrictions that we imposed on the equation of X, we will use the deformation theory that was developed earlier to deform any contraction to one that can be treated as we just explained. Then we will use the fact that a deformation of a terminal singularity is again terminal [Nak98].

Part 1. of the theorem follows from [Tzi02, Theorem 6.1].

To see part 2. By [Tzi02, Proposition 4.8], Γ ⊂ S ⊂ X is given by

x^{2} + y^{2}z + 2xz^{2} + t[xzψ(z, t) + axt + bxt^{k} + ϕ_{≥2}(y, z, t)] = 0,

and I_{Γ} = (x, y, t).

The technical condition that we need to impose at the time is that yt and t^{2} do not appear in ϕ_{≥2}(y, z, t).

If a = 0 then we already have the claimed form. If a ≠ 0 then write axt + bxt^{k} = axt · (unit). The change of variables t ↦ t/\sqrt[2]{(}unit) will give the required normal form.

Start by describing Y = B_{Γ}X. In the chart x = xt, y = yt it is given by

x^{2}t + y^{2}tz + 2xz^{2} + xtzψ(z, t) + axt^{2} + bxt^{k+1} + ϕ_{≥2}(yt, z, t) = 0.

t = 0 gives that E_{1} : x = t = 0, E_{2} : z = t = 0 and L = E_{1} ∩ E_{2} : x = z = t = 0. Now it is easy to see that Y is singular along L. This is what makes this case so different from the D_{2n} cases where Y had only one singularity along L, and much more difficult to work.

As we said before, we will try to show that none of the f_{2}-exceptional curves is contained in F_{1}^{Y_{2}} and also study the singularities of Y_{2} away from E_{2}^{Y_{2}} ∪ F_{1}^{Y_{2}}. Moreover, it is clear that Z can only have isolated cDV point over a cDV point and therefore we restrict our attention to what happens over non cDV points. One can also check the other charts and see that all the non cDV point are contained in the first one given by x = xt, y = yt. So it suffices to do all our calculations in that chart.

Calculate f_{1} : Y_{1} ⟶ Y. In the chart x = xt, Y_{1} is given by

(6) x^{2}t^{2} + y^{2}z + 2xz^{2} + xztψ(z, t) + axt^{2} + bxt^{k+1} + \frac{1}{t}ϕ_{≥2}(yt, z, t) = 0.

Let F_{1} be the f_{1}-exceptional divisor. t = 0 gives that

F_{1} : z = t = 0 (7)

E_{1}^{Y_{1}} : t = y^{2} + 2xz + \frac{1}{z}[\frac{1}{t}ϕ_{≥2}(yt, z, t)]_{t=0} = 0. (8)

Observe that E_{2}^{Y_{2}} does not appear in this chart. Now let

ϕ_{2}(y, z, t) = a_{3}zt.

Then \frac{1}{t}ϕ_{2}(yt, z, t) = a_{3}z. Hence [\frac{1}{t}ϕ_{≥2}(yt, z, t)]_{t=0} = a_{3}z.

The singular locus of Y_{1} is contained in F_{1} ∩ E_{1}^{Y_{1}}. This is the two lines, l_{d}, given by

z = t = 0
y = d

with d such that d^{2} + a_{3} = 0. We will now study the singularities of Y_{1} along these lines. The change of variables y ↦ y + d brings the equations of l_{d} to y = z = t = 0, and

Y_{1} : x^{2}t^{2} + (y + d)^{2}z + 2xz^{2} + xztψ(z, t) + axt^{2} + bxt^{k+1}

+ \frac{1}{t}ϕ_{≥2}((y + d)t, z, t) = 0.

Look what happens along l_{d} by making the change of variables x ↦ x − δ. Y_{1} is given by

(x − δ)^{2}t^{2} + (y + d)^{2}z + 2(x − δ)z^{2} + (x − δ)ztψ(z, t)

+ a(x − δ)t^{2} + b(x − δ)t^{k+1} + a_{3}z + \frac{1}{t}ϕ_{≥3}((y + d)t, z, t) = 0,

and this gives

(9)(x − δ)^{2}t^{2} + y^{2}z + 2dyz + 2(x − δ)z^{2} + (x − δ)ztψ(z, t) + a(x − δ)t^{2}

+ b(x − δ)t^{k+1} + \frac{1}{t}ϕ_{≥3}((y + d)t, z, t) = 0.

It is now clear that Y_{1} is singular along these two lines, and smooth away from them. Therefore we want to calculate Y_{2} in a neighborhood of these lines.

Now it is possible to write

\frac{1}{t}ϕ_{≥3}(yt, z, t) = z^{2}f(y, z) + tϕ(y, z, t)

and

ϕ(y, z, t) = zϕ_{1}(y, z) + tϕ_{2}(y, z, t).

Then (2) becomes

(10) x^{2}t^{2} + y^{2}z + 2xz^{2} + a_{3}z + xztψ(z, t) + axt^{2} + bxt^{k+1}

+ z^{2}f(y, z) + ztϕ_{1}(y, z) + t^{2}ϕ_{2}(y, z, t) = 0.

Moreover,

F_{1} : z = t = 0
E_{1}^{Y_{1}} : t = y^{2} + 2xz + a_{3} + zf(y, z) = 0
∪l_{d} : y^{2} + a_{3} = z = t = 0

Now we want to describe Y_{2}. It can be obtained by blowing up a suitable multiple of E_{1}^{Y_{1}}. To find it look what happens along l_{d}. The change of variables y ↦ y + d and x ↦ x − δ shows that 2E_{1}^{Y_{1}} given by

I_{2E_{1}^{Y_{1}}} = (t^{2}, y^{2} + 2xz + a_{3} + xtψ(z, t) + zf(y, z) + tϕ_{1}(y, z))

is Cartier at all but finitely many points of l_{d}. Therefore we can get Y_{2} by blowing up 2E_{1}^{Y_{1}}. At this point I want to mention that the reason that i have the condition that yt does not exist in the equation of X is that if it appears then I don’t see that 2E_{1}^{Y_{1}} is Cartier at all but finitely many points and therefore i don’t know what needs to be blown up to get Y_{2}.

Therefore Y_{2} ⊂ ℂ_{x,y,z,t}^{4} × ℙ_{u,w}^{1} is given by the equations

wt^{2} − u[y^{2} + 2xz + zf(y, z) + xtψ(z, t) + tϕ_{1}(y, z)] = 0 (11)

x^{2}t^{2} + y^{2}z + 2xz^{2} + a_{3}z + xztψ(z, t) + axt^{2} + bxt^{k+1}

+z^{2}f(y, z) + ztϕ_{1}(y, z) + t^{2}ϕ_{2}(y, z, t) = 0. (12)

In the affine chart u = 1, Y_{2} is given by

y^{2} + 2xz + zf(y, z) + xtψ(z, t) + tϕ_{1}(y, z) − wt^{2} = 0 (13)

x^{2} + wz + ax + bxt^{k−1} + ϕ_{2}(y, z, t) = 0. (14)

Now we study what happens over l_{d}. It is easy to see that C = f_{2}^{−1}(l_{d}) is given by

y^{2} + a_{3} = z = t = 0 (15)

x^{2} + ax |_{t=0} +ϕ_{2}(y, 0, 0) = 0.

It is easy to check that no component of C is contained in F_{1}^{Y_{2}}. Next we want to see what kind of singularities Y_{2} has alog C. To do this we must calculate the jacobian, J, of Y_{2} along C. An easy computation gives that

\begin{aligned}& J = \\ ( & \begin{matrix}0 & 2d & 2x_{0} + f(0, 0) & x_{0}ψ(0, 0) + ϕ_{1}(0, 0) & 0 \\ 2x_{0} + a & \frac{∂ϕ_{2}}{∂y} (d, 0, 0) & w + \frac{∂ϕ_{2}}{∂z} (d, 0, 0) & \frac{∂ϕ_{2}}{∂t} (d, 0, 0) & 0 \end{matrix} & ) \end{aligned}

where d^{2} + a_{3} = 0 and x_{0}^{2} + ax_{0} + ϕ_{2}(d, 0, 0) = 0. If d ≠ 0 then from (9) it follows that Y_{1} is cA_{n} along l_{d} and therefore Y_{2} can have at worst isolated cDV points on C. Hence the case of interest is when d = 0 which implies that a_{3} = 0 as well.

It is easy to see that x_{0}ψ(0, 0) + ϕ_{1}(0, 0) is the coefficient of zt in (9) at any point on l given by x ↦ x + x_{0}. Moreover, x_{0}^{2} + ax_{0} + ϕ_{2}(0, 0, 0) is the coefficient of t^{2}. Hence if it is nonzero then Y_{1} is cA_{n} under C, and therefore Y_{2} has isolated cDV points along C. So we want to investigate what happens if x_{0}ψ(0, 0) + ϕ_{1}(0, 0) = 0. Now by looking at the Jacobian matrix J it is clear that Y_{2} is either singular along C or smooth. In the first case the resulting contraction will be only canonical and in the second we will have to see what happens in the other chart. Y_{2} is singular along C iff

(2x_{0} + a)(2x_{0} + f(0, 0)) = 0 (16)

[(b(k − 1)x_{0}t^{k−2}) |_{t=0} +\frac{∂ϕ_{2}}{∂t} (0, 0, 0)](2x_{0} + f(0, 0)) = 0

\frac{∂ϕ_{2}}{∂y} (0, 0, 0)(2x_{0} + f(0, 0)) = 0

x_{0}ψ(0, 0) + ϕ_{1}(0, 0) = 0

x_{0}^{2} + ax_{0} + ϕ_{2}(0, 0, 0) = 0

A similar calculation in the other affine piece of Y_{2} given by w = 1 leads to the same conclusion about the singularities of Y_{2} and we again obtain the equations (16).

Now we will show that there are no other f_{2}-exceptional curves that appear over the other affine piece of Y_{1} given by t = tx, and therefore the conditions given by (16) are necessary and sufficient.

Y_{1} is given by

(17) x^{2}t + y^{2}tz + 2z^{2} + xztψ(z, xt)

+ ax^{2}t^{2} + bx^{k+1}t^{k+1} + \frac{1}{x}ϕ_{≥2}(xyt, z, xt) = 0.

Moreover, x = 0 gives that F_{1} : z = x = 0, E_{2}^{Y_{1}} : z = t = 0 and

E_{1}^{Y_{1}} = (x, y^{2}t + 2z + \frac{1}{z}[\frac{1}{x}ϕ_{≥2}(xyt, z, xt)]_{x=0}).

The singularities of Y_{1} are along L_{1} ∪ l_{d} = E_{1}^{Y_{1}} ∩ F_{1}. L_{1} : x = z = t = 0 is over L and l_{d} : x = y − d = t = 0, with d^{2} + ad + ϕ_{2}(0, 0, 0) = 0 as before, is over a point and we already studied the part away from zero in the other chart. So we only need to see what happens along L_{1}.

Claim: E_{1}^{Y_{1}} is ℚ-Cartier along L_{1}. Therefore there are no f_{2}- exceptional curves over L_{1} and hence the conditions for nonexistence of a terminal contraction are precisely those given by (16).

We know that 2E_{2}^{Y_{1}} is Cartier. Consider the index 1 cover of E_{2}^{Y_{1}}

π : Ỹ_{1} ⟶ Y_{1}.

We will show that F Ỹ_{1} ; 1 is ℚ-Cartier over L_{1} and therefore F_{1}, and hence E_{1}^{Y_{1}} too, is ℚ-Cartier along L_{1}.

So now we calculate the index 1 cover of E_{2}^{Y_{1}}. By definition [Ko-Mo98],

Ỹ_{1} = Spec(O_{Y_{2}} ⊕ I_{E_{2}^{Y_{1}},Y_{1}})

It is easy to see now that Ỹ_{1} is given by

(18) x^{2} + y^{2}uw + 2u^{2} + xuwψ(uw, xw^{2}) + ax^{2}w^{2}

+ bx^{k+1}w^{2k} + \frac{1}{xw^{2}} ϕ_{≥2}(xyw^{2}, uw, xw^{2}) = 0

and π_{1} is given by t = w^{2} and uw = z. Moreover,

F Ỹ_{1} ; 1 : u = x = 0

and therefore, since t^{2} does not appear in ϕ_{≥2}(y, z, t), it is clear that it is ℚ-Cartier along π_{1}^{−1}(L_{1}) : x = u = w = 0. The claim now follows easily.

Therefore, a terminal contraction does not exist iff (16) are satisfied. Let a_{i,j,k} be the coefficient of y^{i}z^{j}t^{k} in ϕ_{≥2}(y, z, t). Then f(0, 0) = a_{0,2,1}, \frac{∂ϕ_{2}}{∂t} (0, 0, 0) = a_{0,0,4}, \frac{∂ϕ_{2}}{∂y} (0, 0, 0) = a_{1,0,2}, ϕ_{2}(0, 0, 0) = a_{0,0,1} and ϕ_{1}(0, 0) = a_{0,1,2}. Moreover, ab = 0. Substitute these to (16) to get the conditions claimed by the statement of the theorem.

To find the index of the singularities of W in the case that it is terminal, we will do the same as in the proof of [Tzi02, Theorem 5.1] and example 1.6.

Let S be the general hyperplane section of X contaning Γ and S′ its birational transform in Y. Then by [Tzi02] it follows that S′ has exacty one singular point that is A_{4}. So Y has an A_{4} point at the generic point of L and hence 5E_{1} and (5E_{2} are Cartier at the generic point of L. Again as in the proof of [Tzi02, Theorem 5.1] we conclude that

index(E_{1}^{Z}) = index(E_{2}^{Z}) = 5.

Note that the above arguments show that Z = Z′ and E_{2}^{Z} ≅ ℙ^{2}. Let a > 0 such that

(19)  K_{Z} = g′^{∗}K_{W} + aE_{2}^{Z}.

Let l ⊂ E_{2}^{Z} ≅ ℙ^{2} be a line. Then l · E_{2}^{Z} = −b/5, with b a positive integer. We want to find b.

(20)  l · (E_{1}^{Z} + 2E_{2}^{Z}) = l · g^{∗}(E_{1} + 2E_{2}) = −1.

Moreover, l · E_{1}^{Z} = 1/5(5E_{1}^{Z}) · l. This is easily computable. I claim that the scheme theoretic intersection of 5E_{1}^{Z} and E_{2}^{Z} is just 3L. Then it is clear that for general l, l · E_{1}^{Z} = 3/5. To see the claim it suffices to work at the generic point of L. Then as we explained earlier, Z is an A_{4} surface singularity there and E_{1}^{Z}, E_{2}^{Z} are two lines through the singular point. Moreover, by [Tzi02, Theorem 4.1, Proposition 4.6], it is possible to write at the generic point of L, Z as xy − z^{5} = 0, E_{1}^{Z} is given by x − z = y − z^{4} = 0, and E_{2}^{Z} by x − z^{2} = y − z^{3} = 0. Now it is not difficult to see that length(5E_{1}^{Z} ∩ E_{2}^{Z}) = 3 (at the generic point of L), and hence 5E_{1}^{Z} ∩ E_{2}^{Z} = 3L. Therefore from (20) it follows that

l · E_{2}^{Z} = 1/2(−1 − 3/5) = −4/5.

Now from (19) it follows that a = 5/4, and therefore W has index 4.

Now we want to remove the technical condition that we imposed on the equation of X. So, suppose that the monomials yt or t^{2} appear in ϕ_{≥2}(y, z, t). Then write the equation of X as

x^{2} + y^{2}z + xz^{2} + yt^{2} + tψ(x, y, z, t) = 0.

We can now deform X to X_{0} given by

x^{2} + y^{2}z + xz^{2} + yt^{2} = 0.

Let X ⟶ Δ be the deformation. Let ⋄ ⊂ X be the plane x = y = t = 0. Let Y ⟶^{{} X be the blow up of X along Π. Then it is easy to see that Y ⟶ Δ is a deformation of Y = BX_{Γ} with special fiber Y_{0} = BX_{0}_{Γ_{0}}.

Now it is easy to check that Y_{0} satisfies the conditions of theorem 1.3. Therefore there exist a common ℚ-factorialisation Z ⟶ Y of E_{1}, E_{2}, and hence as in the proof of corollary 1.4, the divisorial contractions form a family, i.e., there is a morphism E ⊂ W ⟶^{g} Π ⊂ X, such that E_{0} ⊂ W_{0} ⟶^{g} Π_{0} ⊂ X_{′} is the divisorial contraction E_{0} ⊂ W_{0} ⟶ Γ_{0} ⊂ X_{0}. Therefore W deforms to W_{0}. Hence if W_{0} is terminal, then by [Nak98] W is also terminal. We will now show that indeed W_{0} is terminal. Hence in order to study when there is no terminal contraction, it is no restriction to assume that there is no yt or t^{2} in ϕ_{≥2}(y, z, t) as we did. At this point I would like to mention that all the arguments so far work for the general D_{2n+1} case. However I do not know how to show that W deforms to W_{0} for n ≥ 4. I believe that a more carefull look at theorem 1.3 will treat the general case but the amount of calculations involved exceed the value of the result.

To show that W_{0} is terminal. Observe that the equation of X_{0} is equivalent to

x^{2} + y^{2}z + xz^{2} + yt^{2} + yt^{3} = 0.

Therefore it is possible to deform X_{0} to X′_{0} given by

x^{2} + y^{2}z + xz^{2} + yt^{3} = 0.

Again theorem 1.3 applies and hence W_{0} deforms to W′_{0}. But now X′_{0} satisfies the conditions imposed in the beginning of the proof of theorem 2.3 and the equations (16) show that W′_{0} is terminal. Hence W_{0} is terminal too. A similar argument works for the case that t^{2} appears.

Now the general section Σ of X through 0 is given by z = ax + by + ct. If one looks carefully through the calculations in the proof of theorem 2.3, it can be seen that the equations (16) are exactly the conditions for Σ not to be normal. Hence we get corollary 2.4. It is unfortunate that the only proof I know of this result is a computational one. □

Proof of corollary 2.6. Let Σ′ ⟶^{f} Σ be a birational morphism such that Σ′ is normal, −K_{Σ′} is f-ample, and Σ′ admits a terminal 1parameter deformation. Then there is a morphism Def(Σ′) ⟶^{ϕ} Def(Σ) [Ko-Mo92, Prop. 11.4]. Let Z ⊂ Def(Σ) be the union of the images of all such morphisms. We may consider X as a deformation of Σ along Γ, say over a base Δ. Let W ⟶^{g} X be the canonical contraction that always exists [Tzi02]. Then W can be considered as a deformation of Σ′ = g^{∗}Σ. If W is terminal, then we showed that Σ′ is normal. Hence Im[Δ ⟶ Def(Σ)] ⊂ Z. Conversely, if Im[Δ ⟶ Def(Σ)] ⊂ Z, then there is a morphism Σ′ ⟶ Σ that lifts to a morphism W ⟶ X, where W is a terminal deformation of Σ′, and is the required contraction. □

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Max Planck Institute for Mathematics, Vivatsgasse 7, Bonn 53111, Germany

E-mail address: tziolas@mpim-bonn.mpg.de