along a 0-dimensional scheme $\zeta$. Then $\mathrm{length}\, \zeta\le 2$.

Let $X$ be a non-degenerate irreducible subvariety of $\mathbb{P}^N$ with $\Delta(X,\mathcal{O}(1))=0$ and $c\in \mathbb{P}^N$ be a point outside $X$. Denote $pr_c\colon \mathbb{P}^N\dashrightarrow \mathbb{P}^{N-1}$ as the projection from $c$. Assume $pr_c|_X$ is birational and is not an isomorphism. Denote by $W$ the image of $pr_c|_X$. Then $W$ is non-normal and $pr_c|_X\colon X\to W$ is a normalization map. The non-normal locus $D$ of $W$ is a line in $\mathbb{P}^{N-1}$. The pre-image $\bar{D}\colon = pr_c|_X^{-1}(D)$ is described as follows:

-   if $X$ is $\mathbb{P}^2$ embedded in $\mathbb{P}^5$, then $\bar{D}\in |\mathcal{O}_{\mathbb{P}^2}(1)|$.

-   if $X$ is a cone $C_{N-1}\hookrightarrow \mathbb{P}^N$, then $\bar{D}$ is a sum of two rulings.

-   if $X$ is $\Sigma_d\hookrightarrow \mathbb{P}^N$, either $\bar{D}\in |\Delta_0+\Gamma|$, $N=d+3$, or $\bar{D}=\Delta_0$, $N=d+5$.

$pr_c|_X\colon X\to W$ is a normalization map by Cor \[lineIntersecting\].

We show that $D$ is a line in $\mathbb{P}^{N-1}$ by contradiction hypothesis. Assume $D$ is not a line, then there are two points $p,q\in D$ such that the line $L_{p,q}$ passing $p,q$ is not contained in $D$. Let $H_{p,q}\subset \mathbb{P}^N$ be the pre-image of $L_{p,q}$ with respect to $pr_c$. Then $H_{p,q}\cap X$ has length great than $4$ which contradicts Thm \[linearsectionthm\]. Therefore $D$ is a 