Let $X$ be a nodal maximally non-factorial Fano threefold. If there is $1$-node and no other singularities, they by the work of Kuznetsov-Shinder <https://arxiv.org/pdf/2207.06477.pdf> Lemma 6.18, $D^b(X)$ contains a categorical ordinary double point subcategory $\mathcal{P}\subset D^b(X)$, which is generated by a $\mathbb{P}^\infty$ object $P$, such that the category $\mathcal{P}^{\perp}$ is a smooth category.


I have a question


Let $X$ be a nodal maximally non-factorial Fano threefold with no other singularities, if $D^b(X)$ contains two categorical ordinary double point subcategories $\mathcal{P}$ and $\mathcal{P}'$ such that the $\mathbb{P}^{\infty}$ objects generating $\mathcal{P}$ and $\mathcal{P}'$ are $P$ and $P'$ respectively and $P\not\cong P'$, can we say that $X$ has at least two node?


Assume not, then $X$ has only one node, then by Lemma above, $D^b(X)=\langle\mathcal{P}^{\perp},\mathcal{P}\rangle$ such that $\mathcal{P}^{\perp}$ is a smooth category, if we can further show that $\mathcal{P}'\subset\mathcal{P}^{\perp}$, then this is a contradiction, then $X$ has at least two nodes.


But I am not sure if $\mathcal{P}'\subset\mathcal{P}^{\perp}$ is true.



Semi-orthogonal decomposition for maximally non-factorial Fano threefolds
If you assume that $\mathcal{P}$ and $\mathcal{P'}$ are semiorthogonal, this is true. The easiest way to see this is by looking at the singularity category. If $X$ has one node, (the idempotent completion of) the singularity category of $X$ is the category of $\mathbb{Z}/2$-graded vector spaces, and if
$$
D^b(X) = \langle \mathcal{P}, \mathcal{P}', \dots \rangle,
$$
the singularity category contains a direct sum of two copies of the category of $\mathbb{Z}/2$-graded vector spaces, a contradiction.


