Consider the affine cone $X \subset \mathbb A^6$ over the Veronese surface $V \cong \mathbb P^2 \subset \mathbb P^5$. After blowing up the cone point we get a resolution $Y \rightarrow X$, with exceptional divisor $E \cong \mathbb P^2$.
How do you see that $\mathcal O_E(-E) = \mathcal O(2)$?