Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-26T16:56:07.676Z Has data issue: false hasContentIssue false
  • English
  • Français

Birationally Rigid Complete Intersections with a Singular Point of High Multiplicity

Part of: Birational geometry

Published online by Cambridge University Press:  26 September 2018

A. V. Pukhlikov
A. V. Pukhlikov*
Affiliation:
Department of Mathematical Sciences, Peach Street, The University of Liverpool, Liverpool L69 7ZL, UK (pukh@liverpool.ac.uk)
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove the birational rigidity of Fano complete intersections of index 1 with a singular point of high multiplicity, which can be close to the degree of the variety. In particular, the groups of birational and biregular automorphisms of these varieties are equal, and they are non-rational. The proof is based on the techniques of the method of maximal singularities, the generalized 4n2-inequality for complete intersection singularities and the technique of hypertangent divisors.

Keywords

birational rigiditymaximal singularitymultiplicityhypertangent divisorcomplete intersection singularity

MSC classification

Primary: 14E05: Rational and birational maps
Secondary: 14E07: Birational automorphisms, Cremona group and generalizations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 1 , February 2019 , pp. 221 - 239
Copyright
Copyright © Edinburgh Mathematical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Dedicated to Yu. I. Manin on the occasion of his 80th birthday

References

1.Call, F. and Lyubeznik, G., A simple proof of Grothendieck's theorem on the parafactoriality of local rings, Contemp. Math. 159 (1994), 15–18.Google Scholar
2.Cheltsov, I. and Park, J., Birationally rigid Fano threefold hypersurfaces, Memoires of AMS 246 (2017), 1–117.Google Scholar
3.Corti, A. and Mella, M., Birational geometry of terminal quartic 3-folds. I, Amer. J. Math. 126(4) (2004), 739–761.Google Scholar
4.Corti, A., Pukhlikov, A. and Reid, M., Fano 3-fold hypersurfaces, In Explicit birational geometry of threefolds (eds Corti, A. and Reid, M.), pp. 175–258, London Mathematical Society Lecture Note Series, Volume 281 (Cambridge University Press, 2000).Google Scholar
5.de Fernex, T., Birational geometry of singular Fano hypersurfaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 17 (2017), 911–929.Google Scholar
6.Eckl, Th. and Pukhlikov, A. V., On the locus of non-rigid hypersurfaces, In Automorphisms in birational and affine geometry (eds Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y. G. and Zaidenberg, M.), pp. 121–139, Springer Proceedings in Mathematics and Statistics, Volume 79 (Springer, 2014).Google Scholar
7.Evans, D. and Pukhlikov, A., Birationally rigid complete intersections of codimension two, Bull. Korean Math. Soc. 54(5) (2017), 1627–1654.Google Scholar
8.Iskovskikh, V. A. and Manin, Yu. I., Three-dimensional quartics and counterexamples to the Lüroth problem, Math. USSR Sb. 86(1) (1971), 140–166.Google Scholar
9.Mella, M., Birational geometry of quartic 3-folds. II. The importance of being ℚ-factorial, Math. Ann. 330(1) (2004), 107–126.Google Scholar
10.Mullany, R., Fano double spaces with a big singular locus, Math. Notes 87(3) (2010), 444–448.Google Scholar
11.Okada, T., Birational Mori fiber structures of ℚ-Fano 3-fold weighted complete intersections, Proc. Lond. Math. Soc. 109(6) (2014), 1549–1600.Google Scholar
12.Pukhlikov, A. V., Birational automorphisms of a three-dimensional quartic with an elementary singularity, Math. USSR Sb. 63 (1989), 457–482.Google Scholar
13.Pukhlikov, A. V., Birationally rigid Fano complete intersections, Crelle J. für die reine und angew. Math. 541 (2001), 55–79.Google Scholar
14.Pukhlikov, A. V., Birationally rigid Fano hypersurfaces with isolated singularities, Sb. Math. 193(3) (2002), 445–471.Google Scholar
15.Pukhlikov, A. V., Birationally rigid singular Fano hypersurfaces, J. Math. Sci. 115(3) (2003), 2428–2436.Google Scholar
16.Pukhlikov, A., Birationally rigid varieties, Mathematical Surveys and Monographs, Volume 190 (American Mathematical Society, Providence, RI, 2013).Google Scholar
17.Pukhlikov, A. V., The 4n 2-inequality for complete intersection singularities, Arnold Math. J. 3 (2017), 187–196.Google Scholar
18Shramov, K. A., Birational automorphisms of nodal quartic threefolds, arXiv:0803.4348, 32 p.Google Scholar
19.Suzuki, F., Birational rigidity of complete intersections, Math. Z. 285(1–2) (2017), 479–492.Google Scholar