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

Recently, 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.Comment: 38 pages, v2;major revision. improved the readability of proofs, Appendix C adde

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

Mutations of noncommutative crepant resolutions in geometric invariant theory

Let $X$ be a generic quasi-symmetric representation of a connected reductive group $G$. The GIT quotient stack $\mathfrak{X}=[X^{\rm ss}(\ell)/G]$ with respect to a generic $\ell$ is a (stacky) crepant resolution of the affine quotient $X/G$, and it is derived equivalent to a noncommutative crepant resolution (=NCCR) of $X/G$. Halpern-Leistner and Sam showed that the derived category $\mathrm{D}^b(\mathrm{coh}~\mathfrak{X})$ is equivalent to certain subcategories of $\mathrm{D}^b(\mathrm{coh}~[X/G])$, which are called magic windows. This paper studies equivalences between magic windows that correspond to wall-crossings in a hyperplane arrangement in terms of NCCRs. We show that those equivalences coincide with derived equivalences between NCCRs induced by tilting modules, and that those tilting modules are obtained by certain operations of modules, which is called exchanges of modules. When $G$ is a torus, it turns out that the exchanges are nothing but iterated Iyama--Wemyss mutations. Although we mainly discuss resolutions of affine varieties, our theorems also yield a result for projective Calabi-Yau varieties. Using techniques from the theory of noncommutative matrix factorizations, we show that Iyama--Wemyss mutations induce a group action of the fundamental group $\pi_1(\mathbb{P}^1 \backslash\{0,1,\infty\})$ on the derived category of a Calabi-Yau complete intersection in a weighted projective space.Comment: 41 pages, 3 figures, to appear in Selecta Mathematic

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

非可換クレパント解消の幾何学

早大学位記番号:新8130早稲田大