Let $V$ be a complex vector space. Consider the algebra $D(\mathbb{P}(V),\mathcal{O}(n)))$ of global differential operators from line bundle $\mathcal{O}(n)$ to itself, here $n \in \mathbb{Z}_{\geqslant 0}$.

Can we define this algebra via generators and relations?

It looks like we can indeed describe the kernel of the natural map $U(\mathfrak{gl}(V)) \rightarrow D(\mathbb{P}(V), \mathcal{O}(n))$. This is a quantisation of the set of relations defining the closure of the minimal nilpotent orbit in $\mathfrak{sl}(V) \subset \mathfrak{gl}(V)$. Using now that the kernel of $U(\mathfrak{gl}(V)) \rightarrow D(\mathbb{P}(V), \mathcal{O}(n))$ should be zero being restricted to the affine chart in $\mathbb{P}(V)$ (corresponds to the fact that this kernel is an annihilator of a certain parabolic Verma) we can compute and get \begin{multline*} D((\mathbb{P}(\mathbb{C}^{m+1}), \mathcal{O}(n)))=\\ =U(\mathfrak{gl}_{m+1})/(e_{ki}e_{lj}-e_{li}e_{kj}-\delta_{i,l}e_{kj}+\delta_{i,k}e_{lj},\, (\sum_{p=1}^{m+1}e_{pp})-n ,\, i<j ,\, k<l), \end{multline*} here $e_{ij},\, 1 \leqslant i,j \leqslant m+1$ is the standard basis in $\mathfrak{gl}_{m+1}$.

The relation $\sum_{p=1}^{m+1}e_{pp}=n$ is clear and the relations $e_{ki}e_{lj}-e_{li}e_{kj}-\delta_{i,l}e_{kj}+\delta_{i,k}e_{lj}$ can be easily seen if one considers the natural action $\mathfrak{gl}(V) \curvearrowright V^* \setminus \{0\}$ and looks at the kernel of the natural map $U(\mathfrak{gl}(V)) \rightarrow D(V^* \setminus \{0\})$, now $D(\mathbb{P}(\mathbb{C}^{m+1}),\mathcal{O}(n))$ is just the hamiltonian reduction of $D(V^* \setminus \{0\})$ by the group $\mathbb{C}^\times$.