On the Cynk-Hulek criterion for crepant resolutions of double covers. (English) Zbl 1509.14033
Summary: A collection \(A=\{D_1,\dots,D_n\}\) of divisors on a smooth variety \(X\) is an arrangement if the intersection of every subset of \(A\) is smooth. We show that, if \(X\) is defined over a field of characteristic not equal to 2, 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 are splayed, a property of the tangent spaces of the components first studied by E. Faber [Publ. Res. Inst. Math. Sci. 49, No. 3, 393–412 (2013; Zbl 1277.32032)]. This strengthens a result of S. Cynk and K. Hulek [Can. Math. Bull. 50, No. 4, 486–503 (2007; Zbl 1141.14009)], which requires a stronger hypothesis on the intersection components. Further, we study the singular subscheme of the union of the divisors in \(A\) 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.
MSC:
14E20 | Coverings in algebraic geometry |
14J32 | Calabi-Yau manifolds (algebro-geometric aspects) |
14B05 | Singularities in algebraic geometry |
13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |
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