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

TL;DR

This paper extends the Cynk-Hulek criterion for crepant resolutions of double covers by introducing the splayed intersection condition, allowing for broader applicability in resolving singularities of double covers ramified along divisors.

Contribution

It generalizes previous results by weakening intersection hypotheses and analyzes the singular subscheme structure in the resolution process.

Findings

Crepant resolutions exist under splayed intersection conditions.

The singular subscheme has a primary decomposition supported on blown-up subvarieties.

The results strengthen the criteria for crepant resolutions of double covers.

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.