Let $S$ be a smooth projective surface over $\mathbb{C}$, let $\Sigma$ be the linear system of a very ample divisor $D$ on $S$, and let $d$ be the dimension of $\Sigma$.
I know that if $\mathcal{L}$ is the line bundle associated to $D$, the set $\Sigma$ is in a natural bijection with $(\Gamma(S,\mathcal{L})\setminus\{0\})/\mathbb{C}^*$ and is therefore a projective space. Some time ago, I read in a paper that we can identify $\Sigma$ with $\mathbb{P}^{d*}$, where $\mathbb{P}^{d*}$ is the dual projective space of $\mathbb{P}^{d}$.
I am just learning about linear systems, and is not clear for me how the identification $\Sigma=\mathbb{P}^{d*}$ goes. I will be grateful if someone can illuminate me.