I was recently reading one of the papers of Kapranov on $\mathcal{M}_{0,n}$ and he says that one can see it as a subvariety of the Hilbert scheme parametrizing all subschemes of a certain projective space. What is the precise definition of this Hilbert scheme? Why it is sure it exists? Is it in EGA somewhere or does anybody have a reference?

The construction is done in Geometry of Algebraic Curves by Arbarello et al. 


There is a book by Nakajima "Lectures on Hilbert Schemes of points of surfaces". First chapter will do. 

