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Francesco Polizzi
Francesco Polizzi
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Is the pull-back of canonical sheaf invertible (modulo torsion)?

Let $X$ be a $\mathbb{Q}$-Gorenstein (isolated) singularity of dimension at least $2$ and $f:Y \to X$ be a resolution of singularities. In this case the canonical sheaf $K_X$ is not necessarily invertible, it is only reflexive. Is the pull-back $f^*K_X$ invertible? If not, can we say that $f^*K_X/\mathrm{tors}$ is invertible?

Question. Is the pull-back $f^*K_X$ invertible? If not, can we say that $f^*K_X/\mathrm{tors}$ is invertible?

Is the pull-back of canonical sheaf invertible (modulo torsion)

Let $X$ be a $\mathbb{Q}$-Gorenstein (isolated) singularity of dimension at least $2$ and $f:Y \to X$ be a resolution of singularities. In this case the canonical sheaf $K_X$ is not necessarily invertible, it is only reflexive. Is the pull-back $f^*K_X$ invertible? If not, can we say that $f^*K_X/\mathrm{tors}$ is invertible?

Is the pull-back of canonical sheaf invertible (modulo torsion)?

Let $X$ be a $\mathbb{Q}$-Gorenstein (isolated) singularity of dimension at least $2$ and $f:Y \to X$ be a resolution of singularities. In this case the canonical sheaf $K_X$ is not necessarily invertible, it is only reflexive.

Question. Is the pull-back $f^*K_X$ invertible? If not, can we say that $f^*K_X/\mathrm{tors}$ is invertible?

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user45397
user45397
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Is the pull-back of canonical sheaf invertible (modulo torsion)

Let $X$ be a $\mathbb{Q}$-Gorenstein (isolated) singularity of dimension at least $2$ and $f:Y \to X$ be a resolution of singularities. In this case the canonical sheaf $K_X$ is not necessarily invertible, it is only reflexive. Is the pull-back $f^*K_X$ invertible? If not, can we say that $f^*K_X/\mathrm{tors}$ is invertible?