derived smooth geometry
Ciocan-Fontanine and Kapranov construct , a derived enhancement of the classical Quot scheme parametrizing subsheaves of a given coherent sheaf on a smooth projective variety (1999). Similarly they also construct a dg-scheme , a derived enhancement of the Hilbert scheme parametrizing subschemes? of a given projective scheme? with Hilbert polynomial? (2000). As an application they construct the derived moduli stack of stable maps of curves to a given projective variety.
There is a functor from the category of dg-schemes to the category of derived stacks of Bertrand Toen and Gabriele Vezzosi. It takes values in the full subcategory of 1-geometric derived stacks, but is not known (or expected) to be fully faithful.
In particular, the dg-schemes and , discussed above, also induce derived stacks in the modern sense.
A prediction of derived moduli spaces is spelled out (in a bit different language) in
The first examples of derived moduli spaces, using dg-schemes, are constructed in