higher geometry / derived geometry
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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.
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
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)
Last revised on November 29, 2013 at 06:21:26. See the history of this page for a list of all contributions to it.