On images of Mori dream spaces

Okawa, Shinnosuke

doi:10.1007/s00208-015-1245-5

Cited by 71 publications

(55 citation statements)

References 30 publications

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“…We first treat the case when X is Q-factorial. By [27,Appendix] by Jinhyung Park, X is a Mori dream space, therefore we see that Y is a (non-Q-factorial) Mori dream space by [36,Theorem 9.3]. Moreover, by [1,Theorem 2.3] there is a small Q-factorialization Y ′ → Y .…”

Section: Proof Of Theorem 15 and Theorem 16mentioning

confidence: 95%

On minimal model theory for log abundant lc pairs

Hashizume

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Under the assumption of the minimal model theory for projective klt pairs of dimension n, we establish the minimal model theory for lc pairs (X/Z, ∆) such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most n. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling.Date: 2019/08/25, version 0.56. 2010 Mathematics Subject Classification. 14E30.

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Section: Proof Of Theorem 15 and Theorem 16mentioning

confidence: 95%

On minimal model theory for log abundant lc pairs

Hashizume

2019

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…The cone C(Cox(X)) is divided into some chambers and each chamber corresponds to a projective variety which is birational to X (cf. Hu-Keel [8], Okawa [16], Laface-Velasco [13]). In order to define a chamber, Laface-Velasco studied the ideal generated by elements with degree in a given ray, i.e., for a ∈ Z n , J a (Cox(X)) = the ideal of Cox(X) generated by ∪ r>0 Cox(X) ra .…”

Section: Introductionmentioning

confidence: 99%

Demazure Construction for ℤn-Graded Krull Domains

2018

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For a Mori dream space X, the Cox ring Cox(X) is a Noetherian Z ngraded normal domain for some n > 0. Let C(Cox(X)) be the cone (in R n ) which is spanned by the vectors a ∈ Z n such that Cox(X) a = 0. Then C(Cox(X)) is decomposed into a union of chambers. Berchtold and Hausen [2] proved the existence of such decompositions for affine integral domains over an algebraically closed field. We shall give an elementary algebraic proof to this result in the case where the homogeneous component of degree 0 is a field.Using such decompositions, we develop the Demazure construction for Z ngraded Krull domains. That is, under an assumption, we show that a Z n -graded Krull domain is isomorphic to the multi-section ring R(X; D 1 , . . . , D n ) for certain normal projective variety X and Q-divisors D 1 , . . . , D n on X.2010 Mathematics Subject Classification: 13A02, 14E99. 1 Some people call it the Dolgachev-Pinkham-Demazure construction.1 Proposition 2.4. Let A be a Z n -graded ring.

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“…Let u = e 1 , v = e 2 and w = −4u − 2v + 2a 1 + a 2 + a 3 − a 8 + a 10 = e 3 + e 5 + e 6 . So, we have that 26u + 15v + 7w = 11a 1 + 8a 2 + 4a 3 + a 4 + a 5 + a 6 − a 7 − 3a 8 a 9 = u a 10 = 4u + 2v + w − 2a 1 − a 2 − a 3 + a 8 Let N ′ ⊂ Z 10 be the sublattice generated by a 1 , . .…”

Section: The Moduli Space M 0nmentioning

confidence: 99%

“…. , a 8 . Then Z 10 /N ′ is generated by a 9 , a 10 , both of which can be expressed in terms of u, v, w. The vectors u, v, w satisfy the relation 26u + 15v + 7w = 0 (mod N ′ ).…”

Section: The Moduli Space M 0nmentioning

confidence: 99%