Strong full exceptional collections on certain toric varieties with Picard number three via mutations

In this paper, we study derived categories of certain toric varieties with Picard number three that are blowing-up another toric varieties along their torus invariant loci of codimension at most three. We construct strong full exceptional collections by using Orlov's blow-up formula and mutations.Comment: 17 pages, To apper in Le Matematich

On derived equivalence for Abuaf flop: mutation of non-commutative crepant resolutions and spherical twists

Segal constructed a derived equivalence for an interesting 5-fold flop that was provided by Abuaf. The aim of this article is to add some results for the derived equivalence for Abuaf's flop. Concretely, we study the equivalence for Abuaf's flop by using Toda-Uehara's tilting bundles and Iyama-Wemyss's mutation functors. In addition, we observe a ``flop-flop=twist" result and a ``multi-mutation=twist" result for Abuaf's flop

Rank two weak Fano bundles on del Pezzo threefolds of Picard rank one

We classify rank two vector bundles on a del Pezzo threefold $X$ of Picard rank one whose projectivizations are weak Fano. We also investigate the moduli spaces of such vector bundles when $X$ is of degree five, especially whether it is smooth, irreducible, or fine.Comment: 23 pages; v2: Remark 1.3 added, Proofs in Section 3.5 improve

Classification of rank two weak Fano bundles on del Pezzo threefolds of degree four

We classify rank two vector bundles on a given del Pezzo threefold of degree four whose projectivizations are weak Fano into seven cases. We also give an example for each of these seven cases.Comment: 20 pages; minor typos corrected, exposition improved, section 5.4 adde