Quot scheme

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, the Quot scheme is a scheme parametrizing sheaves on a <a href="/facts/Projective_scheme/aB2XdB7B">projective scheme</a>. More specifically, if X is a projective scheme over a Noetherian scheme S and if F is a <a href="/facts/Coherent_sheaf/h2NoUN7e">coherent sheaf</a> on X, then there is a scheme 
 
 
 
 
 Quot
 
 F
 
 
 ⁡
 (
 X
 )
 
 
 {\displaystyle \operatorname {Quot} _{F}(X)}
 
 whose set of <a href="/facts/Functor_of_points/LwtxagnV">T-points</a> 
 
 
 
 
 Quot
 
 F
 
 
 ⁡
 (
 X
 )
 (
 T
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 =
 
 Mor
 
 S
 
 
 ⁡
 (
 T
 ,
 
 Quot
 
 F
 
 
 ⁡
 (
 X
 )
 )
 
 
 {\displaystyle \operatorname {Quot} _{F}(X)(T)=\operatorname {Mor} _{S}(T,\operatorname {Quot} _{F}(X))}
 
 is the set of isomorphism classes of the <a href="/facts/Quotient_by_an_equivalence_relation/Y1v4oW2A">quotients</a> of 
 
 
 
 F
 
 ×
 
 S
 
 
 T
 
 
 {\displaystyle F\times _{S}T}
 
 that are flat over T. The notion was introduced by <a href="/facts/Alexander_Grothendieck/jLQGPY7M">Alexander Grothendieck</a>.
It is typically used to construct another scheme parametrizing geometric objects that are of interest such as a <a href="/facts/Hilbert_scheme/kJcbts4e">Hilbert scheme</a>. (In fact, taking F to be the structure sheaf 
 
 
 
 
 
 
 O
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathcal {O}}_{X}}
 
 gives a Hilbert scheme.)

Quot scheme open-in-new

Quot scheme