Maximal lengths of exceptional collections of line bundles
type: Item
instance of: scholarly article
title: Maximal lengths of exceptional collections of line bundles
zbMATH Open document ID: 1318.14047
author: Alexander I. Efimov
publication date: 2014-10-15T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.3755
Mathematics Subject Classification ID: 14M25
Mathematics Subject Classification ID: 14F05
Mathematics Subject Classification ID: 14J45
zbMATH DE Number: 6355575
zbMATH Keywords: toric Fano varieties
zbMATH Keywords: line bundle
zbMATH Keywords: exceptional collection
MaRDI profile type: Publication
OpenAlex ID: W2151697929
arXiv ID: 1010.3755
cites work: On the classification of smooth projective toric varieties
cites work: Mellin--Barnes integrals as Fourier--Mukai transforms
cites work: On the conjecture of King for smooth toric Deligne-Mumford stacks
cites work: Tilting sheaves on toric varieties
cites work: The coherent-constructible correspondence for toric Deligne-Mumford stacks
cites work: A counterexample to King's conjecture
cites work: Exceptional sequences of invertible sheaves on rational surfaces
cites work: Derived categories of toric varieties
cites work: Family of counterexamples to King's conjecture
DOI: 10.1112/JLMS/JDU037
Recommended article: On the conjecture of King for smooth toric Deligne-Mumford stacks
Recommended article: Derived category of toric varieties with Picard number three
Recommended article: Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two
Recommended article: Frobenius splitting and derived category of toric varieties
Recommended article: On the full, strongly exceptional collections on toric varieties with Picard number three
Recommended article: Derived Categories of Toric Fano 3-Folds via the Frobenius Morphism
Recommended article: Exceptional collections on toric Fano threefolds and birational geometry
Recommended article: Q5483292
Recommended article: Smooth projective toric varieties whose nontrivial nef line bundles are big
Recommended article: Smooth complete toric threefolds with no nontrivial nef line bundles
published in: Journal of the London Mathematical Society. Second Series
review text: 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 \(\frac{3}{4} \,\mathrm{rk} K_0(Y)\).
reviewed by: Maria Donten-Bury
description: scientific article; zbMATH DE number 6355575