Maximal lengths of exceptional collections of line bundles

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 $c>\frac{3}{4}$ 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 $c\rk K_0(Y).$ 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 $\frac{3}{4}$ is thus maximal with this property.

27 pages, no figures; misprints and typos corrected, an arithmetic mistake in the proof of Theorem 6.2 corrected, consequently Theorem 6.3 slightly modified, new Lemma 4.4 added, description of the constructed varieties extended, references added

References in corpus (3)

• Dimer models and exceptional collections
• The Coherent-Constructible Correspondence for Toric Deligne-Mumford Stacks
• Family of counterexamples to King's conjecture

Cited by in corpus (16)

• On the full, strongly exceptional collections on toric varieties with Picard number three
• Dimer models and exceptional collections
• Resolutions of toric subvarieties by line bundles and applications
• On exceptional collections of line bundles and mirror symmetry for toric Del-Pezzo surfaces
• Exceptional collections on toric Fano threefolds and birational geometry
• The nonequivariant coherent-constructible correspondence and tilting
• High Frobenius pushforwards generate the bounded derived category
• Monodromy of monomially admissible Fukaya-Seidel categories mirror to toric varieties
• Equivariant exceptional collections on smooth toric stacks
• Derived Category of Fibrations
• On Landau-Ginzburg Systems and ${\mathcal{D}}^b(X)$ of projective bundles
• Twisted Polytope Sheaves and Coherent-Constructible Correspondence for Toric Varieties
• Classification of full exceptional collections on smooth toric Fano varieties with Picard rank two
• On Landau-Ginzburg systems and ${\mathcal{D}}^b(X)$ of various toric Fano manifolds with small picard group
• On $\mathrm{H}-$trivial line bundles on toric DM stacks of dim $\ge 3$
• Strong full exceptional collections on certain toric varieties with Picard number three via mutations