Abstract

Using the Lascar inequalities, we show that any finite rank $\delta$-closed subset of a quasiprojective variety is definably isomorphic to an affine $\delta$-closed set. Moreover, we show that if X is a finite rank subset of the projective space $\mathbb{P}^n$ and a is a generic point of $\mathbb{P}^n$, then the projection from a is injective on X. Finally we prove that if RM = RC in DCF$_0$, then RM = RU.