Abel maps for integral curves via a derived perspective
Qingyuan Jiang

TL;DR
This paper introduces a derived algebraic geometry framework for Abel maps on integral curves, providing new insights into moduli spaces and their derived enhancements, with broad generalizations and novel tools.
Contribution
It develops a unified derived approach to Abel maps and moduli spaces of sheaves on integral curves, extending previous results and introducing new derived tools.
Findings
Derived enhancements of Quot and Hilbert schemes
Semiorthogonal decompositions for derived categories
Generalization to higher rank sheaves
Abstract
We develop a general framework for Abel maps associated with a family of integral curves using derived algebraic geometry. For compactified Picard schemes, our approach yields relative quasi-smooth derived enhancements of the Quot schemes and, in the Gorenstein case, of the Hilbert schemes of points on . 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 -complex, a derived generalization of Grothendieck's -module and the…
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques · Algebraic Geometry and Number Theory
