Abstract: The framework of Derived Algebraic Geometry (DAG), developed by Toen-Vezzosi, Lurie and many others, allows us to extend Grothendieck’s theory of projectivizations and Grassmannians of sheaves to the cases of complexes. This derived extension is useful for constructing and studying moduli spaces, especially when the spaces are singular and difficult to analyze in the classical framework. We will discuss the constructions of derived projectivizations and Grassmannians as well as their properties, with a focus on applications to Abel maps for singular curves and Hecke correspondences for surfaces. The talk will be based on papers arXiv:2202.11636 and arXiv:2212.10488 and works in preparation.