Quot/Hilbert schemes

It is standard practice to construct the Hilbert scheme as a special case of the Quot scheme. Often you can construct the Quot scheme out of a Hilbert scheme too.

Namely, suppose you have X —> S a flat, proper morphism of finite presentation and suppose that F is a finitely presented O_X-module. Then you can consider

Y = Spec(O_X[F]) —> X

where O_X[F] = O_X ⊕ F is the O_X-algebra where F is a square zero ideal. We have a section σ : X —> Y. Then we can consider the closed subscheme Q of Hilb_{Y/S} parametrizing families of closed subschemes of Y which contain σ. If I am not mistaken, then Q = Quot_{F/X/S}.

This only works because we assumed X —> S is proper and flat!