Maximal lengths of exceptional collections of line bundles
Alexander I. Efimov
TL;DR
The paper tackles King's conjecture by constructing infinite families of toric Fano varieties with Picard number three that lack full strong exceptional collections of line bundles, establishing that the maximal length of such collections can be strictly smaller than a $\frac{3}{4}$-multiple of the Grothendieck group rank $\rk K_0(Y)$ for these examples. It develops a framework based on Gale duality, stacky fans, and polytope-volume arguments to both produce counterexamples and prove limits on possible collection lengths. Conversely, it proves a positive result: every toric nef-Fano DM stack with Picard number three admits a strong exceptional collection of length at least $\frac{3}{4}\rk K_0(Y)$, highlighting a sharp threshold in this setting. The work combines explicit constructions (the $Y_{n,k,a}$ family) with combinatorial and geometric methods to separate the existence of long exceptional collections from the Picard-number constraints, providing both infinite counterexamples and a universal lower bound.
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 $c>\frac34$ 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 $c\rk K_0(Y).$ 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 $\frac34 \rk K_0(Y).$ The constant $\frac34$ is thus maximal with this property.