TY - JOUR
TI - On exceptional collections of line bundles on toric deligne-mumford stacks
DO - https://doi.org/doi:10.7282/t3-6388-cs26
PY - 2020
AB - We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $mathbb{P}_{mathbf{Sigma}}$. We prove that when the rank of Picard group is no more than two, any strong exceptional collection of line bundles generates the derived category of $mathbb{P}_{mathbf{Sigma}}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $mathbb{P}_{mathbf{Sigma}}$.
Moreover, we consider generalized Hirzebruch surfaces $mathbb{F}_{alpha,n}$ which are not Fano and have Picard rank two. We give a classification of all (strong) exceptional collections of line bundles of maximum length and show they generate the derived category, which is a generalization for the results of Hirzebruch surfaces. We show that any exceptional collections of line bundles on $mathbb{F}_{alpha,n}$ can be extend to maximum length $2(alpha+1)$ which is the rank of $K$-theory.
We give examples of strong exceptional collections of line bundles on $mathbb{F}_{alpha,n}$ which cannot be extended to strong exceptional collections of line bundles of length $2(alpha+1)$, but can be extend to exceptional collections of line bundles of maximum length $2(alpha+1)$.
KW - Mathematics
LA - English
ER -