An upper bound for the generalized greatest common divisor of rational points

Let $X$ be a smooth projective variety defined over a number field $K$. We give an upper bound for the generalized greatest common divisor of a point $x\in X$ with respect to an irreducible subvariety $Y\subseteq X$ also defined over $K$. To prove the result, we stablish a rather uniform Riemann--Roch type inequality.