Let $X$ be a projective variety. Suppose $\mathcal{L}$ is a basepoint free globally generated line bundle so we get a map $\pi:X\to \mathbb{P}^N$ induced by $\mathcal{L}$.
Let $R_n=H^0(X,\mathcal{L}^{\otimes{n}})$ and $R=\oplus_{n=0}^{\infty}R_n$. I vaguely remember that $\overline{\pi(X)}\cong \text{Proj} (R)$ (the grading is by natrual number) Is it true? In particular, when $\mathcal{L}$ is very ample,do we have $X\cong Proj(R)$? Thank you!
For very ample, this is true and if not, in general it is false. Just as an example, take a degree 2 line bundle $L$ on an elliptic curve $X$. Then $N=1$ in your notation, so $\pi(X)=\mathbb{P}^1$. But, $\mathrm{Proj}\, R$ is just $X$, since $L$ is ample.