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?)