2018
DOI: 10.4310/mrl.2018.v25.n3.a3
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Iitaka’s $C_{n,m}$ conjecture for $3$-folds in positive characteristic

Abstract: In this paper, we prove that for a fibration f : X → Z from a smooth projective 3-fold to a smooth projective curve, over an algebraically closed field k with chark = p > 5, if the geometric generic fiber X η is smooth, then subadditivity of Kodaira dimensions holds, i.e. κ(X) ≥ κ(X η ) + κ(Z).

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Cited by 10 publications
(22 citation statements)
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“…Applying the proof of [, Theorem 2.8] we can show that fscriptOXfalse(m(KX/Y+B)false) contains a nef sub‐bundle of rank cm for some c>0 and any sufficiently divisible m>0. Then the assertion follows from the arguments of [, Section 4, Step 1–4] by replacing KX with KX+B. In case (iii), combining results of [, Corollary 2.23], we see that all the conditions of [, Theorem 1.4] are satisfied, hence the assertion follows.…”
Section: Preliminariesmentioning
confidence: 72%
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“…Applying the proof of [, Theorem 2.8] we can show that fscriptOXfalse(m(KX/Y+B)false) contains a nef sub‐bundle of rank cm for some c>0 and any sufficiently divisible m>0. Then the assertion follows from the arguments of [, Section 4, Step 1–4] by replacing KX with KX+B. In case (iii), combining results of [, Corollary 2.23], we see that all the conditions of [, Theorem 1.4] are satisfied, hence the assertion follows.…”
Section: Preliminariesmentioning
confidence: 72%
“…So KX+B is nef, and false(KX+Bfalse)|Xη¯ is semi‐ample (Theorem (3.2, 3.4)) and induces an elliptic fibration Iη¯:Xη¯Cη¯ to a normal curve Cη¯. Applying the proof of [, Theorem 2.8] we can show that fscriptOXfalse(m(KX/Y+B)false) contains a nef sub‐bundle of rank cm for some c>0 and any sufficiently divisible m>0. Then the assertion follows from the arguments of [, Section 4, Step 1–4] by replacing KX with KX+B.…”
Section: Preliminariesmentioning
confidence: 90%
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“…Up to now, the following have been proved (i) W C n,n−1 and C 2,1 ([10]); (ii) W C 3,1 (overF p , p > 5 by [6], over general k with char k > 5 by [15] and [16]); (iii) C 3,1 under the situation that K Xη is big, g(Y ) > 1 and char k > 5 ( [36]).…”
Section: Introductionmentioning
confidence: 99%