Effective log Iitaka fibrations for surfaces and threefolds

Todorov, Gueorgui

doi:10.1007/s00229-010-0370-4

Cited by 13 publications

(14 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is done in Theorem 3.1 and the hardest case there is when X itself is a ruled surface over a curve of positive genus. This completes the effectiveness of log Iitaka fibration in dimension two (for the other cases, see [Todorov 2008]). For this case, it relies on results from [Alexeev 1994] and [Alexeev and Mori 2004].…”

Section: Introductionmentioning

confidence: 53%

See 1 more Smart Citation

Effectiveness of the log Iitaka fibration for 3-folds and 4-folds

Todorov

2009

ANT

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 53%

“…Then the usual argument of cutting log canonical centers [Todorov 2008] works as long as we can prove that the volume of K W + (1−x j )b j B j + L has a uniform lower bound. Now we run the minimal model program, f :…”

Section: Effectiveness Of the Iitaka Fibrationmentioning

confidence: 99%

Effectiveness of the log Iitaka fibration for 3-folds and 4-folds

Todorov

2009

ANT

View full text Add to dashboard Cite

show abstract

“…In [7], by a different method, we found such integer which is considerably smaller than the one in [22]. For the reader convenience we present here an argument, due to Todorov [22,Theorem 3.2], valid in the general case.…”

Section: Bounding the Denominators Of The Moduli Partmentioning

confidence: 80%

“…The result was proved in [22,Theorem 3.2] when the fibre is a rational curve. In [7], by a different method, we found such integer which is considerably smaller than the one in [22].…”

Section: Bounding the Denominators Of The Moduli Partmentioning

confidence: 97%

“…Moreover Todorov and Todorov-Xu prove some unconditional uniformity results for the Iitaka fibration of varieties of Kodaira dimension at most 2 and Kodaira codimension 1 (Todorov [22,Theorem 1.2]) and 2 (Todorov and Xu [23,Theorem 1.2]). Their proofs relie on the existence of a bound on the denominators of the moduli part and, for the case of Kodaira codimension one, on the result by Prokhorov and Shokurov that proved conjecture EbS for all k in the case where the fibres are curves (see [20,Theorem 8.1]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Inductive Approach to Effective b-Semiampleness

Floris

2012

International Mathematics Research Notices

View full text Add to dashboard Cite

Abstract. An lc-trivial fibration is the data of a pair (X, B) and a fibration such that (K X + B)| F is torsion where F is the general fibre. For such a fibration we have an equality K X + B + 1/r(ϕ) = f * (K Z + B Z + M Z ) where B Z is the discriminant of f , M Z is the moduli part and ϕ is a rational function. It has been conjectured by Prokhorov and Shokurov that there exists an integer m = m(r, dim F ) such that mM Z ′ is base-point-free on some birational model Z ′ of Z. In this work we reduce this conjecture to the case where the base Z has dimension one. Moreover in the case where M Z ≡ 0 we prove the existence of an integer m, that depends only on the middle Betti number of a canonical covering of the fibre, such that mM Z ∼ 0. IntroductionThe canonical bundle formula is a tool for studying the properties of the canonical bundle of a variety X, such that there exists a fibration f : X → Z, in terms of the properties of the canonical bundle of Z, of the singularities of the fibration and of the birational variation of the fibres of f .More precisely we consider a pair (X, B) such that there exists an lc-trivial fibration f : (X, B) → Z, that is a fibration such that (K X + B)| F is a torsion divisor, where F is a general fibre (see Section 2 for a complete definition). Then we can writewhere ϕ is a rational function and r is the minimum integer such that r(K X + B)| F ∼ 0. The divisor B Z is called the discriminant and it is defined in terms of some log-canonical thresholds with respect to the pair (X, B). Precisely we have B Z = (1 − γ p )p whereThe divisor M Z , called the moduli part, is a Q-Cartier divisor and it is nef on some birational modification of Z by [2, Theorem 0.2]. The moduli part should be related to the birational variation of the fibres of f . This is true for instance in the case of elliptic fibrations. Indeed, if f : X → Z is an elliptic fibration, we have Kodaira's canonical bundle formulaDate: July 30, 2017.where j : Z → P 1 = M 1 is the application induced by f to the moduli space of elliptic curves. In particular M Z is semiample.In [20] Prokhorov and Shokurov state the following conjecture (in our statement we specify the dimension of the base).EbS(k) 1.1 (Effective b-Semiampleness, Conjecture 7.13.3, [20]). There exists an integer number m = m(d, r) such that for any lc-trivial fibration f : (X, B) → Z with dimension of the generic fibre F equal to d, dimension of Z equal to k and Cartier index of (F, B| F ) equal to r there exists a birational morphism ν : Z ′ → Z such that mM Z ′ is base point free.The relevance of the above conjecture is well illustrated for instance by a result due to X. It is worth noticing that the proofs of semiampleness in these cases use the existence of a moduli space for the fibres.The main goal of this work is to develop an inductive approach to the conjecture EbS. Our first result is the following. Theorem 1.2. EbS(1) implies EbS(k).An inductive approach on the dimension of the base as in Theorem 1.2 allows us to prove a result of effective semiam...

show abstract

On the pluricanonical maps of varieties of intermediate Kodaira dimension

Jiang

2012

Math. Ann.

View full text Add to dashboard Cite

show abstract

Effective log Iitaka fibrations for surfaces and threefolds

Cited by 13 publications

References 21 publications

Effectiveness of the log Iitaka fibration for 3-folds and 4-folds

Effectiveness of the log Iitaka fibration for 3-folds and 4-folds

Inductive Approach to Effective b-Semiampleness

On the pluricanonical maps of varieties of intermediate Kodaira dimension

Contact Info

Product

Resources

About