Threefolds with big and nef anticanonical bundles II

Jahnke, Priska; Peternell, Thomas; Radloff, Ivo

doi:10.2478/s11533-011-0023-1

Cited by 37 publications

(83 citation statements)

References 14 publications

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“…Thanks to recent work of various authors -including Arap-Cutrone-Marshburn [2], Blanc-Lamy [8], Cutrone-Marshburn [18], Jahnke-Peternell-Radloff [40,41], Kaloghiros [45] and Takeuchi [93] -class (a) is known to consist of over 150 distinct deformation classes of semi-Fano 3-folds; many of these can be obtained by blowing up an appropriate smooth irreducible curve in an appropriate smooth rank one Fano 3-fold. This makes it relatively straightforward to determine many of the basic topological properties of such weak Fano 3-folds.…”

Section: Introductionmentioning

confidence: 99%

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

et al. 2013

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We prove the existence of asymptotically cylindrical (ACyl) Calabi-Yau 3-folds starting with (almost) any deformation family of smooth weak Fano 3-folds. This allow us to exhibit hundreds of thousands of new ACyl Calabi-Yau 3-folds; previously only a few hundred ACyl Calabi-Yau 3-folds were known. We pay particular attention to a subclass of weak Fano 3-folds that we call semi-Fano 3-folds. Semi-Fano 3-folds satisfy stronger cohomology vanishing theorems and enjoy certain topological properties not satisfied by general weak Fano 3-folds, but are far more numerous than genuine Fano 3-folds. Also, unlike Fanos they often contain P 1 s with normal bundle O(−1) ⊕ O(−1), giving rise to compact rigid holomorphic curves in the associated ACyl Calabi-Yau 3-folds.We introduce some general methods to compute the basic topological invariants of ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds, and study a small number of representative examples in detail. Similar methods allow the computation of the topology in many other examples.All the features of the ACyl Calabi-Yau 3-folds studied here find application in [17] where we construct many new compact G 2 -manifolds using Kovalev's twisted connected sum construction. ACyl Calabi-Yau 3-folds constructed from semi-Fano 3-folds are particularly well-adapted for this purpose.

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Section: Introductionmentioning

confidence: 99%

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

et al. 2013

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“…(vi) Case (g, d) = (8,9) corresponds to a Sarkisov link of type I to a fibration in del Pezzo surfaces of degree 5 (after one flop): see [17,Proposition 6.5(25)].…”

Section: Existence Of Sarkisov Linksmentioning

confidence: 99%

“…Recently, several articles appeared which contain lists of numerical possibilities for Sarkisov links between threefolds. In a series of two papers [16,17], Jahnke, Peternell and Radloff embark on the classification of smooth threefolds X with big and nef. (but not ample) anticanonical divisor, and Picard number equal to 2.…”

Section: Introductionmentioning

confidence: 99%

“…There are several cases depending on the type of the Mori contractions on X and X . In [17] are classified all possible links where at least one of the contractions is of fibring type (conic bundle or del Pezzo fibration). In the paper by Cutrone and Marshburn [4], the classification is completed with a list of all possible links where both contractions are divisorial.…”

Section: Introductionmentioning

confidence: 99%

“…As a final remark, note that the truly new result in this paper lies in the last five lines in Theorem 1.1, about the generic existence of these curves. Indeed, it would be possible to recover the possible (g, d) from the papers [4,17] cited above. However, we shall derive the bound on g and d starting from scratch, and it is interesting to keep in mind that both conditions (i) and (ii) are improvements on the following easy estimates.…”

Section: Introductionmentioning

confidence: 99%

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Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

Blanc

Lamy

2012

Proceedings of the London Mathematical Society

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We characterize smooth curves in P 3 whose blow-up produces a threefold with anticanonical divisor big and nef.These are curves C of degree d and genus g lying on a smooth quartic, such that (i) 4d − 30 g 14 or (g, d) = (19, 12), (ii) there is no 5-secant line, 9-secant conic nor 13-secant twisted cubic to C. This generalizes the classical similar situation for the blow-up of points in P 2 .We describe then Sarkisov links constructed from these blow-ups, and are able to prove the existence of Sarkisov links which were previously only known as numerical possibilities.We obtain in fact a more precise result, which is our main theorem below. Here, we denote by H S g,d the Hilbert scheme of smooth irreducible curves of genus g and degree d in P 3 . We always work over the base field C of complex numbers.Theorem 1.1 (see Table 1). Let C ∈ H S g,d be a curve of genus g and degree d. We denote by X the blow-up of P 3 along C and by −K X its anticanonical divisor. Consider the sets A 0These pairs form a partition of all (g, d) corresponding to non-empty H S g,d with 4d − 30 g 14 or (g, d) = (19, 12), and we have the following characterizations. or (g, d) ∈ A 2 and there is no 4-secant line to C;(ii) The variety X is weak Fano, that is, −K X is big and nef., if and only if one of the following condition holds:and there is no 4-secant line to C; (c) (g, d) ∈ A 3 , there is no 5-secant line to C, and C is contained in a smooth quartic; (d) (g, d) ∈ A 4 , there is no 5-secant line, 9-secant conic, nor 13-secant twisted cubic to C, and C is contained in a smooth quartic. Moreover, Condition (i) corresponds to a non-empty open subset of H S g,d if (g, d) ∈ A 2 , and Condition (ii) corresponds to a non-empty open subset of H S g,d if (g, d) ∈ A 3 ∪ A 4 . Furthermore, for a general curve in these sets, there are finitely many irreducible curves intersecting trivially K X , except for (g, d) ∈

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On the existence of almost Fano threefolds with del Pezzo fibrations

Fukuoka

2016

Math. Nachr.

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Threefolds with big and nef anticanonical bundles II

Cited by 37 publications

References 14 publications

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

Asymptotically cylindrical Calabi–Yau 3–folds from weak Fano 3–folds

Weak Fano threefolds obtained by blowing-up a space curve and construction of Sarkisov links

On the existence of almost Fano threefolds with del Pezzo fibrations

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