I am reading lecture note 15 of Vakil's intersection theory course (Ref: https://virtualmath1.stanford.edu/~vakil/245/)
Let $\mathcal{L}$ be a line bundle on a scheme (let's make everything easier: A smooth projective variety) $X$ and $V\subset H^0(X,\mathcal{L})$. Let $B$ be the base locus of $V$. Let $\tilde X=Bl_BX$ be the blow up and $E$ be the exceptional divisor. Then it is stated in the notes that $\pi^\ast\mathcal{L}-E$ is base locus free.
Q1. I want to prove this in this way: Let $x\in \tilde X$ be a point, we want to show that there's a section $s$ of $\pi^\ast\mathcal{L}-E$ which does not vanish at $x$. For $x\notin E$, this is easy since such a section already exists in $V$ on $X$. But what if $x\in E$?
Q2 Can we generalize this to vector bundles? Let me try to formulate a theorem: Let $\mathcal F$ be a vector bundle of rank 2 and $s\in H^0(X,\mathcal{F})$. Let $B=Z(s)$ and $\tilde X=Bl_BX$. Then $\pi^\ast det \mathcal{F}-2E$ is base point free. (The reason for guessing in this way is to think of what happens for $\mathcal{F}=\mathcal{L}^{\oplus 2}$).
Thank you sincerely for your kind help!