Assume that $U$ is an affine quasi-projective variety, so $U\subseteq Y$ for some projective variety $Y$ (say everythings is defined over an algebraically closed field $K$). Let $D_+(f)$ be a principal basic open subset of $Y$ ($f$ is homogeneous) and assume that $D_+(f)\subseteq U$.
Is $D_+(f)$ a principal open subset of $U$, i.e., does there exists $g\in K[U]$ such that $D_+(f)=\operatorname{Spec} K[U]_g$?