Exceptional collections on toric Fano threefolds and birational geometry

Abstract

Bernardi and Tirabassi show the existence of full strong exceptional collections consisting of line bundles on smooth toric Fano $3$-folds under assuming Bondal's conjecture, which states that the Frobenius push-forward of the structure sheaf $\mc O_X$ generates the derived category $D^b(X)$ for smooth projective toric varieties $X$. In this article, we show Bondal's conjecture for smooth toric Fano $3$-folds and also improve their result, using birational geometry.

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References in corpus (5)

• Maximal lengths of exceptional collections of line bundles
• On the full, strongly exceptional collections on toric varieties with Picard number three
• Dimer models and exceptional collections
• Derived category of toric varieties with Picard number three
• Tilting bundles via the Frobenius morphism

Cited by in corpus (3)

• On Landau-Ginzburg Systems and ${\mathcal{D}}^b(X)$ of projective bundles
• On Landau-Ginzburg systems and ${\mathcal{D}}^b(X)$ of various toric Fano manifolds with small picard group
• Frobenius morphisms and derived categories on two dimensional toric Deligne-Mumford stacks