Projective morphism defined by linear systems

Question

Let $X$ be a normal variety and $D$ be a Cartier divisor, suppose $\sigma, \delta$ are two basepoint free linear systems in $|D|$, then we have two morphisms defined by these two linear system: $$\phi_{\sigma}: X \to \mathbb{P}^n,\qquad \phi_{\delta}: X \to \mathbb{P}^m. $$ My question is: are the images $\phi_{\sigma}(X), \phi_{\delta}(X) $ the same (maybe after a normalization)?

Answer 1

No. For example, let $X$ be a smooth quartic surface in $\mathbb{P}^3$ and let $L$ be the restriction of $\mathcal{O}(1)$.

Now $L$ is very ample, and its complete linear system is the (isomorphic) embedding of $X$. On the other hand, if we project from a point $p$ not on $X$, the corresponding linear system is still basepoint free, and gives a surjective map $X \to \mathbb{P}^2$. So the images are not isomorphic (from the adjunction formula, the canonical divisor on $X$ is trivial, so $X \not\cong \mathbb{P}^2$.)