# Quot scheme

In algebraic geometry, the **Quot scheme** is a scheme parametrizing locally free sheaves on a projective scheme. More specifically, if *X* is a projective scheme over a Noetherian scheme *S* and if *F* is a coherent sheaf on *X*, then there is a scheme whose set of *T*-points is the set of isomorphism classes of the quotients of that are flat over *T*. The notion was introduced by Alexander Grothendieck.^{[1]}

It is typically used to construct another scheme parametrizing geometric objects that are of interest such as a Hilbert scheme. (In fact, taking *F* to be the structure sheaf gives a Hilbert scheme.)

## References

- ↑ Grothendieck, Alexander. Techniques de construction et théorèmes d'existence en géométrie algébrique III : préschémas quotients. Séminaire Bourbaki, 6 (1960-1961), Exposé No. 212, 20 p.

- Nitsure, N.
*Construction of Hilbert and Quot schemes.*Fundamental algebraic geometry: Grothendieck’s FGA explained, Mathematical Surveys and Monographs 123, American Mathematical Society 2005, 105–137. - https://amathew.wordpress.com/2012/06/02/the-stack-of-coherent-sheaves/

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