Narrow Results

For a smooth projective variety $X$, we study analogs of Quot schemes using hearts of non-standard $t$-structures of $D^b(\mathrm{\text{Coh}}(X))$. The technical framework uses families of $t$-structures as studied in A. Bayer, M. Lahoz, E. Macrì, H. Nuer, A. Perry and P. Stellari, Stability conditions in families, preprint (2019), arXiv:1902.08184. We provide several examples and suggest possible directions of further investigation, as we reinterpret moduli spaces of stable pairs, in the sense of M. Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117(2) (1994) 317–353; D. Huybrechts and M. Lehn, Stable pairs on curves and surfaces, J. Algebraic Geom. 4(1) (1995) 67–104, as instances of Quot schemes.

For a finite semigroup S and pseudovariety V, (Y, T) is a V-stable pair of S iff Y ⊆ S, T ≤ S and for any relational morphism R : S ⇝ V with V ∈ V there exists a v ∈ V such that Y ⊆ R^-1(v) and T ≤ R^-1(Stab(v)). X ≤ S is stable if it is generated by an -chain {a_i} with a_ia_j = a_i for j < i. Given a relation R : S ⇝ A ∈ A (where A denotes the pseudovariety of aperiodic semigroups) that computes Pl_A(S), we construct a new relation R^∞ : S ⇝ (A^(M))^# that computes A-stable pairs. This proves the main result of this paper: (Y, T) is an A-stable pair of S iff T ≤ ∪ X for some stableX ≤ Pl_A(S) and Y ⊆ Y' for some Y' ∈ Pl_A(S) with Y'x = Y' for all x ∈ X. As a corollary we get that if V is a local pseudovariety of semigroups, then V * A has decidable membership problem.

We give a short proof, using profinite techniques, that idempotent pointlikes, stable pairs and triples are decidable for the pseudovariety of aperiodic monoids. Stable pairs are also described for the pseudovariety of all finite monoids.