Interactions between autoequivalences, stability conditions, and moduli problems

TL;DR

This paper explores the interaction between autoequivalences and stability conditions in derived categories, applying the theory to singular genus 1 curves called n-gons, and explicitly computing their moduli spaces of stable objects.

Contribution

It introduces a compatibility criterion for autoequivalences and stability conditions, and explicitly computes moduli spaces for derived categories of n-gons, including the autoequivalence group structure.

Findings

Explicit moduli space of stable objects on E_n computed

Aut-equivalence group characterized as an extension of the modular group

Compatibility criterion for autoequivalences and stability conditions established

Abstract

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.