Suppose $X$ is a smooth variety, $X\to \mathbf{P}^n$ is a closed immersion. For the fixed imbedding, we call $X$ is linearly normal if $\Gamma(\mathbf{P}^n,O(1))\to\Gamma(X,O_X(1))$ is surjective; we call $X$ is projectively normal if $\Gamma(\mathbf{P}^n,O(k))\to\Gamma(X,O_X(k))$ is surjective for all $k$.
Is there an example for a linearly normal but not projectively normal variety. (Is there example for curves and surfaces?)
There are many examples! Just take a very ample line bundle $L$ on $X$, and the embedding $\ X\hookrightarrow |L|^*$ defined by its global sections. It is linearly normal by construction, but there is no reason why for instance the map $\mathrm{Sym}^2H^0(X,L)\rightarrow H^0(X,L^2)$ should be surjective. Here is a concrete example : take a curve $C$ of genus 4 and a line bundle $L$ on $C$ of degree 7 which is very ample (that means that $L$ is not of the form $K_C(p+q-r)$ for $p,q,r\in C$). Then $L$ embeds $C$ in $\mathbb{P}^3$, $\dim H^0(\mathbb{P}^3,\mathcal{O}_{\mathbb{P}^3}(2))=10$ and $\dim H^0(C,L^2)= 11$, so the above map cannot be surjective.
