Let $X$ be a nonsingular projective variety over an algebraically closed field of characteristic $0,$ let $H$ be an ample line bundle that is generated by its global sections. If $x$ is any closed point of $X$ with ideal sheaf $\mathcal{J},$ is there a sufficiently large $m>>0$ such that $H^{\otimes m}\otimes \mathcal{J}^m$ is globally generated?
For $m>>0,$ the divisor $H^{\otimes m}$ becomes very ample, so it separates points. Does this suffices to conclude that $H^{\otimes m}\otimes \mathcal{J}^m$ is globally generated?