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arXiv · 2006.14981

Crepant resolutions of double covers: On the Cynk-Hulek criterion for crepant resolutions of double cover

Colin Ingalls·Adam Logan
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Abstract

A collection $S = \{D_1,\ldots, D_n\}$ of divisors in a smooth variety $X$ is an {\em arrangement} if intersections of all subsets of $S$ are smooth. We show that a double cover of $X$ ramified on an arrangement has a crepant resolution under additional hypotheses. Namely, we assume that all intersection components that change the canonical divisor when blown up satisfy are {\em splayed}, a property of the tangent spaces of the components first studied by Faber. This strengthens a result of Cynk and Hulek, which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in $S$ and prove that it has a primary decomposition where the primary components are supported on exactly the subvarieties which are blown up in the course of constructing the crepant resolution of the double cover.

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Colin Ingalls, Adam Logan. 2020-06-26. Crepant resolutions of double covers: On the Cynk-Hulek criterion for crepant resolutions of double cover. https://arxiv.org/abs/2006.14981

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