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.



what is the linear system on a cubic surface giving the blow-down map to the plane
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.


