Duality of the cones of divisors and curves

Choi, Sung Rak

doi:10.4310/mrl.2012.v19.n2.a12

Cited by 7 publications

(8 citation statements)

References 12 publications

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“…B0Amp(X) = H0Amp(X) = Amp(X), while B(n − 1)Amp(X) = Big(X). The cones BqAmp(X) are open [8,Theorem 4.5], and B(n − 2)Amp(X) coincides with the interior of the cone of mobile divisors:…”

Section: Remark 6 It Is Easily Seen Thatmentioning

confidence: 99%

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Some Remarks on Cones of Partially Ample Divisors

LATERVEER¹

2016

Kyushu J. Math.

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Abstract. We study the cones of q-ample divisors qAmp. In favourable cases, we identify a part where the closure qAmp and the nef cone have the same boundary. This is especially interesting for Fano (or almost Fano) varieties. IntroductionTotaro's landmark paper [22] has given a new impetus to the study of partially ample divisors. Let X be a smooth projective complex variety of dimension n, and let L on X be a line bundle. We recall that L is called q-ample if for every coherent sheaf F there exists an integer m 0 such that (X) in N 1 (X) (the space of R-divisors modulo numerical equivalence). We thus obtain a series of conesWhile the ample cone Amp(X) and the cone (n − 1)Amp(X) are fairly well understood, the intermediate cones qAmp(X) for 0 < q < n − 1 are still quite elusive and mysterious (see for instance [22, Section 11] for some fundamental open questions). The modest goal of this paper is to identify a part of these cones qAmp. Indeed, it turns out that in favourable cases, part of the boundary of the closed cone qAmp coincides with the boundary of the nef cone. To start with, let us restrict attention to the case that is easiest to state, that of the cone of 1-ample divisors 1Amp. Let ∂Nef(X) denote the boundary of the nef cone, and let K X ∈ N 1 X denote the class of the canonical divisor. We defineto be the part of the boundary that is visible from K X ; cf. Definition 17 for the precise definition. (We note that when K X is nef, we have ∂Nef(X) visible = ∅!)This 'K X -visible part' of the boundary turns out to be closely related to the boundary of 1Amp(X). This is detailed in the following result, where Mob(X) and Big(X) denote the cone of mobile divisors and big divisors, respectively.

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Section: Remark 6 It Is Easily Seen Thatmentioning

confidence: 99%

“…The cones BqAmp (or rather, their closure) have been studied by Payne [22] and Choi [8]. It is established by Choi [8,Theorem 4.5] that the closure of BqAmp(X) can be described in terms of the diminished base locus: [20]) Let X be a smooth projective variety. For any 0 ≤ q ≤ n − 1, there are inclusions of cones…”

Section: Remark 6 It Is Easily Seen Thatmentioning

confidence: 99%

Some Remarks on Cones of Partially Ample Divisors

LATERVEER¹

2016

Kyushu J. Math.

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“…Hence it is a contradiction. Sommese [21] proved that for any semiample (not necessarily big) line bundles, the conditions (4), (5) and cohomological k-ampleness condition are equivalent.…”

Section: Definition 31mentioning

confidence: 99%

On partially ample divisors

Choi

2014

Mathematische Nachrichten

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“…Nevertheless, Debarre and Lazarsfeld have asked whether one can formulate a duality statement for movable divisors and mov 1 -curves. This has been accomplished for toric varieties in [Pay06] and for Mori Dream Spaces in [Cho10] by taking other birational models of X into account. Our main theorem proves an analogous statement for all smooth varieties.…”

Section: Introductionmentioning

confidence: 99%