Exceptional collections on toric Fano threefolds and birational geometry

Abstract

Bondal's conjecture states that the Frobenius push-forward of the structure sheaf O X generates the derived category D b (X) for smooth projective toric varieties X.Bernardi and Tirabassi exhibit a full strong exceptional collection consisting of line bundles on smooth toric Fano 3-folds assuming Bondal's conjecture. In this article, we prove Bondal's conjecture for smooth toric Fano 3-folds and improve upon their result using birational geometry.

The toric Frobenius morphism and a conjecture of Orlov

“…Proof. One can now apply Theorem 2.11 thanks to [Ueh14] which guarantees that T bu (Y ) is tilting for Y a smooth toric Fano variety of dimension at most 3.…”

Tilting bundles on toric Fano fourfolds

“…Smooth toric Fano varieties are of particular interest; there are a finite number of these varieties in each dimension and they have been classified in dimension 3 by Watanabe-Watanabe and Batyrev [WW82,Bat82b], dimension 4 by Batyrev and Sato [Bat99,Sat00], dimension 5 by Kreuzer-Nill [KN09], whilst Øbro [Øb07] provided a general classification algorithm. King [Kin97] has exhibited full strong exceptional collections of line bundles for the 5 smooth toric Fano surfaces, and by building on work by Bondal [Bon06], Costa-Miró-Roig [CMR04] and Bernardi-Tirabassi [BT09], Uehara [Ueh14] provided full strong exceptional collections of line bundles for the 18 smooth toric Fano threefolds. The main theorem of this paper is as follows:…”

On Landau–Ginzburg systems, quivers and monodromy

“…(b) does X admit a full strongly exceptional collection of line bundles in P ic(X)? A class of manifolds on which these questions have been extensively studied in recent years is the class of toric manifolds and, specifically, the class of toric Fano manifolds, see [12,13,14,15,16,30,31,42,48]. Question (a) was answered affirmatively by Kawamata which showed that any toric manifold admits a full exceptional collection of objects in D b (X), see [30].…”

“…For instance, the full strongly exceptional collections E described in examples 3.6-8 are, in fact, Frobenius collections. Many further examples of toric manifolds whose Frobenius collections are full strongly exceptional collections were found, see [15,48]. On the other hand, the condition |D X | = ρ(X) does not always hold.…”

“…For instance, when X is a toric Fano threefold |D X | = ρ(X) for sixteen of the eighteen Fano toric threefolds. Let us note that it is shown by Uehara in [48] that in the cases when |D X | = ρ(X), the Frobenius collection D X ⊂ P ic(X) is a full strongly exceptional collection of line bundles. Moreover, Uehara shows that, in the remaining two cases, there is a subset E ⊂ D X which is a full strongly exceptional collection.…”