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Non-rigid quartic $3$ -folds

  • Hamid Ahmadinezhad (a1) and Anne-Sophie Kaloghiros (a2) (a3)

Abstract

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 $8$ .

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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.

References

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[Ahm12]Ahmadinezhad, H., On del Pezzo fibrations that are not birationally rigid, J. Lond. Math. Soc. (2) 86 (2012), 3662; MR 2959294.
[Ahm14]Ahmadinezhad, H., On pliability of del Pezzo fibrations and Cox rings, J. reine angew. Math. (2014), doi:10.1515/crelle-2014-0095.
[AGV85]Arnol’d, V. I., Guseĭn-Zade, S. M. and Varchenko, A. N., Singularities of differentiable maps: Vol. I: the classification of critical points, caustics and wave fronts, Monographs in Mathematics, vol. 82 (Birkhäuser, Boston, MA, 1985); translated from the Russian by Ian Porteous and Mark Reynolds.
[BCHM10]Birkar, C., Cascini, P., Hacon, C. D. and McKernan, J., Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), 405468.
[Bro99]Brown, G., Flips arising as quotients of hypersurfaces, Math. Proc. Cambridge Philos. Soc. 127 (1999), 1331.
[Cor95]Corti, A., Factoring birational maps of threefolds after Sarkisov, J. Algebraic Geom. 4 (1995), 223254.
[CM04]Corti, A. and Mella, M., Birational geometry of terminal quartic 3-folds. I, Amer. J. Math. 126 (2004), 739761.
[CPR00]Corti, A., Pukhlikov, A. and Reid, M., Fano 3-fold hypersurfaces, in Explicit birational geometry of 3-folds, London Mathematical Society Lecture Note Series, vol. 281 (Cambridge University Press, Cambridge, 2000), 175258.
[GRD]The graded ring database, http://www.grdb.co.uk/.
[HM13]Hacon, C. D. and McKernan, J., The Sarkisov program, J. Algebraic Geom. 22 (2013), 389405.
[HK00]Hu, Y. and Keel, S., Mori dream spaces and GIT, Michigan Math. J. 48 (2000), 331348.
[IM71]Iskovskikh, V. A. and Manin, Ju. I., Three-dimensional quartics and counterexamples to the Lüroth problem, Mat. Sb. (N.S.) 86 (1971), 140166.
[IP99]Iskovskikh, V. A. and Prokhorov, Yu. G., Fano varieties, in Algebraic geometry, V, Encyclopaedia of Mathematical Sciences, vol. 47 (Springer, Berlin, 1999), 1247.
[Kal11]Kaloghiros, A.-S., The defect of Fano 3-folds, J. Algebraic Geom. 20 (2011), 127149.
[Kal12]Kaloghiros, A.-S., A classification of terminal quartic 3-folds and applications to rationality questions, Math. Ann. 354 (2012), 263296.
[Kal13]Kaloghiros, A.-S., Relations in the Sarkisov program, Compositio Math. 149 (2013), 16851709.
[KKL14]Kaloghiros, A.-S., Küronya, A. and Lazić, V., Finite generation and geography of models, in Minimal models and extremal rays, Advanced Studies in Pure Mathematics (Mathematical Society of Japan, Tokyo, 2014), to appear.
[Kaw01]Kawakita, M., Divisorial contractions in dimension three which contract divisors to smooth points, Invent. Math. 145 (2001), 105119.
[Kaw02]Kawakita, M., Divisorial contractions in dimension three which contract divisors to compound A 1 points, Compositio Math. 133 (2002), 95116.
[Kaw03]Kawakita, M., General elephants of three-fold divisorial contractions, J. Amer. Math. Soc. 16 (2003), 331362 (electronic).
[Kol98]Kollár, J., Real algebraic threefolds. I. Terminal singularities, Collect. Math. 49 (1998), 335360; dedicated to the memory of Fernando Serrano.
[KM98]Kollár, J. and Mori, S., Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134 (Cambridge University Press, Cambridge, 1998).
[Mar96]Markushevich, D., Minimal discrepancy for a terminal cDV singularity is 1, J. Math. Sci. Univ. Tokyo 3 (1996), 445456.
[Mel04]Mella, M., Birational geometry of quartic 3-folds. II. The importance of being ℚ-factorial, Math. Ann. 330 (2004), 107126.
[Nam97]Namikawa, Y., Smoothing Fano 3-folds, J. Algebraic Geom. 6 (1997), 307324.
[NS95]Namikawa, Y. and Steenbrink, J. H. M., Global smoothing of Calabi–Yau threefolds, Invent. Math. 122 (1995), 403419.
[Pet98]Pettersen, K. F., On nodal determinantal quartic hypersurfaces in $\mathbf{P}^{4}$, PhD thesis, University of Oslo (1998).
[Rei87]Reid, M., Young person’s guide to canonical singularities, in Algebraic geometry (Bowdoin, Brunswick, Maine, 1985), Proceedings of Symposia in Pure Mathematics, vol. 46 (American Mathematical Society, Providence, RI, 1987), 345414.
[Tod33]Todd, J. A., Configurations defined by six lines in space of three dimensions, Math. Proc. Cambridge Philos. Soc. 29 (1933), 5268.
[Tod35]Todd, J. A., A note on two special primals in four dimensions, Q. J. Math. 6 (1935), 129136.
[Tod36]Todd, J. A., On a quartic primal with forty-five nodes, in space of four dimensions, Q. J. Math. 7 (1936), 169174.
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Non-rigid quartic $3$ -folds

  • Hamid Ahmadinezhad (a1) and Anne-Sophie Kaloghiros (a2) (a3)

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