Linear system on $\mathbb P^2$ I am not sure about it : this is a probably a really naive question. I saw in a lecture note the author talking about the linear system $ |L - p_1 - p_2 - p_3|$ where $L \subset \mathbb P^2$ is a line and the $p_i$ are points. Since $L - p_1 - p_2 - p_3$ is not a divisor I don't understand what does it mean. Is this all the rational map $f \in \mathbb C(\mathbb P^2)$ such that $f$ has a pole at $L$ and a zero at the $p_i$ ? 
If this is the case, then such map should be $f/l$ with $l$ a linear polynomial which is zero on $L$. The points were assumed non-collinear, so there is no such $f$ and so $|L - p_1 - p_2 - p_3| = \emptyset$. Does this make sense ? Am I mistaken somewhere ? Thanks a lot for any comments ! 
 A: Think of $|2L - p_1 - p_2 - p_3|$ as the sublinear system of $|2L|$ on $\mathbb P^2$ consisting of all quadrics on $\mathbb P^2$ that vanish at $p_1, p_2, p_3$. This is a linear system, but it is not a complete linear system.
Let's do a dimension count. The linear system of quadrics on $\mathbb P^2$ has dimension 5, so after imposing the constraint that the quadrics pass through $p_1, p_2, p_3$, you end up with a linear system of dimension 2. Thus your linear system defines a rational map from $\mathbb P^2$ to $\mathbb P^2$. If the three points are $[1:0:0]$, $[0:1:0]$, $[0:0:1]$, for example, then the equation for this rational map is
$$ [x_0 : x_1: x_2] \mapsto [x_1x_2:x_0x_2:x_0x_1].$$
In your notes, this map is called the elementary quadratic transformation, but I have heard people calling it a Cremona transformation.
Note that if $\pi : Bl_{(p_1, p_2, p_3)}\mathbb P^2 \to \mathbb P^2$ is the blow-up of $\mathbb P^2$ at $p_1, p_2, p_3$, then $\pi^\star |2L-p_1-p_2-p_3|$ is the complete linear system $|2L-E_1-E_2-E_3|$, where $E_i$ is the exceptional divisor above the point $p_i$. My guess would be that the sections of $\pi^\star |2L-p_1-p_2-p_3|$ define a morphism from $Bl_{(p_1, p_2, p_3)}\mathbb P^2$ to $\mathbb P^2$ but $|2L-p_1-p_2-p_3|$ only defines a rational map from $\mathbb P^2 $ to $\mathbb P^2$.
