Exceptional collections on toric Fano threefolds and birational geometry

Abstract

Bondal's conjecture states that the Frobenius push-forward of the structure sheaf 𝒪X generates the derived category Db(X) for smooth projective toric varieties X. Bernardi and Tirabassi exhibit a full strong exceptional collection consisting of line bundles on smooth toric Fano 3-folds assuming Bondal's conjecture. In this paper, we prove Bondal's conjecture for smooth toric Fano 3-folds and improve upon their result using birational geometry.