= Canonical bundle formula for a point blowup
{title2=$K_{\widetilde M}\cong\phi^*K_M\otimes[E]^{\otimes(n-1)}$}
For the <blowup of a complex manifold at a point> in complex dimension $n$, the differential gives a section of $K_{\widetilde M}\otimes(\phi^*K_M)^{-1}$ whose <divisor on a complex manifold> is $(n-1)E$. Indeed, a local chart has $x_i=u$ and $x_j=ut_j$ for $j\ne i$, so its <Jacobian determinant> is $\pm u^{n-1}$. The <holomorphic line bundle associated to a divisor> then gives the formula for the <canonical bundle>.
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