We will discuss the paper:

Maximal lengths of exceptional collections of line bundles

Alexander I. Efimov

arXiv:1010.3755

Author´s abstract: In this paper we construct infinitely many examples of toric Fano varieties with Picard number three, which do not admit full exceptional collections of line bundles. In particular, this disproves King´s conjecture for toric Fano varieties.

More generally, we prove that for any constant c>3/4 there exist infinitely many toric Fano varieties $Y$ with Picard number three, such that the maximal length of exceptional collection of line bundles on $Y$ is strictly less than c times the rank of K_0(Y). To obtain varieties without exceptional collections of line bundles, it suffices to put c=1.

On the other hand, we prove that for any toric nef-Fano DM stack $Y$ with Picard number three, there exists a strong exceptional collection of line bundles on $Y$ of length at least 3/4 times rank(K_0(Y)). The constant 3/4 is thus maximal with this property.