Intersection of hypersurfaces in the projective space

Question

Fix an integer $n>0$. Is it true that for any $k>0$ and a closed point $x \in \mathbb{P}^n$, there exist hypersurface sections of degree $k$ (i.e., global sections of $\mathcal{O}_{\mathbb{P}^n}(k)$), say $Z_1,...,Z_m$ such that $Z_1 \cap Z_2 \cap ... \cap Z_m=\{x\}$? Note that I am not putting any constraint on $m$.