Maximal lengths of exceptional collections of line bundles

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 $c>\frac{3}{4}$ 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 $c\text{rk}{K}_0(Y)\mathrm{.}$ 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 $\frac{3}{4}\text{rk}{K}_0(Y)\mathrm{.}$ The constant $\frac{3}{4}$ is thus maximal with this property.