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Let $\phi:X\rightarrow Y$ be a birational regular map between projective varieties where $Y$ is non-singular. Define $C=\{q\in Y:\dim(\phi^{-1}(q))>0)\}$. Let $G=\phi^{-1}(C)$. I saw the following statement:

"Irreducible components of $G$ are sub-varieties of codimension $1$".

Could someone please explain or give a hint of why this should be true.

Thanks in advance.

PS: This subject is completely new to me. It would be really helpful if someone explain the answer with more details.

Cusp
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  • The intuition is that contracting a variety of codimension $>1$ always produces singularities. In fact one can show that if a component of $C$ has codimension $>1$ then $Y$ is not $\mathbb{Q}$-factorial, which means that all Weil divisors are $\mathbb{Q}$-Cartier (on a smooth variety all Weil divisors are Cartier divisors, in particularly $\mathbb{Q}$-Cartier). This is worked out in https://mathoverflow.net/questions/31696/best-strategy-for-small-resolutions – Pol van Hoften Jul 26 '19 at 12:15
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    The basic reason is `Hartog's theorem'. If $Z=\mathrm{Spec A}$ is a normal affine variety and $T\subset Z$ is a closed subset of codimension at least 2, then $\Gamma(Z-T,\mathcal{O})=A$. – Mohan Jul 26 '19 at 14:31

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