Let $X$ be a projective variety over $\mathbb{C}$ of dimension $n$ and $L$ be an invertible sheaf on $X$ with no fixed part and $h^0(L) \ge n+1$. Choose $n+1$ linearly independent global sections $s_0,...,s_n$ on $X$. Is the rational map $f:X \dashrightarrow \mathbb{P}^n$ defined by these $n+1$-sections, dominant? In other words, is it sufficient to choose $n+1$ linearly independent global sections to guarantee dominant rational map?
NB. If necessary assume that $X$ is non-singular.
The rational map $f$ is of course dominant to its image, but in genegral not to the $P^N$.For example, if $L$ is very ample, $dim H^0(L)$ might be very large and the map associated to this divisor is an embedding, not dominant.