Quot scheme

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Published on November 16, 2024 12:00:00 AM ● Video Link: https://www.youtube.com/watch?v=SGrpgI3xCUY

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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 south and if Fahrenheit 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 Fahrenheit to be the structure sheaf 𝒪_(X) gives a Hilbert scheme.)

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