Beauville Exercise II.20.
Let C be curve on a smooth, projective surface S. Show that there is a morphism of smooth surfaces $\tilde{S}\rightarrow S$ such that the strict transform $\tilde{C}\subset\tilde{S} $ of $C$ is smooth.
If $C$ is smooth then there is nothing to prove, so assume C has at least one singular point, call it $p$. Let $\pi :{\text{Bl}_pS}\rightarrow S$ be the blowup of $S$ at $p$. It follows from the adjunction formula that,
$$g_{\tilde{C}} =\frac{(\tilde{C}).(K_{\text{Bl}_pS}+\tilde{C})+2}{2}=\frac{(\pi^*C-mE).(\pi^*(K_S+C)-(m-1)E)+2}{2}=g_C-\frac{m(m-1)}{2},$$
with $m:=\text{mult}_p(C)$, E the exceptional divisor of the blowup. We have used that for any divisors $D,D’$ on $S$, $(\pi^*D).(\pi^*D’)=D.D’$, $E.(\pi^*D)=0$, $E^2=-1$, $\pi^*C=\tilde{C}+mE$, and $K_{\text{Bl}_pS}=\pi^*K_S+E$. Note that, because $C$ is singular at $p$, $m>1$, thus $g_{\tilde{C}}<g_C$. If $\tilde{C}$ is smooth we are done, else repeat the process above, replacing the pair $(S,C)$ with $(\text{Bl}_pS,\tilde{C})$. With each repetition the genus of the strict transform strictly decreases, and since the arithmetic genus of a projective curve is non-negative, this process must eventually terminate. We thus obtain a smooth surface $\tilde{S}$ and a smooth curve $\tilde{C}$ on S that is the strict transform of a finite number of blowups, which is the desired result.
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