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 together with a sheaf of nonnegatively graded commutative differential graded -algebras, such that is surjective.
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.
Relation with derived stacks
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
- M. Kontsevich, Enumeration of rational curves via torus actions. The moduli space of curves (Texel Island, 1994), 335–368, Progr. Math. 129, Birkhäuser 1995. MR1363062 (97d:14077), hep-th/9405035.
The first examples of derived moduli spaces, using dg-schemes, are constructed in
- M. Kapranov, Injective resolutions of BG and derived moduli spaces of local systems, J. Pure Appl. Algebra 155 (2001), no. 2-3, 167–179; math/alg-geom/9710027, MR1801413 (2002b:18017)
Revised on November 29, 2013 06:21:26
by Adeel Khan