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 Quot F ⁡ ( X ) whose set of T-points Quot F ⁡ ( X ) ( T ) = Mor S ⁡ ( T , Quot F ⁡ ( X ) ) is the set of isomorphism classes of the quotients of F × S T that are flat over T. The notion was introduced by Alexander Grothendieck.

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 O X gives a Hilbert scheme.)