higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
The notion of dg-schemes was introduced by Maxim Kontsevich as the first approach to derived algebraic geometry, and was further developed by Mikhail Kapranov and Ionut Ciocan-Fontanine.
A differential graded scheme (dg-scheme) is a scheme $(X, O_X)$ together with a sheaf $O_X^\bullet$ of nonnegatively graded commutative differential graded $O_X$-algebras, such that $O_X \to H^0(O_X^\bullet)$ is surjective.
Ciocan-Fontanine and Kapranov construct $RQuot(X, F)$, a derived enhancement of the classical Quot scheme parametrizing subsheaves of a given coherent sheaf $F$ on a smooth projective variety $X$ (1999). Similarly they also construct a dg-scheme $RHilb_h(X)$, a derived enhancement of the Hilbert scheme parametrizing subschemes? of a given projective scheme? $X$ with Hilbert polynomial? $h$ (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 $RQuot(X, F)$ and $RHilb_h(X)$, 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
Ionut Ciocan-Fontanine, Mikhail Kapranov, Derived Quot schemes, 1999, arXiv:math/9905174.
Ionut Ciocan-Fontanine, Mikhail Kapranov, Derived Hilbert schemes, 2000, arXiv:math/0005155.
Kai Behrend, Differential graded schemes I: prefect resolving algebras (arXiv:0212225)
Kai Behrend, Differential Graded Schemes II: The 2-category of Differential Graded Schemes (arXiv:0212226)