Thank you Dori. I am trying to understand your proof which actually is very much like unfolding definitions. I wanted to know if you would have any reference for an good introduction to G-torsors?

Oh yes I'm sorry. I typed the wrong symbols. I meant $\times_S$. I am not really at ease with proving things using stacks. Could you show me Carnahan how to proceed? Yes Sasha, $X$ and $Y$ are schemes and more precisely you can assume that they are open subschemes of affine schemes. And yes again, $G$ acts on $X$ and $H$ on $Y$.

What I would like to do is to find a one to one correspondence between a point in the dual my wps and an surface in my original wps for example. Or more generally to be able to understand the dual of the wps through the original wps. Such a correspondence exists between point of the dual of the regular projective space and its hyperplanes. I am at the moment reading some theory of geometric invariant as I think that this should have led me to a dualise th wps , I don't know anything yet about toric varieties but I think that this is really interesting way to look on how to dualise a wps.