I have to prove that the Grassmannian is a smooth projective variety.
I was able to show that it is a projective variety using this script: https://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Hudec.pdf, but I have trouble with the smooth part and I only find proofs showing that the Grassmannian is a smooth manifold. I am terrible in Algebraic Geometry, but I think I cannot use these proofs since the Grassmanian has the Zarisky topology as a variety and there it is not Hausdorf. Is there an easy proof to show that the Grassmannian is also a smooth variety?