Tensor triangulated categories associated to schemes and their families

Question

For any essentially small, rigid and idempotent-complete tensor triangulated (TT for short) category $\mathcal{T}$ Balmer (The spectrum of prime ideals in tensor triangulated categories) constructs a locally ringed topological space Spec($\mathcal{T}$).

When $\mathcal{T}$ is the derived category of perfect complexes, $D^o(S)$, over a quasi-compact quasi-separated (qcqs) scheme $S$, the construction recovers $S$ as a locally ringed space together with its structure sheaf from the tensor triangular structure of $D^o(S)$.

My questions are:

(1) Given a TT category, can we tell when it is equivalent to $D^o(X)$ for a qcqs scheme $X$?

Let's call the TT categories that fulfill the criterion schematic.

(2) If we take an essentially small additive category and put different schematic tensor triangulated structures on it, we'd get different schemes via Balmer's construction. Can all flat families of schemes of a certain type (e.g., smooth curves of genus $g$ over a field $k$) be obtained by varying the tensor structure alone?

(3) What are some natural discrete invariants of TT-categories and how can we build their moduli spaces after fixing values of the discrete invariants?

Answer 1

I don't have an answer to this, but this is too long for a comment.

As for your fist question, one thing to check is whether the resulting locally ringed space is a scheme. This doesn't happen always, as Balmer showed in his paper Spectra, spectra, spectra Proposition 9.7.

He concretely shows that the locally ringed space associated to $SH^{fin}_{(p)}$, the topological stable homotopy category of finite spectra localized at p with smash product as the monoidal structure, is not a scheme. He does so by checking that if it were a scheme it would be the spec of the ring of global sections, which he shows is local and then compares the number of points of the underlying space.

I believe that you could abstract a bit how this counterexample goes and arrive to a statement with a somewhat contrived test to check whether the space is a scheme. It is my impression that the general suspicion is that triangulated categories of topological nature would hardly give you a scheme, but then again I don't know of very general theory on this idea.

However even if the space is a scheme I think it wont guarantee you the TT-category is a derived category. A semisimple abelian category is triangulated, so the usual tensor product on something like $k-Mod$ will yield $Spec(k)$ under Balmer's reconstruction, but $k-Mod$ is not (I don't think?) equivalent to a derived category of a scheme. But this example looks a bit extreme.

As for your second question this seems even tougher, the extreme case I can think of is when the derived category already determines the scheme as the Bondal-Orlov reconstruction setting, some discussion on the tensor products that arise in this situation was discussed in this MO question although this doesn't address your question per se.