Interactions between autoequivalences, stability conditions, and moduli problems

We begin by discussing various ways autoequivalences and stability conditions associated to triangulated categories can interact. Once an appropriate definition of compatibility is formulated, we derive a sufficiency criterion for this compatibility. We next apply this criterion to derived categories associated to Galois covers of the Weierstrass nodal cubic, known as n-gons and denoted by E_n. These are singular non-irreducible genus 1 curves naturally arising in variety of contexts, including as certain degenerations of elliptic curves. In particular, fixing the stability condition to be the natural extension of classical slope to E_n, we explicitly compute the moduli space of stable objects and its compactification (given by S-equivalence). The compactification of stable objects with a fixed slope is isomorphic to a disjoint union of E_m and Z/nZ where m|n; m varies as the slope varies and all such m occur. This computation is made possible by explicitly constructing the group of all autoequivalences compatible with the choice of stability condition. It is found that this group is an extension of the modular group \Gamma_0(n) by a direct product of Aut(E_n), Pic^0(E_n), and Z.

💡 Research Summary

The paper investigates the interplay between autoequivalences of triangulated categories, Bridgeland stability conditions, and the associated moduli problems. It begins by introducing a precise notion of “compatibility” between an autoequivalence Φ and a stability condition σ = (𝒜, Z) on a triangulated category 𝒟. Compatibility requires that Φ sends the heart 𝒜 to another heart (often the same) and that the central charge transforms by an element g ∈ GL⁺(2,ℝ), i.e. Z ∘ Φ⁻¹ = g·Z. This condition guarantees that Φ preserves the slicing defined by σ and thus maps σ‑stable objects to σ‑stable objects (up to a possible shift). The authors then prove a sufficiency criterion: if Φ induces an integral matrix A ∈ SL(2,ℤ) on the Grothendieck group K₀(𝒟) and the central charge satisfies Z ∘ Φ⁻¹ = A⁻¹·Z, then Φ is compatible with σ. In particular, matrices whose lower‑left entry is divisible by a fixed integer n form the congruence subgroup Γ₀(n) ⊂ SL(2,ℤ); these matrices give rise to a distinguished family of compatible autoequivalences.

Having set up the abstract framework, the authors apply it to the derived category Dᵇ(Coh(Eₙ)) of a singular, non‑irreducible genus‑one curve Eₙ, called an “n‑gon”. An n‑gon consists of n copies of ℙ¹ arranged in a cycle, each meeting its neighbours transversely at a node; it can be viewed as an n‑fold Galois cover of the nodal cubic. The natural stability condition on Eₙ is obtained by extending the classical slope stability for vector bundles on smooth curves. Concretely, the central charge is \