On a summer school for undergraduate and graduate students Okounkov gave the following exercise (without hints): Prove that the Hilbert scheme of points on $\mathbb C^2$ is smooth.

Only a definition of the scheme was given. I don't really understand how to solve this exercise.

I remember Beauville was saying in one of his courses on a school (for graduate students and postdocs), that he is not aware of an elementary proof of the fact that the Hilbert scheme of points on a smooth surface is smooth (a standard proof uses $Ext$ groups).

So I would like to know, who is right, Okunkov or Beauville? (or maybe both?...)

On a summer school for undergraduate and graduate students Okounkov gave the following exercise (without hints): Prove that the Hilbert scheme of points on $\mathbb C^2$ is smooth.

Only a definition of the scheme was given. I don't really understand how to solve this exercise.

I remember Beauville was saying in his course on a school at Lac de Garde (while he was commenting page 14 here: http://math1.unice.fr/~beauvill/conf/lacgarde2.pdf), that he is not aware of an elementary proof of the fact that the Hilbert scheme of points on a smooth surface is smooth (a standard proof uses $Ext$ groups).

So I would like to know, who is right, Okounkov or Beauville? (or maybe both?...)