Moduli of sheaves on ribbons
Michele Savarese, Filippo Viviani
TL;DR
This work develops the geometry of the moduli stack 𝓜X(r,D) of torsion-free sheaves on ribbons X by stratifying it via complete type (r0,r1; d0,d1) and analyzing invariants such as the generalized rank R and generalized degree Deg. It establishes non-emptiness criteria, smooth irreducibility and precise dimensions of strata, and clarifies how strata specialize within the stack, yielding a complete description of irreducible components in three regimes determined by δ relative to 2ar g-2. The paper also characterizes the intersection of strata with the (semi)stable loci, provides a concrete semistability criterion for torsion-free sheaves of type (n,1), and computes tangent spaces and Ext-structures to understand infinitesimal deformations. Overall, it extends Drézet’s and related works to the full moduli stack, identifies rigidity phenomena in the Fano case, and lays groundwork for higher-multiplicity primitive curves. The results have implications for understanding nilpotent cones via spectral correspondences and for the broader study of moduli of sheaves on non-reduced curves.
Abstract
We study the geometry of the moduli stack of torsion-free sheaves on ribbons. We introduce a stratification of the stack by the complete type of the sheaves, and we investigate the geometric properties of the strata and their closure relation, and which strata intersect the (semi)stable locus. Then we describe the irreducible components of the stack, by revealing an interesting trichotomy between Fano, Calabi-Yau and canonically polarized cases. Finally, we compute the tangent space of the moduli stack at a given sheaf.