Iitaka’s $C_{n,m}$ conjecture for $3$-folds in positive characteristic

2018

J. London Math. Soc.

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“…Applying the proof of [, Theorem 2.8] we can show that

f_{*} {scriptO}_{X} false(m (K_{X / Y} + B) false)

contains a nef sub‐bundle of rank

⩾ c m

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

K_{X}

with

K_{X} + 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

K_{X} + B

is nef, and

false(K_{X} + B false) |_{X_{\bar{η}}}

is semi‐ample (Theorem (3.2, 3.4)) and induces an elliptic fibration

I_{\bar{η}} : X_{\bar{η}} \to C_{\bar{η}}

to a normal curve

C_{\bar{η}}

. Applying the proof of [, Theorem 2.8] we can show that

f_{*} {scriptO}_{X} false(m (K_{X / Y} + B) false)

contains a nef sub‐bundle of rank

⩾ c m

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

K_{X}

with

K_{X} + B

.…”

Section: Preliminariesmentioning

confidence: 90%

“…In case (i), by Theorem (3.3)

K_{X / Y} + B \sim_{double-struckQ} f^{*} A

, and

A

is nef by [, Theorem 1.5]. If

g (Y) > 1

then the assertion follows by

κ false(X, K_{X} + B false) = κ false(Y, K_{Y} + A false) = 1

; and if

g (Y) = 1

then the assertion follows by [, Theorem 3.2]. In case (ii), by running a relative LMMP over

Y

, we may assume

K_{X} + B

f

‐nef.…”

Section: Preliminariesmentioning

confidence: 99%

“…Remark For the condition (ii) of the theorem above, the reason why we assume

K_{X_{\bar{η}}} + B_{\bar{η}}

induces an elliptic fibration lies in that, to show

f_{*} {scriptO}_{X} false(m (K_{X / Y} + B) false)

contains a nef sub‐bundle, the proof of [, Theorem 2.8] uses ‘canonical bundle formula’: for a fibration

h : X \to Z

, if

K_{X / Z}

double-struckQ

‐trivial over

Z

, then

K_{X / Z} \sim_{double-struckQ} h^{*} Δ

for some effective

double-struckQ

‐divisor

normalΔ

Z

, which is true for elliptic fibrations by the following theorem.…”

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

Abundance for non‐uniruled 3‐folds with non‐trivial Albanese maps in positive characteristics

2018

J. London Math. Soc.

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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%

A Note on Iitaka's Conjecture $C_{3,1}$ in Positive Characteristics

2017

Taiwanese J. Math.

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Subadditivity of Kodaira dimensions for fibrations of three-folds in positive characteristics

Advances in Mathematics

2019

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In this paper, we will prove subadditivity of Kodaira dimensions for a fibration with possibly singular geometric generic fiber, under certain nefness and relative semi-ampleness conditions. As an application, for a fibration f : X → Y of a smooth projective threefold over an algebraically closed field of characteristic p > 5, under the assumption that Y is of general type and non-uniruled, we prove subadditivity of Kodaira dimensions when general fibers are smooth or when K X/Y is relatively big over Y .