What is the corresponding linear system to the blowup of a surface?

Question

Suppose we have a smooth projective surface $S$, $p$ is a point on this surface. We blowup $S$ at this point by $\pi:\tilde{S}\to S$. Then $\pi$ is a birational morphism. By which linear system $\pi$ is given?

Answer 1

The morphism coming from a linear system is a morphism from to the Blow-up of the base locus of the system, so you have to take the system of divisors going through $p$. See the wikipedia entry for linear systems, for instance