Quot scheme

In algebraic geometry, the Quot scheme is a scheme parametrizing 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 ${\displaystyle \operatorname {Quot} _{F}(X)}$ whose set of T-points ${\displaystyle \operatorname {Quot} _{F}(X)(T)=\operatorname {Mor} _{S}(T,\operatorname {Quot} _{F}(X))}$ is the set of isomorphism classes of the quotients of ${\displaystyle F\times _{S}T}$ 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 ${\displaystyle {\mathcal {O}}_{X}}$ gives a Hilbert scheme.)