Maximal lengths of exceptional collections of line bundles (Q2922842)

A.~D.~King conjectured that every smooth projective toric variety has a full strong exceptional collection of line bundles. Infinitely many counterexamples to this statement were constructed (e.g. by \textit{L. Hille} and \textit{M. Perling} [Compos. Math. 142, No. 6, 1507--1521 (2006; Zbl 1108.14040)] and by \textit{M. Michałek} [C. R., Math., Acad. Sci. Paris 349, No. 1--2, 67--69 (2011; Zbl 1211.14056)]), but none of them was a nef-Fano variety (i.e. a variety with the anticanonical class big and nef). These results led L. Borisov and Z. Hua to propose a conjecture that every smooth nef-Fano toric DM stack has a full strong exceptional collection of line bundles, and L. Costa and R. M. Miró-Roig to consider a weaker version, that any smooth complete toric Fano variety has such a collection.NEWLINENEWLINEIn the article under review these conjectures are disproved. The author constructs a family \(Y_{n,k,a}\), parametrized by integers \(k,n\geq 2\) and \(a\geq 1\), of toric Fano varieties with Picard number~3 by describing their fans using Gale duality. Let \(l(Y)\) denote the maximal length of an exceptional collection of line bundles on a variety \(Y\). The main result is that for any constant \(c > \frac{3}{4}\), any positive integer \(a\), and \(n,k\) large enough (lower bound on \(k\) depending on \(n\)) one has NEWLINE\[NEWLINEl(Y_{n,k,a}) < c \,\mathrm{rk} K_0(Y_{n,k,a}).NEWLINE\]NEWLINE Thus, setting \(c=1\), the author obtains toric Fano varieties without a full exceptional collection of line bundles; note that collections are not required to be strong. More precisely, \(Y_{n,k,a}\) for \(a=1\), \(n=16\), \(k \geq 386\) have this property.NEWLINENEWLINEIn the last section of the paper it is shown that any toric nef-Fano DM stack~\(Y\) with Picard number~3 has a strong exceptional collection of line bundles of length at least \(\frac{3}{4} \,\mathrm{rk} K_0(Y)\).