# Semi-orthogonal decomposition for maximally non-factorial Fano threefolds

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.

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.
• I completely agree with you that if $\mathcal{P},\mathcal{P}'$ are semi-orthogonal, then it is true. In fact, what I want to show is a little bit weaker. Is it possible that $X$ is a 1-nodal maximally non-factorial Fano threefold and $D^b(X)$ contains two categorical ordinary double point $\mathcal{P},\mathcal{P}'$ but this two category are NOT semi-orthogonal to each other? I think it is not possible but since they are not semi-orthogonal, I do not know how to prove it. Commented Oct 3, 2022 at 21:16
• Without the semiorthogonality condition this is of course possible --- just apply any autoequivalence (e.g. a line bundle twst) to $\mathcal{P}$, this will give you another copy. Commented Oct 4, 2022 at 6:06
• sorry about that, is this the only exceptional case that $\mathcal{P},\mathcal{P}'$ appear, i.e, one is created from the other by auto-equivalence of $D^b(X)$? Commented Oct 4, 2022 at 8:24
• Another possibility is to replace $\mathcal{P} = \langle P \rangle$ by $\mathcal{P}' := \langle P^\vee \rangle$ or by the image of the latter under an autoequivalence. Commented Oct 4, 2022 at 8:52