Let $X$ be a smooth projective surface and $K$ a canonical divisor on $X$.
Suppose $V$ and $W$ are subspaces of $H^0(X, \mathcal{O}_X(nK))$ (for $n$ large).
Q: How can we blow-up $X$ to obtain $\pi: Y \rightarrow X$ so that the subsheaves of $\mathcal{O}_Y(n\pi ^*K)$ generated by the sections in $V$ and $W$ are invertible?
It seems that $\pi$ is blow-up along ideal sheaves $I,J$ associated to $V,W$. $\underline{\text{Right}}$?
If it's true, then $\underline{\text{what are} \ I \ \text{and} \ J}$?
and let $I':=\pi^{-1}I\cdot \mathcal{O}_Y$. Then $I'$ is invertible. $\underline{\text{Is} \ I' \ \text{subsheaf of} \ \mathcal{O}_Y(n\pi ^*K)}$?