Reading through the literature I see that there are two proofs for the Noether normalization for projective varieties (over infinite fields).
The first, "geometric", by iterating a projection away from a point and with some hyperplane center.
The second, applying the algebraic Noether normalization but for graded algebras.
My question is, why does the the injection of graded algebras given by the algebraic Noether normalization give a map between projective varieties? In general a map between graded rings does not induce a map between the varieties