This question is about the definition and properties of a linear system of divisors $g^r_d$ of dimension $r$ and degree $d$. Where can I find an extended exposition of $g^r_d$ and linear systems in general? Does the notation $g^r_d$ apply only to linear systems on curves rather than on higher-dimensional varieties?
We have that a $g^1_d$ on a curve $C$ gives a map $C \to \mathbb P^1$ of degree $d$. Intuitively the above map is given by $(s_1:s_2)$ where $s_1$,$s_2$ are sections of the line bundle in the definition of $g^1_d$ and $s_1$,$s_2$ have $d$ zeroes each, so the map has degree $d$. But how do I see this rigorously?
It is known that a $g^r_d$ on a curve $C$ gives a map from $C$ to $\mathbb P^r$. Let the curve $C^′$ be the image of $C$ in $\mathbb P^r$. How does one prove that $d$ is the degree of the map from $C$ to $C^′$ times the degree of $C^′$ in $\mathbb P^r$ if the linear system is base point free (my guess is that this should follow from the formula $deg(f^*L)=deg(f)deg(L)$ for a finite map $f$ and a line bundle $L$, but I don't know an intuitive proof of this formula)? More generally, if we have a linear system of dimension $r$ and degree $d$ on a variety $X$, what does the number $d$ tell us about the resulting map from $X$ to $\mathbb P^r$?