Explicit birational geometry of threefolds of general type, I

Chen, Jungkai Alfred; Meng, Chen

doi:10.24033/asens.2124

Cited by 58 publications

(122 citation statements)

References 34 publications

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“…From non-positive linearized equation forms LE 1,7 and LE 1,9 on (0, 1 2 ], we obtain p 7 = 0 and p 9 = 0 respectively.…”

Section: Definitionmentioning

confidence: 99%

“…J. A. Chen and M. Chen showed that h 0 (X, O X (nK X )) ≥ 1 for every integer n ≥ 27 and that h 0 (X, O X (24K X )) ≥ 2 and h 0 (X, O X (n 0 K X )) ≥ 2 for some integer n 0 ≤ 18 (see Chen and Chen [1,2]). …”

Section: Dong-kwan Shinmentioning

confidence: 99%

“…As an example, to express L(7) explicitly, it is enough to consider all the subintervals of (0, 1 2 ] determined by…”

Section: Rmentioning

confidence: 99%

“…A. Chen and M. Chen (see Chen and Chen [1,2]). They inspect linear combinations of p n and baskets of singularities.…”

Section: Dong-kwan Shinmentioning

confidence: 99%

“…(2) Suppose that n j=1 c j p j is expressed explicitly over the subintervals of (0, 1 2 ] determined by ED n as follows:…”

Section: Rmentioning

confidence: 99%

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On a Computation of Plurigenus of a Canonical Threefold

Shin¹

2016

Bulletin of the Korean Mathematical Society

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Abstract. For a canonical threefold X, it is known that pn does not vanish for a sufficiently large n, where pn = h 0 (X, O X (nK X )). We have shown that pn does not vanish for at least one n in {6, 8, 10}. Assuming an additional condition p 2 ≥ 1 or p 3 ≥ 1, we have shown that p 12 ≥ 2 and pn ≥ 2 for n ≥ 14 with one possible exceptional case. We have also found some inequalities between χ(O X ) and K 3 X .Throughout this paper X is assumed to be a projective threefold with only canonical singularities and an ample canonical divisor K X over the complex number field C, i.e., a canonical threefold.It is well known that H 0 (X, O X (mK X )) does not vanish and generates a birational map for a sufficiently large m. If there exists a positive integer n such that h 0 (X, O X (nK X )) ≥ 2, then by using Kollár's technique we can find the integer m which generates a birational map (see Kollár [4]. Shin [6] improved the above results. J. A. Chen and M. Chen showed that h 0 (X, O X (nK X )) ≥ 1 for every integer n ≥ 27 and that h 0 (X, O X (24K X )) ≥ 2 and h 0 (X, O X (n 0 K X )) ≥ 2 for some integer n 0 ≤ 18 (see Chen and Chen [1, 2]).Plurigenus p n of canonical threefolds were extensively studied by J. A. Chen and M. Chen (see Chen and Chen [1, 2]). They inspect linear combinations of p n and baskets of singularities. In this paper, we study also linear combinations of p n . But our approach is slightly different and includes less complex calculations. To find special linear combinations of p n , our strategy is searching linear combinations which satisfy the following (1) or (2):(1) linear combinations of p n are non-positive at every point in (0, 1 2 ]. (2) linear combinations of p n can be expressed as pure linear forms a i b+d i r of singularity type b r on some partition of (0, 1 2 ], where a i , d i are integers.From (1), we may obtain information of p n in linear combinations.

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“…From non-positive linearized equation forms LE 1,7 and LE 1,9 on (0, 1 2 ], we obtain p 7 = 0 and p 9 = 0 respectively.…”

Section: Definitionmentioning

confidence: 99%

Section: Dong-kwan Shinmentioning

confidence: 99%

“…As an example, to express L(7) explicitly, it is enough to consider all the subintervals of (0, 1 2 ] determined by…”

Section: Rmentioning

confidence: 99%

“…A. Chen and M. Chen (see Chen and Chen [1,2]). They inspect linear combinations of p n and baskets of singularities.…”

Section: Dong-kwan Shinmentioning

confidence: 99%