On some positive embedding of P^d

Abstract

We prove that any two embeddings P^d ∼ Y → X_1 , P^d ∼ Y → X_2 ,  d ≥ 3, in two n-folds projective varieties X_1 , X_2 with normal bundle N_{Y |X_1} ∼ X_{Y |X_2} ∼ (n − d)O_{P^d} (1) are formally equivalent .
Moreover, we see that Y is G2 in both X_1 and X_2. As an immediate consequence of this result and if furthermore Y is G3 in both X_1 and X_2 then we deduce that the two embeddings are Zariski equivalent.