ZU′
oo
XU oo XU′
commutes, the H-orbit on ZU′ maps to a H-orbit on ZU. Thus we have a natural map
F × U′ ∼= ZU′ /H → ZU/H = XU.
The morphism F × U′ → XU factors through the fiber product XU′ → XU. Since both mor-phisms are finite with the same degree, we have XU′ ∼= F × U′. Let G = Gal(C′/C) = Gal(U′/U ), then there exists a G-action on XU′ such that XU′ /G ∼= XU. Thus X is birational to (F × C′)/G.
Now we have the étale covering Z0 ∼= ( ¯F × C′)/G → (F × C′)/G. Since Z0 is terminal, we have (F × C′)/G is terminal. Replacing X by (F × C′)/G one may assume that our given bielliptic fibration is isotrivial. In this case using the same argument as in the proof of Proposition 4.3.2 one can show that|mKX| is birational to the Iitaka fibration if m ≥ 86 and is divisible by 12, or equivalently, m≥ 96 and divisible by 12.
4.5 Boundedness of Iitaka fibration for Kodaira dimension one
Theorem 4.5.1. Let X be a smooth complex projective threefold of Kodaira dimension one.
Then|mKX| defines the Iitaka fibration if m ≥ 96 and is divisible by 12. More precisely, let F be a general fiber of the Iitaka fibration of X, we have
1. If F is birational to a K3 surface, then|mKX| defines the Iitaka fibration if m ≥ 86.
2. If F is birational to an Enriques surface, then|mKX| defines the Iitaka fibration if m ≥ 42 and is even.
3. If F is birational to an abelian surface, then|mKX| defines the Iitaka fibration if m ≥ 86.
Moreover, assume the Iitaka fibration is not isotrivial, then |mKX| defines the Iitaka fibration if m = 2 or m≥ 4.
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4. If F is birational to a bielliptic surface, then|mKX| defines the Iitaka fibration if m ≥ 96 and is divisible by 12.
Proof. If C is not rational, this follows from Proposition 4.1.9. The K3 or Enriques cases follow by Proposition 4.2.2. The isotrivial abelian fibration case follows from Proposition 4.3.2 and the non-isotrivial abelian fibration case follows from Proposition 4.3.7. The bielliptic case follows from Proposition 4.4.2.
Combining the work of J. A. Chen-M. Chen [CM14] and Ringler [Rin07], we have the following effective bound for threefolds of positive Kodaira dimension.
Corollary 4.5.2. Let X be a smooth complex projective threefold of positive Kodaira dimension.
Then|mKX| defines the Iitaka fibration if m ≥ 96 and is divisible by 12.
We remark that we do not know whether our estimate is optimal or not. However, as in Example 4.5.6, one can construct a threefold of Kodaira dimension one, such that|iKX| is not birational to the Iitaka fibration for all i < 42. Since the optimal value of the Iitaka fibration for threefolds of Kodaira dimension one should be divisible by 12, we have the following estimate.
Corollary 4.5.3. If m is the smallest integer such that|mKX| is birational to the Iitaka fibration for all smooth projective threefold of Kodaira dimension one, then 48≤ m ≤ 96.
In the remaining part we will compute several examples.
Example 4.5.4. The first example is a trivial example. Let F be a bielliptic curve such that
|6KF| is non-empty but |iKF| is empty for all i ≤ 5 and let C be a curve of general type. Then X = F × C is a smooth threefold of Kodaira dimension one such that |6KX| defines the Iitaka fibration but|iKX| is empty for all i ≤ 5.
Example 4.5.5. Let E be an elliptic curve. Pick two different points P and Q on E. One can find a line bundle L such that L2 =OE(P + Q). Let C be the cyclic cover corresponds to L2. Then C is a curve of genus two and ϕ : C → E is a double cover ramified at P and Q. Let G = Aut(C/E), which is a cyclic group of order two and let F be an abelian surface. One can define a G-action on F via−Id. Let X = (F × C)/G.
The singular points of X are of the type 12(1, 1, 1), hence X has terminal singularities. We want to show that|4KX| defines the Iitaka fibration, and |iKX| does not define the Iitaka fibra-tion for i≤ 3. One has
H0(X, mKX) = H0(F × C, mKF mKC)G = H0(C, mKC)G
doi:10.6342/NTU201801712 since the unique section in H0(F, mKF) is fixed by G for all m. To compute H0(C, mKC)G,
note that ϕ∗OC =OE⊕L−1andOC(2KC) = ϕ∗OE(2KE+P +Q) = ϕ∗L2, hence ϕ∗OC(2kKC) = L2k ⊕ L2k−1 by the projection formula. The G-invariant part of H0(C, 2kKC) is H0(E, L2k) and L2k is very ample if and only if k ≥ 2. Hence |2KX| does not define the Iitaka fibration, but|4KX| does.
On the other hand, by Grothendieck duality we have
ϕ∗(2k + 1)KC = ϕ∗HomOC(−2kKC, KC) ∼= HomOE(ϕ∗(−2kKC), KE) ∼= L2k⊕ L2k+1.
This shows that h0(C, 3KC)G = h0(E, L2) = 2 and hence |3KX| do not define the Iitaka fibration.
We remark that this is the worst example we know for abelian fibrations.
Example 4.5.6. Let C be the Klein quartic
(x3y + y3z + z3x = 0)⊂ P2.
It is known that
|G| = |Aut(C)| = 168 = 42(2g(C) − 2) (c.f. [Dol12, Section 6.5.3]). Let
F = (x3y + y3z + z3x + u4 = 0) ⊂ P3,
which is a K3 surface. Define the G-action on F by g([x : y : z : u]) = [g([x : y : z]) : u]. Let X = (F × C)/G. We will prove the following:
(1) X has terminal singularities.
(2) H0(X, iKX)≤ 1 for i ≤ 41 and H0(X, 42KX) = 2.
Hence the smooth model of X is a threefold of Kodaira dimension one, such that|42KX| defines the Iitaka fibration, but|iKX| do not define the Iitaka fibration for i ≤ 41.
First we prove (1). Since |Aut(C)| = 168 = 42(2g(C) − 2), it is well-known that the morphism C → C/G ramified at three points P2, P3and P7 ∈ C/G and the stabilizer of points over Pr is a cyclic group of order r for r = 2, 3 and 7 (cf. also [Elk99, Proposition in Section 2.1]). Let Fr ⊂ X be the fiber over Pr. We need to compute the singularities of Fr.
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(i) r = 7. Note that any order 7 subgroup of G is a Sylow-subgroup, which is unique up to conjugation. To compute the singularities we may assume the stabilizer is the cyclic group generated by the element (please see [Elk99, Section 1.1] for the description of elements in G) and 17(1, 6) respectively. The conclusion is that X has three singular points over P7which are cyclic quotient points of the form 17(1, 6, 1), 17(2, 5, 1) and 17(3, 4, 1) respectively.
(ii) r = 3. As before any order 3 subgroup of G is a Sylow-subgroup and hence we may assume the stabilizer is generated by coordinates near P and the defining equation of F3near P can be written as
(1 + 3ω)α + higher order terms,
doi:10.6342/NTU201801712 singularity of P ∈ X is a terminal cyclic quotient13(1, 2, 1). A similar computation (simply interchange ω and ω2in the calculation) shows that P′ = [1 : ω2 : ω : 0]∈ F3 ⊂ X is also
Using the same technique above we can say that the singularity of Qi ∈ F3is of the form
1
3(1, 2), hence Qi ∈ X is also a terminal cyclic quotient point for i = 1, ..., 4.
(iii) r = 2. Let µ∈ G be an order two element. We have to compute the singularities of F2/⟨µ⟩.
By [Elk99, Proposition in Section 2.1], µ fixes a line and a point inP2. By the character table of G (cf. [Elk99, Section 1.1]), we know that the three-dimensional character of µ is equal to−1. This implies the fixed line of µ in P2corresponds to the two-dimensional eigenspace with eigenvalue −1, and the fixed point of µ in P2 corresponds to the one-dimensional eigenspace with eigenvalue 1. Assume that L is the fixed line of µ in P2, L∩ C = [xi : yi : zi]i=1,...,4and the fixed point of µ inP2 is [x5 : y5 : z5]. One can check that the fixed point of µ on F2is [xi : yi : zi : 0] for i = 1, ..., 4 and [x5 : y5 : z5 : uj]j=1,...,4, where ujare the roots of the equation u4+ x35y5+ y35z5+ z35x5 = 0. The conclusion is that
74
there are eight cyclic quotient points of indices two on F2. Since they are isolated, all the singular points should be the from 12(1, 1). The conclusion is that there are eight singular points on X which is of the from 12(1, 1, 1).
Note that the Iitaka fibration of X is a K3 fibration, and the basket data of X is
{(2, 1) × 8, (3, 1) × 6, (7, 1), (7, 2), (7, 3)}.
It is the worst case in Section 4.2.
Now we prove (2). We need to compute H0(X, mKX) = H0(F × C, mKF mKC)G. Consider the long exact sequence
0→H0(P3, mKP3 + (m− 1)F ) → H0(P3, m(KP3 + F )) → H0(F, mKF)→ H1(P3, mKP3 + (m− 1)F ) → · · · .
Since Hi(P3, mKP3 + (m− 1)F ) = 0 for i = 0, 1, we have
H0(F, mKF) = H0(P3, m(KP3 + F )) = H0(P3,OP3).
Thus any section in H0(F, mKF) is G-invariant. This tell us that
H0(X, mKX) = H0(F × C, mKF mKC)G= H0(C, mKC)G.
One can consider the following long exact sequence
0→H0(P2, mKP2 + (m− 1)C) → H0(P2, m(KP2 + C))→ H0(C, mKC)→ H1(P2, mKP2 + (m− 1)C) → · · · .
Since H1(P2, mKP2 + (m− 1)C) = 0, the restriction map
H0(P2, m(KP2 + C)) = H0(P2,OP2(m))→ H0(C, mKC)
is surjective. Thus to find G-invariant sections in H0(C, mKC) is equivalent to find G-invariant polynomials of degree m on C. It is known that (c.f. [Elk99, Section 1.2]) the G-invariant polynomials are generated by three elements f6, f14and f21, where fdis a polynomial of degree
doi:10.6342/NTU201801712 d, satisfying f212 = f143 − 1728f67. Hence h0(C, iKC)G≤ 1 for all i ≤ 41 and H0(C, 42KC)Gis
spanned by f67and f143 . Thus h0(X, iKX)≤ 1 for i ≤ 41 and h0(X, 42KX) = 2.
76
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