Tilting bundles on toric Fano fourfolds

Prabhu-Naik, Nathan

doi:10.1016/j.jalgebra.2016.09.007

Cited by 13 publications

(17 citation statements)

References 26 publications

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“…Example 2.2 (Dimension 4). The variety 𝑉 4 is exactly (116) in the enumeration of [34] or (118) in the enumeration of [12].…”

Section: Examplesmentioning

confidence: 99%

“…Throughout, we let Δ denote the fan corresponding to 𝑉 𝑛 . Note that 𝑉 2 is the del Pezzo surface 𝖽𝖯 6 of degree 6; and 𝑉 4 is the variety (116) in the enumeration of [34] or (118) in the enumeration of [12]. Any odd-dimensional centrally symmetric toric Fano variety has ℙ 1 as a factor and there are no generalized del Pezzo surfaces of odd degree.…”

Section: Introductionmentioning

confidence: 99%

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Derived categories of centrally‐symmetric smooth toric Fano varieties

Ballard

Duncan

McFaddin

2022

Mathematische Nachrichten

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“…Example 2.2 (Dimension 4). The variety 𝑉 4 is exactly (116) in the enumeration of [34] or (118) in the enumeration of [12].…”

Section: Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Derived categories of centrally‐symmetric smooth toric Fano varieties

Ballard

Duncan

McFaddin

2022

Mathematische Nachrichten

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“…After the first version of this paper was announced, Professor Alastair Craw introduced to the author of the paper [Na17] which constructs tilting bundles on every smooth toric Fano fourfolds by using the strong full exceptional collections of line bundles.…”

Section: Introductionmentioning

confidence: 99%

Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two

Lee¹

2020

Preprint

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“…Throughout, we let ∆ denote the fan corresponding to V n . Note that V 2 is the del Pezzo surface dP 6 of degree 6; and V 4 is the variety (116) in the enumeration of [PN17] or (118) in the enumeration of [Bat99]. The variety V n admits a natural (S n+1 × C 2 )-action, given by an action on the rays e i , e i .…”

Section: Introductionmentioning

confidence: 99%

On derived categories of arithmetic toric varieties

2019

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We exhibit full exceptional collections of vector bundles on any smooth, Fano arithmetic toric variety whose split fan is centrally symmetric. 1 Lemma 5.5. On W , the ideal of the contracting locus W + J := W + λ J is (y j | j ∈ J), and the ideal of the repelling locus W − J := W − λ J is (x i | i ∈ J). The ideal of the fixed locusProof. This is obvious from the definitions.In light of Theorem 4.7, we also record t ± and µ for each J.Lemma 5.6. Let t ± J := t ± λ J and µ J := µ(λ J ) for J ⊆ {0, ..., n}. We have the following:The following lemma is the key observation in proving that F n generates D b (V n ). Given a set J ⊆ {0, . . . , n}, we denote the Koszul complex associated to the set {y i | i ∈ J} by K(J).We let X ± J be a GIT quotient for a linearization in Σ ± J , and X 0 J the GIT quotient corresponding to the generic linearization in the wall for J [CLS11, Def. 14.3.13]. The sequence of birational maps given by crossing walls according to this ordering begins at V n and terminates at P n .As described in Section 4, passing through the wall corresponding to J yields a diagram where we replace the subscripts λ J with J for brevity: 21 Lemma 5.10. We have isomorphismsWe handle the (+) claims. The statements for the (−) side are proven completely analogously.On W , the ideal (y j | j ∈ J) defines the contracting locus W + J , so that functions on W + J are given by k[x 0 , . . . , x n ] ⊗ k k[y i | i ∈ J]. Assume that y i = 0 for some point p ∈ W + J and some l ∈ J. Then λ J∪{l} destabilizes p: λ J∪{l} is negative on this chamber and p lies in the contracting locus of λ J∪{l} , since λ J∪{l} has positive weights on k[x 0 , . . . ,Assume that x j = 0 for j ∈ J for some point p ∈ W + J . Then λ J λ −1 J\{j} destabilizes p. The weights x l for l = j and y i for i ∈ J are non-negative. The chamber Σ + J lies in the positive half-spaces for both λ J and λ J\{j} . But λ J = 0 intersects the closure of Σ + J . Thus, Σ + J lies in the negative half-space associated to λ J λ −1 J\{j} .n+1 m with trivial λ J -action. So we have Z 0 J ∼ = BG m . It then follows from Lemma 4.3 that Z 0,rig J = Spec k. Notation 5.11. Following the identification in Lemma 5.10, we will write O Z + J (a) for the sheaf corresponding to O(a) on P |J c |−1 .Proposition 5.12. Let J ⊆ {0, 1, . . . , n} with |J| ≤ n 2 . For any d ∈ Z, there is a semiorthogonal decompositionUsing Proposition 5.12, we have a semi-orthogonal decompositionBy Lemma 5.14, the collection F n , viewed as line bundles, generates the components O Z + J n + 2 4 − |J c | , . . . , O Z + J n + 2 4 − 1 − |J| .To show that F n generates D b (V n ), we work via (downward) induction on the lexicographic ordering given above on J ⊆ {0, . . . , n} with |J| ≤ n 2 . Using Lemma 4.10 and the

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Tilting bundles on toric Fano fourfolds

Cited by 13 publications

References 26 publications

Derived categories of centrally‐symmetric smooth toric Fano varieties

Derived categories of centrally‐symmetric smooth toric Fano varieties

Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two

On derived categories of arithmetic toric varieties

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