Abel maps for integral curves via a derived perspective

Abstract

We develop a general framework for Abel maps associated with a family $X/S$ of integral curves using derived algebraic geometry. For compactified Picard schemes, our approach yields relative quasi-smooth derived enhancements of the Quot schemes $\mathrm{Quot}_{\omega/X/S}^d$ and, in the Gorenstein case, of the Hilbert schemes of points $\mathrm{Hilb}_{X/S}^d$ on $X/S$. These constructions naturally generalize to higher rank torsion-free sheaves and their coherent systems. We obtain unified semiorthogonal decompositions for the derived categories of these derived moduli spaces, broadly extending previous results for symmetric powers, varieties of linear series, and Thaddeus pairs to torsion-free sheaves on integral curves. Central to our approach are two novel tools of independent interest: the $\mathcal{Q}$-complex, a derived generalization of Grothendieck's $Q$-module and the Altman--Kleiman $H$-module, and a derived theory of moduli of extensions that extends Lange's classical framework.