Reference for the Hilbert scheme of points as GIT quotient

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Could you provide me some ressources concerning the construction of the Hilbert scheme of n ponts in $\mathbb{C} ^2$ as a GIT quotient.
There is this article of Barbara Bolognese and Ivan Losev but the proof of the proposition 3.4 doesn't prove the good inclusion or at least, I don't understand it.
Best, Rhylx