Yesterday Francesco Bastianelli, Shinnosuke Okawa, Andrea Ricolfi and I uploaded a preprint to the arXiv, titled Indecomposability of derived categories in families. In it we discuss how to check that a derived category of a smooth projective variety does not admit any semiorthogonal decompositions. Such varieties are "atomic" in a certain sense, and knowing when (not) to look for semiorthogonal decompositions is an important question.

The main idea here is that decomposability is a property which can be extended (based on the ideas from arXiv:2002.03303, with a subset of the authors), so if you have a family of varieties which is indecomposable in a dense subset, then you know all members must be indecomposable.

We then study instances where this tool gives new and interesting indecomposability results. In particular,

• roughly half of the symmetric powers of curves which are expected to have indecomposable derived categories now provably have so;
• we searched the literature on algebraic surfaces for interesting smooth projective surfaces of general type with $p_{\mathrm{g}}\geq 1$ (because then conjecturally they have an indecomposable derived category), and came up with two interesting families to study;
• we discussed indecomposability for Hilbert schemes of points on surfaces of general type.

The one thing which you can help us with, is by telling us about your favourite surface $S$ of general type with $p_{\mathrm{g}}\geq 1$ for which the base locus of $|\omega_S|$ is 1-dimensional, such that the intersection matrix of its irreducible components is not negative definite. By a result of Kawatani–Okawa surfaces with finite canonical base locus, or negative definite intersection matrix, have an indecomposable derived category.

We have found only 1 family of such surfaces, introduced by Ciliberto in The degree of the generators of the canonical ring of a surface of general type, and by extending their definition and combining their geometry with the deformation theory of semiorthogonal decompositions were able to prove their indecomposability.

Are there others? Likely so, but if there are, can we always apply the deformation trick? Or are there maybe rigid surfaces with this property? In that case we'd need to develop new tools to prove their indecomposability!