Categorical absorption of singularities and nodal nonfactorial Fano threefolds

If $X$ is a threefold with a nonfactorial ordinary double point, we show that there is a semiorthogonal decomposition of the derived category of $X$ with two components, a "small" component responsible for the singularity (we say it "absorbs" the singularity of the derived category of $X$), and a "big" component that deforms to the derived category of a smoothing of $X$. We use this construction to relate the derived categories of Fano threefolds of index $2$ and degree $d$ to derived categories of Fano threefolds of index $1$ and genus $g = 2d + 2$. This is joint work in progress with Evgeny Shinder.