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Singularities on the base of a Fano type fibration

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Published/Copyright: May 16, 2014
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2016 Issue 715

Abstract

Let f:X→Z be a Mori fibre space. McKernan conjectured that the singularities of Z are bounded in terms of the singularities of X. Shokurov independently proposed a more general conjecture in the setting of pairs: let (X,B) be a pair with klt singularities and f:X→Z be a contraction such that KX+B∼ℝ0/Z and that the general fibres of f are Fano type varieties; adjunction for fibre spaces produces a discriminant divisor BZ and a moduli divisor MZ on Z. It is then conjectured that the singularities of (Z,BZ+MZ) are bounded in terms of the singularities of (X,B). We prove Shokurov's conjecture when (F,SuppBF) belongs to a bounded family where F is a general fibre of f and KF+BF=(KX+B)|F.

Thanks to Yuri Prokhorov for useful discussions on his results with Mori. Thanks to Florin Ambro and Vyacheslav V. Shokurov for their historical comments. Also thanks to the referee for valuable corrections and suggestions.

Received: 2013-5-9
Revised: 2014-3-16
Published Online: 2014-5-16
Published in Print: 2016-6-1

© 2016 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. Global gradient estimates for elliptic equations of p(x)-Laplacian type with BMO nonlinearity
  3. Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras
  4. Mixing operators and small subsets of the circle
  5. Singularities on the base of a Fano type fibration
  6. Stable representation homology and Koszul duality
  7. An inverse spectral problem for a star graph of Krein strings
  8. Uniqueness of self-similar shrinkers with asymptotically cylindrical ends
  9. Ultraproducts, QWEP von Neumann algebras, and the Effros–Maréchal topology
https://doi.org/10.1515/crelle-2014-0033

Articles in the same Issue

  1. Frontmatter
  2. Global gradient estimates for elliptic equations of p(x)-Laplacian type with BMO nonlinearity
  3. Universal and homogeneous embeddings of dual polar spaces of rank 3 defined over quadratic alternative division algebras
  4. Mixing operators and small subsets of the circle
  5. Singularities on the base of a Fano type fibration
  6. Stable representation homology and Koszul duality
  7. An inverse spectral problem for a star graph of Krein strings
  8. Uniqueness of self-similar shrinkers with asymptotically cylindrical ends
  9. Ultraproducts, QWEP von Neumann algebras, and the Effros–Maréchal topology
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