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Let $X$ be a conifold defined by the equation $xy-zw=0$ in $\mathbb{C}^4$ and $\tilde{X}$ its crepant resolution, which is isomorphic to $\mathcal{O}_{\mathbb{P^1}}(-1)^{\oplus 2}$. Then there is a map $\tilde{X} \rightarrow X \rightarrow \mathbb{C}$, where the first map is the resolution of the singularity and the second map is $(x,y,z,w)\mapsto xy$ (or equivalently $zw$).

My question is, what is the fiber of the map over $0 \in \mathbb{C}$? Does it depend on the crepant resolution (divisor we blow-up along) we take?

Thank you in advance

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  • $\begingroup$ What is your problem in describing the fiber? $\endgroup$
    – Sasha
    Feb 26, 2015 at 7:55
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    $\begingroup$ I love the typo in the title. I now want to name all singularities after types of trees. $\endgroup$
    – Aaron Bergman
    Feb 26, 2015 at 18:46

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