On derived categories of arithmetic toric varieties

Abstract

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

Semiorthogonal decompositions in families

“…Some results in this direction can be found in [60,61,16,14]. There is also some progress extending results about derived categories known over an algebraically closed field to more general fields, [4,7,64].…”

Derived categories of centrally‐symmetric smooth toric Fano varieties

“…A more natural definition: an object of 𝖣 𝖻 (𝑋) is exceptional if its endomorphism algebra is a division 𝑘-algebra (concentrated in homological degree zero) [7,9]. An exceptional collection is then given by a totally ordered set 𝖤 = { 𝐸 1 , … , 𝐸 𝑠 } of exceptional objects in 𝖣 𝖻 (𝑋) satisfying Ext 𝑛 ( 𝐸 𝑖 , 𝐸 𝑗 ) = 0 for all integers 𝑛 whenever 𝑖 > 𝑗.…”

“…In [9], the authors showed that a smooth projective 𝑘-variety 𝑋 (not required to be toric) admits a full exceptional collection if and only if 𝑋 𝑘 sep admits a full exceptional collection which is Galois-stable, i.e., objects of the collection are permuted by the action of Gal ( 𝑘 sep ∕𝑘 ) . Exhibiting a full exceptional collection over 𝑘 therefore requires that one produce a collection over 𝑘 sep which is highly symmetric with respect to the Galois action.…”

“…Since Aut ( 𝑉 𝑛 ) ≅ 𝑆 𝑛+1 ⋊ 𝐶 2 , this applies to the varieties 𝑉 𝑛 . In fact, [9,Lemma 3.11] allows us to extend this to products of such varieties. Since any centrally symmetric toric Fano variety is a product of projective lines and the varieties 𝑉 𝑛 , the aforementioned results yield the following:…”

Separable algebras and coflasque resolutions

“…End(E i ) is a separable algebra with a G-action (see [BDM19]). However, B X := End( n i=1 E i ) is not a separable algebra in general and A X is only its semisimplification.…”