Exceptional collections on toric Fano threefolds and birational geometry (Q2878773)

Let \(X\) be a smooth projective toric threefold, and let \(F_m\) be the \(m\)-th Frobenius (or multiplication) map on \(X\). It is conjectured by \textit{A. I. Bondal} (see [\textit{K. Altmann} (ed.) et al. [Oberwolfach Rep. 3, No. 1, 253--316 (2006; Zbl 1110.14300)]) that (for \(m\) sufficiently divisible) the push-forward \({\mathfrak{D}_X}:=F_{m*} {\mathcal O}_X\) of the structure sheaf is a classical generator of the derived category \(D^b(X)\), that is, the only sheaf which is (left or right) orthogonal to \(\mathfrak{D}_X\) is the trivial one. This conjecture is related to full exceptional collections on \(X\): the sheaf \(\mathfrak{D}_X\) is split into a direct sum of line bundles and the direct sum of the components of a full exceptional collection is known to be a classical generator. Hence if a subset of the set of line bundles splitting \(\mathfrak{D}_X\) forms a full exceptional collection, then Bondal's conjecture holds.NEWLINENEWLINEIn this paper, the author constructs full and strong exceptional collections of line bundles on all toric Fano threefolds. There are 18 such threefolds, and for 16 (resp. 2) of them the collection consists exactly (resp. of a proper subset) of all the line bundles splitting \(\mathfrak{D}_X\). The main idea is to reduce the study to the birationally maximal such threefolds and study these 3 cases. This is achieved by showing that, given a toric birational contraction \(f: X \to Y\) and assuming that the line bundles splitting \(\mathfrak{D}_X\) form an exceptional collection, then the line bundles splitting \(\mathfrak{D}_Y\) also form a full strong exceptional collection.