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Fundam. Prikl. Mat., 1995, Volume 1, Issue 1, Pages 263–280 (Mi fpm56)  

This article is cited in 1 scientific paper (total in 1 paper)

On general elephant problem for three-dimensional $\mathbf{Q}$-Fano fiber spaces over a surface

Yu. G. Prokhorov

M. V. Lomonosov Moscow State University

Abstract: We consider $\mathbf{Q}$-Fano fiber spaces $X/S$ over a surface, i. e., a three-dimensional variety $X$ with terminal $\mathbf{Q}$-factorial singularities and a projective morphism $\varphi:X\to S$ onto a normal surface $S$ such that $\varphi_*\mathcal{O}_X=\mathcal{O}_S$, $\rho(X/S)=1$ and $-K_X$ $\varphi$-ample. In this situation we discuss Reid's conjecture on general elephants, i. e. on general members of the linear system $|-K_X+\varphi^*h|$. We prove that the surface $S$ has only cyclic quotient singularities, besides if for $X/S$ the elephants conjecture is true, then singularities of $S$ are Du Val singularities of the type $A_n$. In the last case some conditions on singularities of $X$ and $S$ are obtained.

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Received: 01.01.1995

Citation: Yu. G. Prokhorov, “On general elephant problem for three-dimensional $\mathbf{Q}$-Fano fiber spaces over a surface”, Fundam. Prikl. Mat., 1:1 (1995), 263–280

Citation in format AMSBIB
\Bibitem{Pro95}
\by Yu.~G.~Prokhorov
\paper On general elephant problem for three-dimensional $\mathbf{Q}$-Fano fiber spaces over a surface
\jour Fundam. Prikl. Mat.
\yr 1995
\vol 1
\issue 1
\pages 263--280
\mathnet{http://mi.mathnet.ru/fpm56}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1789364}
\zmath{https://zbmath.org/?q=an:0878.14030}
\elib{http://elibrary.ru/item.asp?id=9163042}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. G. Prokhorov, “On the existence of complements of the canonical divisor for Mori conic bundles”, Sb. Math., 188:11 (1997), 1665–1685  mathnet  crossref  crossref  mathscinet  zmath  isi
  • Фундаментальная и прикладная математика
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