PomeCorollary 9.10.10.178E0605C99D Let $X$ be an irreducible compact complex space and $E$ be a prime divisor over $X$. Then there is a modification $\pi \colon Y\rightarrow X$ such that $Y$ is smooth and $E$ can be realized as a prime divisor on $Y$.
Corollary 9.10.10178E0605C99D
Proof. Choose a smooth model $X'\rightarrow X$ on which $E$ is realized. By theorem 9.10.8, this proper bimeromorphic morphism is dominated by a modification $Y\rightarrow X$. After further resolving the pair if necessary, we may assume that $Y$ is smooth and that the strict transform of $E$ on $Y$ is smooth. This strict transform realizes the same prime divisor over $X$. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.