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Relative desingularization and principalization of ideals
Dan Abramovich1; Michael Temkin2; Jarosław Włodarczyk3
1Department of Mathematics Box 1917 Brown University Providence, RI 02912, USA
2Einstein Institute of Mathematics The Hebrew University of Jerusalem Edmond J. Safra Campus Giv’at Ram, Jerusalem, 91904, Israel
3Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907-2067, USA
Publications Mathématiques de l'IHÉS, Online first, pp. 1-103
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In characteristic $0$, we construct relative principalization of ideals for logarithmically regular morphisms of logarithmic schemes, and use it to construct logarithmically regular desingularization of morphisms. These constructions are relatively canonical and even functorial with respect to logarithmically regular morphisms and arbitrary base changes. Relative canonicity means that the principalization requires a fine enough non-canonical modification of the base, and once it is chosen the process is canonical. As a consequence we deduce the semistable reduction theorem over arbitrary valuation rings.

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DOI: 10.5802/pmihes.29

Dan Abramovich  1 ; Michael Temkin  2 ; Jarosław Włodarczyk  3

1 Department of Mathematics Box 1917 Brown University Providence, RI 02912, USA
2 Einstein Institute of Mathematics The Hebrew University of Jerusalem Edmond J. Safra Campus Giv’at Ram, Jerusalem, 91904, Israel
3 Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907-2067, USA 
Dan Abramovich; Michael Temkin; Jarosław Włodarczyk. Relative desingularization and principalization of ideals. Publications Mathématiques de l'IHÉS, Online first, pp. 1-103
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