Generating systems, generalized Thomsen collections and derived categories of toric varieties

Abstract: Bondal claims that for a smooth toric variety $X$, its bounded derived category of coherent sheaves $D_{c}^{b}(X)$ is generated by the Thomsen collection $T(X)$ of line bundles obtained as direct summands of the pushforward of $\mathcal{O}{X}$ along a Frobenius map with sufficiently divisible degree. The claim is confirmed recently. In this article, we consider a generalized Thomsen collection of line bundles $T(X,D)$ with a $\mathbb{Q}$-divisor $D$ as an auxiliary input, and we prove $T(X,D)$ still generates $D{c}^{b}(X)$. For $D=0$ we recover Bondal's oringinal claim. We give a stacky interpretation of this generalization and then we stick to a proof without using stacks and of different nature from existing works. To do this we introduce the notion of generating systems, and prove a theorem on the generation of $\mathcal{O}_{X}$ in terms of many line bundles arising from a given generating system, with an insight to the combinatorics of some cone decomposition or even real hyperplane arrangement attached to the generating system.