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Resolution of singularities of pairs preserving semi-simple normal crossings

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Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas Aims and scope Submit manuscript

Abstract

Let X denote a reduced algebraic variety and D a Weil divisor on X. The pair (X, D) is said to be semi-simple normal crossings (semi-snc) at \({a \in X}\) if X is simple normal crossings at a (i.e., a simple normal crossings hypersurface, with respect to a local embedding in a smooth ambient variety), and D is induced by the restriction to X of a hypersurface that is simple normal crossings with respect to X. We construct a composition of blowings-up \({f:\tilde{X}\rightarrow {X}}\) such that the transformed pair \({(\tilde{X}, \tilde{D})}\) is everywhere semi-simple normal crossings, and f is an isomorphism over the semi-simple normal crossings locus of (X, D). The result answers a question of Kollár.

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Authors and Affiliations

  1. The Fields Institute, 222 College Street, Toronto, ON, M5T 3J1, Canada

    Edward Bierstone & Franklin Vera Pacheco

  2. Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, M5S 2E4, Canada

    Edward Bierstone & Franklin Vera Pacheco

Authors
  1. Edward Bierstone
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  2. Franklin Vera Pacheco
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Correspondence to Edward Bierstone.

Additional information

In honour of Heisuke Hironaka on his eightieth birthday.

Research supported in part by NSERC Grants OGP0009070 and MRS342058.

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Bierstone, E., Vera Pacheco, F. Resolution of singularities of pairs preserving semi-simple normal crossings. RACSAM 107, 159–188 (2013). https://doi.org/10.1007/s13398-012-0092-4

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  • DOI: https://doi.org/10.1007/s13398-012-0092-4

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