hyperplanes ${\bf L}=(L_1,\dots,L_n)$ on ${\mathbb{P}}^d$ as well as weights ${\bf p}=(p_1,\dots,p_n)$ with $p_i\in{\mathbb{Z}}_{\ge0}$. We assume that the hyperplanes are in general position in the following sense:

\[general position\] For any subset $\{i_1,\ldots,i_m\}$, the intersection $\bigcap_{k=1}^mL_{i_k}$ is codimension $m$ in ${\mathbb{P}}^d$ or is empty if $m>d$.

For a triple $({{\mathcal O}},I,n)$ of a sheaf of rings ${{\mathcal O}}$ (or a ring), an ideal sheaf $I$ of ${{\mathcal O}}$ (or an ideal) and a positive integer $n$, let ${\rm T}_n({{\mathcal O}},I)$ be the subsheaf $${\rm T}_n({{\mathcal O}},I)=\left[\begin{array}{ccccc}
        {{\mathcal O}}&I&\cdots&I&I\\
        {{\mathcal O}}&{{\mathcal O}}&\cdots&I&I\\
        \vdots&\vdots&\ddots&\vdots&\vdots\\
        {{\mathcal O}}&{{\mathcal O}}&\cdots&{{\mathcal O}}&I\\
        {{\mathcal O}}&{{\mathcal O}}&\cdots&{{\mathcal O}}&{{\mathcal O}}\end{array}\right]$$ of the sheaf ${\rm M}_n({{\mathcal O}})$ of full matrix rings.

For the structure sheaf ${{\mathcal O}}:={{\mathcal O}}_{{\mathbb{P}}^d}$ of ${\mathbb{P}}^d$, let $$\begin{aligned}
\Lambda_i&:=&{\rm T}_{p_i}({{\mathcal O}},{{\mathcal O}}(-L_i))\\
\Lambda=\Lambda({\bf L}, {\bf p})&:=& \Lambda_1\otimes_{{{\mathcal O}}}\cdots\otimes_{{{\mathcal O}}}\Lambda_n\end{aligned}$$ which can be regarded as a suborder of ${\rm M}_{p_1\cdots p_n}({{\mathcal O}}_{{\mathbb{P}}^d})$. We call $\Lambda$ a *Geigle-Lenzing* (*GL*) *order* on ${\mathbb{P}}^d$ of *type* $({\bf L},{\bf p})$. Note that the authors of [@CI] call the transpose of this $\Lambda$ the *canonical matrix form* of its Morita equivalence class.

\[tilting\] Let $\Lambda$ be a GL order on ${\mathbb{P}}^d$.

-   There exists a tilting bundle $T$ in ${\operatorname{\mathsf{mod}}}\Lambda$ given in below.

