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Abstract
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It is known that there exists a Mori dream space such that the Mori chamber
decomposition of its effective cone is strictly finer than the stable base locus
decomposition. In other words, stable base loci of line bundles do not contain enough
information to separate different Mori chambers in general. Here we show, however, that
different Mori chambers can be separated if the scheme structures of the base loci are
taken into account. More precisely, we show that for any two distinct Mori chambers
and
, the
asymptotic order of vanishing along some divisorial valuation is linear on
and
on
respectively, but not simultaneously on their union
. Two
toric examples are given to illustrate our result: the first one exhibits two adjacent
Mori chambers where the base schemes have the same underlying set but different
embedded components; the second one shows that it is not always possible to
separate two adjacent Mori chambers by the asymptotic order of vanishing along an
associated component of the base schemes.
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Keywords
Mori dream space, Mori chamber, stable base locus,
asymptotic order of vanishing
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Mathematical Subject Classification 2010
Primary: 14C20
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Milestones
Received: 30 January 2019
Revised: 12 August 2019
Accepted: 20 August 2019
Published: 18 January 2020
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