Birationally rigid complete intersections of high codimension
- Authors: Evans D.1, Pukhlikov A.V.1
-
Affiliations:
- Department of Mathematical Sciences, University of Liverpool
- Issue: Vol 83, No 4 (2019)
- Pages: 100-128
- Section: Articles
- URL: https://journals.rcsi.science/1607-0046/article/view/133787
- DOI: https://doi.org/10.4213/im8782
- ID: 133787
Cite item
Open Access
Access granted
Subscription Access
Abstract
We prove that a Fano complete intersection of codimension $k$ and index $1$in the complex projective space ${\mathbb P}^{M+k}$ for $k\ge 20$ and$M\ge 8k\log k$ with at most multi-quadratic singularities is birationallysuperrigid. The codimension of the complement of the set of birationallysuperrigid complete intersections in the natural moduli space is shown tobe at least $(M-5k)(M-6k)/2$. The proof is based on the technique ofhypertangent divisors combined with the recently discovered$4n^2$-inequality for complete intersection singularities.
About the authors
Daniel Evans
Department of Mathematical Sciences, University of Liverpool
Email: sgdevans@liverpool.ac.uk
Aleksandr Valentinovich Pukhlikov
Department of Mathematical Sciences, University of Liverpool
Email: pukh@liv.ac.uk
Doctor of physico-mathematical sciences, no status
References
- A. Pukhlikov, Birationally rigid varieties, Math. Surveys Monogr., 190, Amer. Math. Soc., Providence, RI, 2013, vi+365 pp.
- A. V. Pukhlikov, “Birationally rigid Fano complete intersections”, J. Reine Angew. Math., 2001:541 (2001), 55–79
- A. V. Pukhlikov, “Birationally rigid Fano complete intersections. II”, J. Reine Angew. Math., 2014:688 (2014), 209–218
- F. Call, G. Lyubeznik, “A simple proof of Grothendieck's theorem on the parafactoriality of local rings”, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992), Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994, 15–18
- A. V. Pukhlikov, “Birationally rigid complete intersections with a singular point of high multiplicity”, Proc. Edinb. Math. Soc. (2), 62:1 (2019), 221–239
- A. V. Pukhlikov, “The $4n^2$-inequality for complete intersection singularities”, Arnold Math. J., 3:2 (2017), 187–196
- D. Evans, A. Pukhlikov, “Birationally rigid complete intersections of codimension two”, Bull. Korean Math. Soc., 54:5 (2017), 1627–1654
- Th. Eckl, A. Pukhlikov, “On the locus of nonrigid hypersurfaces”, Automorphisms in birational and affine geometry, Springer Proc. Math. Stat., 79, Springer, Cham, 2014, 121–139
- A. V. Pukhlikov, “Birational automorphisms of Fano hypersurfaces”, Invent. Math., 134:2 (1998), 401–426
- В. А. Исковских, А. В. Пухликов, “Бирациональные автоморфизмы многомерных алгебраических многообразий”, Итоги науки и техн. Сер. Соврем. мат. и ее прил. Темат. обз., 19, ВИНИТИ, М., 2001, 5–139
- I. Cheltsov, M. Grinenko, “Birational rigidity is not an open property”, Bull. Korean Math. Soc., 54:5 (2017), 1485–1526
- А. В. Пухликов, “Бирационально жесткие полные пересечения квадрик и кубик”, Изв. РАН. Сер. матем., 77:4 (2013), 161–214
- А. В. Пухликов, “Бирационально жесткие расслоения Фано. II”, Изв. РАН. Сер. матем., 79:4 (2015), 175–204
- А. В. Пухликов, “Бирациональная геометрия алгебраических многообразий, расслоенных на двойные пространства Фано”, Изв. РАН. Сер. матем., 81:3 (2017), 160–188
- Ю. Джонстон, “Бирационально жесткие вырожденные двойные квадрики и двойные кубики”, Матем. заметки, 102:4 (2017), 549–558
- F. Suzuki, “Birational rigidity of complete intersections”, Math. Z., 285:1-2 (2017), 479–492
- H. Ahmadinezhad, T. Okada, “Birationally rigid Pfaffian Fano $3$-folds”, Algebr. Geom., 5:2 (2018), 160–199
- И. А. Чельцов, Л. Воцлав, “Нерациональные полные пересечения”, Алгебраическая геометрия: методы, связи и приложения, Сборник статей. Посвящается памяти члена-корреспондента РАН Андрея Николаевича Тюрина, Тр. МИАН, 246, Наука, МАИК “Наука/Интерпериодика”, М., 2004, 316–320
- И. А. Чельцов, “Нерациональность четырехмерного гладкого полного пересечения квадрики и квартики, не содержащего плоскости”, Матем. сб., 194:11 (2003), 95–116
- T. Okada, “Birational Mori fiber structures of ${mathbb Q}$-Fano $3$-fold weighted complete intersections”, Proc. Lond. Math. Soc. (3), 109:6 (2014), 1549–1600
- T. Okada, “Birational Mori fiber structures of $mathbb Q$-Fano $3$-fold weighted complete intersections. II”, J. Reine Angew. Math., 2018:738 (2018), 73–129
- T. Okada, Birational Mori fiber structures of ${mathbb Q}$-Fano $3$-fold weighted complete intersections. III, 2014
- Ziquan Zhuang, Birational superrigidity and $K$-stability of Fano complete intersections of index one, 2018
Supplementary files
