If $M$ is a projectively normal projective variety in $\mathbb{C}\mathrm{P}^n$ and we intersect it with a hyperplane $\mathrm{P}V \subseteq \mathbb{C}\mathrm{P}^n$, is the result $M \cap \mathrm{P}V$ a projectively normal variety in the projective space $\mathrm{P}V$?
(This intersection is called a hyperplane section.)
Not necessary. For instance, if $M$ is a surface then any its singular hyperplane section is not normal (because it is singular in codimension 1).