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?...)



Proving that the Hilbert scheme of points on $\mathbb C^2$ is smooth
A way to get the smoothness fro free is by using the description of the Hilbert scheme as a quiver variety. Since it is a result of the hyperkahler reduction applied to a flat variety and the parameter of the reduction is regular (in this case regularity is just its nonvanishing), it is smooth.


