= Anticanonical strict transform under a point blowup
{title2=$[-\widetilde Y]\cong K_{\widetilde X}\Longleftrightarrow\dim_{\mathbb C}X=2$}
Let $Y$ be a smooth <complex analytic hypersurface> through the blowup centre with $[-Y]\cong K_X$. Its multiplicity at that point is one, so $\sigma^*Y=\widetilde Y+E$. Combining this with the <canonical bundle formula for a point blowup> gives $[-\widetilde Y]\cong K_{\widetilde X}\otimes[E]^{\otimes(2-n)}$. The <nontrivial powers of the exceptional line bundle of a point blowup> prove the equivalence for $n\ge2$. On a <compact> complex curve the premise cannot hold for an irreducible hypersurface: it is one point, so its negative divisor has degree $-1$, whereas the <canonical bundle> has even degree $2g-2$.
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