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Non-rigid quartic $3$-folds

Part of: Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  22 December 2015

Hamid Ahmadinezhad  and
Anne-Sophie Kaloghiros
Hamid Ahmadinezhad
Affiliation:
School of Mathematics, University of Bristol, BristolBS8 1TW, UK email h.ahmadinezhad@bristol.ac.uk
Anne-Sophie Kaloghiros
Affiliation:
Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK email anne-sophie.kaloghiros@brunel.ac.uk Current address:Brunel University London, Uxbridge UB8 3PH, UK
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Abstract

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Let $X\subset \mathbb{P}^{4}$ be a terminal factorial quartic $3$-fold. If $X$ is non-singular, $X$ is birationally rigid, i.e. the classical minimal model program on any terminal $\mathbb{Q}$-factorial projective variety $Z$ birational to $X$ always terminates with $X$. This no longer holds when $X$ is singular, but very few examples of non-rigid factorial quartics are known. In this article, we first bound the local analytic type of singularities that may occur on a terminal factorial quartic hypersurface $X\subset \mathbb{P}^{4}$. A singular point on such a hypersurface is of type $cA_{n}$ ($n\geqslant 1$), or of type $cD_{m}$ ($m\geqslant 4$) or of type $cE_{6},cE_{7}$ or $cE_{8}$. We first show that if $(P\in X)$ is of type $cA_{n}$, $n$ is at most $7$ and, if $(P\in X)$ is of type $cD_{m}$, $m$ is at most $8$. We then construct examples of non-rigid factorial quartic hypersurfaces whose singular loci consist (a) of a single point of type $cA_{n}$ for $2\leqslant n\leqslant 7$, (b) of a single point of type $cD_{m}$ for $m=4$ or $5$ and (c) of a single point of type $cE_{k}$ for $k=6,7$ or $8$.

Keywords

birational mapsquartic hypersurfacesbirational rigidity

MSC classification

Primary: 14E05: Rational and birational maps 14J30: $3$-folds 14J45: Fano varieties
Type
Research Article
Information
Compositio Mathematica , Volume 152 , Issue 5 , May 2016 , pp. 955 - 983
Creative Commons
Creative Common License - CCCreative Common License - BY
This article is distributed with Open Access under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided that the original work is properly cited.
Copyright
© The Authors 2015

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