As explained in the comments of [this answer](https://mathoverflow.net/q/371880/458355), given a smooth Fano 3-fold of index 1 and genus $g \geq 6$, we have two semiorthogonal decompositions $$\langle \text{Ku}(X), \mathcal{E}, \mathcal{O}\_X\rangle$$
and
$$
\langle \text{Ku}’(X), \mathcal{O}\_X, \mathcal{E}^\vee \rangle
$$


obtained from effectively dualizing the rectangular Lefschetz decomposition. The functor $\mathbb{L}\_{\mathcal O}(- \otimes \mathcal{O}(H) )$ in this case gives an equivalence between $\text{Ku}(X)$ and $\text{Ku}’(X)$. It is also obvious that any two exceptional collections related by a series of mutations give the same Kuznetsov component. Are there a priori any reasons in general one should expect the Kuznetsov component to be independent of the choice of exceptional collection?



Should we expect Kuznetsov component to be independent of exceptional collection
In general the orthogonal complement to a maximal exceptional collection does depend on the collection. A simple example of this sort is given here <https://arxiv.org/abs/1304.0903>.


