2007
DOI: 10.1007/s00209-007-0186-4
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Characterization of the 4-canonical birationality of algebraic threefolds

Abstract: In this article we present a 3-dimensional analogue of a well-known theorem of Bombieri (Inst Hautes Etudes Sci Publ Math 42:171-219, 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let X be a projective minimal threefold of general type with Q-factorial terminal singularities and the geometric genus p g (X ) ≥ 5. We show that the 4-canonical map ϕ 4 is not birational onto its image if and only if X is birationally fibred by a family C of irreducible curves of geometric g… Show more

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Cited by 17 publications
(27 citation statements)
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“…If either ξ = 1 or ξ = 6 5 and degg(X ) = 3, then ϕ 4,X is not birational.Proof The proof is similar in the spirit to that of Chen-Zhang[8, Proposition4.6].…”
mentioning
confidence: 86%
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“…If either ξ = 1 or ξ = 6 5 and degg(X ) = 3, then ϕ 4,X is not birational.Proof The proof is similar in the spirit to that of Chen-Zhang[8, Proposition4.6].…”
mentioning
confidence: 86%
“…We have an induced fibration f : X −→ whose general fiber is F. By [8,Lemma 4.7] and Corollary 2.5, we have π * (K X )| F ≥ 3 4 σ * (K F 0 ). Still consider the curveĈ on X with π(Ĉ) = C 0 .…”
Section: Part Iii: D 1 =mentioning
confidence: 99%
“…Following the techniques developed by M. Chen [3,7,8], we can prove the next couple of lemmas which are used to show our main theorem. The work on this section relies on a key technical result, Theorem 2.4 in [8].…”
Section: Preliminariesmentioning
confidence: 99%
“…Now l(2) = 359 120 which implies K 3 X = 1 60 . In the Case (ii) through (v), one has, respectively: 5, 3, 12), (6,4,7,21), (1,8,4,6), (2,7,8,15).…”
Section: Proof Of Main Theoremmentioning
confidence: 99%
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