Strongly exceptional sequences on toric quiver varieties (Q2794778)

The thesis of Möller looks for full, strongly exceptional sequences of line bundles on toric quiver varieties. There are two possible sources one can try to take the lineNEWLINEbundles from. First, one can look at the direct summands of the universal bundle. It is known that this does often yield the property ``exceptional'', but usually thereNEWLINEare not sufficiently many of them to reach the property ``full''. Second, and this is the source Möller is exploiting, one can look at the Frobenius images of the structure sheaf. Since toric varieties are binomial, the Frobenius morphisms \(F_k\) exist even in characteristic zero. The set of direct summands of the image of the structure sheaf under \(F_k\) stabilizes for \(k\gg 0\), and it is a conjecture of Bondal that, for general toric varieties, these summands generate the derived category. In the present thesis, Möller has shown that this does indeed hold true for toric quiver varieties.NEWLINENEWLINEMöller does also present a comparison of these two approaches (universal bundle vs. Frobenius). Moreover, in his thesis he describes under which conditions the setNEWLINEof ``Frobenius summands'' is strongly exceptional, too. And, for a general toric Fano variety, he presents an algorithm for deciding if this variety is induced from a quiverNEWLINE-- and providing the quiver if it does.