标题： 更高遗传代数和Picard数为一或二的环面Fano堆栈 显示英文标题

标题： Higher hereditary algebras and toric Fano stacks of Picard number one or two

摘要： 我们证明了存在性并给出了所有$d$-tilting丛（因此是几何螺旋）的分类，这些丛由线丛组成，在Picard数为一或二的维数为$d$的光滑toric Fano DM堆栈上。在这里，$d$-tilting丛是指其自同态代数的全局维数不超过$d$的tilting丛。在Picard数为一的情况下，由线丛组成的tilting丛与在其Picard群上赋予某种偏序的非平凡上集之间存在双射对应关系。此外，它们都是$d$-tilting丛，其自同态代数成为类型为$\tilde{A}$的$d$-表示无限代数。相反，所有这样的代数都以这种方式出现。 从这个意义上说，我们可以将Picard数为一的光滑toric Fano DM堆栈视为高表示无限代数类型的几何模型，即类型$\tilde{A}$。 利用这个几何模型，我们给出了它们的$d$-APR倾斜模的新组合描述。 在Picard数为二的情况下，由线丛组成的$d$-倾斜丛与对$(I,I')$一一对应，其中$I$和$I'$是某些偏序集中的非平凡上集。 这里，$I$对应于一个具有秩为一的除子类群的Gorenstein环面奇点的非交换Crepant分解（NCCR），而$I'$对应于这个NCCR的箭图的一个切割。 此外，这些$d$-倾斜丛的自同态代数也变为$d$-表示无限代数。 显示英文摘要

摘要： We prove the existence and give a classification of all $d$-tilting bundles (and thus geometric Helices) consisting of line bundles on $d$-dimensional smooth toric Fano DM stacks of Picard number one or two. Here, a $d$-tilting bundle is a tilting bundle whose endomorphism algebra has global dimension $d$ or less. In the case of Picard number one, tilting bundles consisting of line bundles correspond bijectively to non-trivial upper sets in its Picard group equipped with a certain partial order. Moreover, all of them are $d$-tilting bundles and their endomorphisms algebras become $d$-representation infinite algebras of type $\tilde{A}$. Conversely, all such algebras arise in this way. In this sense, we can think of smooth toric Fano DM stacks with Picard number one as geometric models of higher representation infinite algebras of type $\tilde{A}$. Using this geometric model, we give a new combinatorial description to $d$-APR tilting modules of them. In the case of Picard number two, $d$-tilting bundles consisting of line bundles correspond bijectively to pairs $(I,I')$, where $I$ and $I'$ are non-trivial upper sets in certain partially ordered sets. Here, $I$ corresponds to a non-commutative crepant resolution (NCCR) of a certain Gorenstein toric singularity with divisor class group of rank one and $I'$ corresponds to a cut of the quiver of this NCCR. Moreover, the endomorphism algebras of these $d$-tilting bundles also become $d$-representation infinite algebras.