The weak Lefschetz principle is false for ample cones

Hassett, Brendan; Lin, Hui-Wen; Wang, Chin-Lung

doi:10.4310/ajm.2002.v6.n1.a5

Cited by 16 publications

(24 citation statements)

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“…Remark 5 In Corollary 2, the part of the result about the preservation of nef cones has previously been obtained by Hassett et al [15,Theorem 4.1], which they called "the weak Lefschetz principle for ample cones". See also the results of Kollár [4,Appendix] and Wiśniewski [24,Theorem 2.1].…”

Section: Examplementioning

confidence: 96%

A Lefschetz hyperplane theorem for Mori dream spaces

Jow

2010

Math. Z.

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Let X be a smooth Mori dream space of dimension ≥ 4. We show that, if X satisfies a suitable GIT condition which we call small unstable locus, then every smooth ample divisor Y of X is also a Mori dream space. Moreover, the restriction map identifies the Néron-Severi spaces of X and Y , and under this identification every Mori chamber of Y is a union of some Mori chambers of X , and the nef cone of Y is the same as the nef cone of X . This Lefschetz-type theorem enables one to construct many examples of Mori dream spaces by taking "Mori dream hypersurfaces" of an ambient Mori dream space, provided that it satisfies the GIT condition. To facilitate this, we then show that the GIT condition is stable under taking products and taking the projective bundle of the direct sum of at least three line bundles, and in the case when X is toric, we show that the condition is equivalent to the fan of X being 2-neighborly.Keywords Mori dream space · Lefschetz theorem · Nef cone · Mori chamber · m-neighborly fan · Cox ring · GIT quotient IntroductionThe main purpose of this paper is to prove an analog of the Lefschetz hyperplane theorem for Mori dream spaces.Let X be a smooth complex projective variety, and let N 1 (X ) be the group of numerical equivalence classes of line bundles on X . Recall from [17] that X is called a Mori dream space if Pic(X ) Q = N 1 (X ) Q (equivalently H 1 (X, O X ) = 0), and X has a finitely generated Cox ring (Definition 1.2). As the name might suggest, Mori dream spaces are very special varieties on which Mori theory works extremely well (see the nice survey article of Hu [16]). On the other hand, not many classes of examples of them are known. It has been understood for a while that toric varieties are Mori dream spaces; indeed their Cox rings are polynomial

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Section: Examplementioning

confidence: 96%

A Lefschetz hyperplane theorem for Mori dream spaces

Jow

2010

Math. Z.

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“…The result in this case follows from [15,Lemma]. Remark 3.6 After this paper was written we found out that the result in Theorem 3.2 is also in [13,Theorem 4.3]. Note that in [13, Theorem 4.3] a slightly more general result is stated, though not proved: the fact that every ray of NE(X) in −(…”

Section: Proposition 34 ([25])mentioning

confidence: 74%

“…For r = 1 a different example can be found in [13,Section 2.1]; the example in [13,Section 3] is similar to the one in [2].…”

Section: Introductionmentioning

confidence: 99%

Connections between the geometry of a projective variety and of an ample section

Andreatta

Novelli

Occhetta

2006

Mathematische Nachrichten

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Let X be a smooth complex projective variety and let Z =(s =0) be a smooth submanifold which is the zero locus of a section of an ample vector bundle E of rank r with dim Z =dim X –r.We show with some examples that in general the Kleiman–Mori cones NE(Z) and NE(X) are different. We then give a necessary and sufficient condition for an extremal ray in NE(X) to be also extremal in NE(Z).We apply this result to the case r =1 and Z a Fano manifold of high index;in particular we classify all X with an ample divisor which is a Mukai manifold of dimension ≥4.In the last section we prove a general result in case Z is a minimal variety with 0 ≤κ (Z) <dim Z. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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“…147 Corollary]. This provided the starting-block for much further work concerning weak Lefschetz for the ample cone ( [15], [18], [1], [2], [6], [24]).…”

Section: Proofmentioning

confidence: 99%

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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The weak Lefschetz principle is false for ample cones

Cited by 16 publications

References 5 publications

A Lefschetz hyperplane theorem for Mori dream spaces

A Lefschetz hyperplane theorem for Mori dream spaces

Connections between the geometry of a projective variety and of an ample section

Some Remarks on Cones of Partially Ample Divisors

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