Consider $X$ a smooth cubic surface in $\mathbb{P}^3$, and let $l_1,…,l_6$ be six disjoint lines contained in $X$.
What is the linear system giving the blow-down map $X \to \mathbb{P}^2$, so that the lines $l_k$ are contracted to points ?
The other way round is well-known : if $p_1,\dots,p_6$ are six points in general position, the rational map $\mathbb{P}^2 \to \mathbb{P}^3$ obtained by the linear system of cubic containing the six points has image a cubic surface.
Best Answer
As you probably know, if a representation of $X$ as blowup is given, $$ K_X = -3h + \sum l_i, $$ where $h$ is the line class. Consequently, the linear system $$ |-K_X + \sum l_i| $$ gives the required contraction.