Let $D$ be a base-point-free divisor on a normal projective variety $X$, and let $Y$ be the image of the morphism $f\_{D}:X\rightarrow Y$ induced by $D$. Assume that $f\_D$ is birational.


Now, let $X(D)=Proj\left(\bigoplus\_{k\geq 0}H^{0}(X,kD)\right)$. Is $X(D)$ the normalization of $Y$ ?



Ring of sections and normalization
The modified assertion is true. For a field $k$, for every proper $k$-scheme $X$, for every $k$-morphism $$f:X\to \mathbb{P}^n,$$ the irreducible curves in $X$ that are contracted by $f$ are precisely the irreducible curves having degree $0$ with respect to the invertible sheaf $\mathcal{L}:=f^\*\mathcal{O}(1)$. These are also the curves that are contracted by the natural morphism $$g:X \to X(\mathcal{L}),$$ where $X(\mathcal{L})$ is $\text{Proj} \bigoplus\_{d\geq 0} H^0(X,\mathcal{L}^{\otimes d})$. Thus, the natural $k$-morphism $$h:X(\mathcal{L})\to f(X)$$ is a finite morphism. If $X$ is normal, then also $X(\mathcal{L})$ is normal. If also $h$ is birational, e.g., this holds if $f$ is birational, then $h$ is the normalization of the image of $f$.


