본문 바로가기
KERD
논문KCI 등재

On cylindrical smooth rational Fano fourfolds

Nguyen Thi Anh Hang|Michael Hoff|Hoang Le Truong
대한수학회 (The Korean Mathematical Society)|2022.01|대한수학회지 (Journal of the KMS)|제59권 제1호|pp. 87-103|DOI 10.4134/JKMS.j210140

1

조회

0

다운로드

0

인용

0

참고문헌

키워드

Fano varietycylinders
주제분류수학

초록

We construct new families of smooth Fano fourfolds with Picard rank $1$ which contain open $\mathbb A^1$-cylinders, that is, Zariski open subsets of the form $Z \times \mathbb A^1$, where $Z$ is a quasiprojective variety. In particular, we show that every Mukai fourfold of genus $8$ is cylindrical and there exists a family of cylindrical Gushel-Mukai fourfolds.

Abstract

We construct new families of smooth Fano fourfolds with Picard rank $1$ which contain open $\mathbb A^1$-cylinders, that is, Zariski open subsets of the form $Z \times \mathbb A^1$, where $Z$ is a quasiprojective variety. In particular, we show that every Mukai fourfold of genus $8$ is cylindrical and there exists a family of cylindrical Gushel-Mukai fourfolds.

참고문헌 (30)

  1. T. Ando, "On extremal rays of the higher-dimensional varieties", Invent. Math, 81(2), pp.347~357 (1985)[DOI]
  2. A. Alzati, "Special birational transformations of projective spaces", Adv. Math, 289, pp.567~602 (2016)[DOI]
  3. M. Andreatta, "On contractions of smooth varieties", J. Algebraic Geom, 7(2), pp.253~312 (1998)
  4. A. Borel, "Characteristic classes and homogeneous spaces. I", Amer. J. Math, 80, pp.458~538 (1958)[DOI]
  5. I. Cheltsov, "Cylinders in Fano varieties", EMS Surv. Math. Sci, 8(1-2), pp.39~105 (2021)[DOI]
  6. I. Cheltsov, "Cylinders in singular del Pezzo surfaces", Compos. Math, 152(6), pp.1198~1224 (2016)[DOI]
  7. I. Cheltsov, "Affine cones over smooth cubic surfaces", J. Eur. Math. Soc. (JEMS), 18(7), pp.1537~1564 (2016)[DOI]
  8. O. Debarre, "Recent advances in algebraic geometry", pp.123~155 (2015), Cambridge Univ. Press
  9. I. V. Dolgachev, "Classical Algebraic Geometry" (2012), Cambridge University Press
  10. H. Finkelnberg, "Small resolutions of nodal cubic threefolds", Nederl. Akad. Wetensch. Indag. Math, 51(2), pp.185~198 (1989)
  11. N. P. Gushel’, "Fano varieties of genus 6", Izv. Akad. Nauk SSSR Ser. Mat, 46(6), pp.1159~1174 (1982)
  12. B. Hassett, "Special cubic fourfolds", Compositio Math, 120(1), pp.1~23 (2000)[DOI]
  13. B. Hassett, "Cremona transformations and derived equivalences of K3 surfaces", Compos. Math, 154(7), pp.1508~1533 (2018)[DOI]
  14. M. Hoff, "New examples of rational Gushel-Mukai fourfolds", Math. Z, 296(3-4), pp.1585~1591 (2020)[DOI]
  15. P. Ionescu, "Algebraic geometry, Bucharest 1982", pp.142~186 (1984), Springer
  16. V. A. Iskovskih, "Fano threefolds. I", Izv. Akad. Nauk SSSR Ser. Mat, 41(3), pp.516~562 (1977)
  17. V. A. Iskovskikh, "Algebraic geometry, V", pp.1~247 (1999), Springer
  18. T. Kambayashi, "On flat fibrations by the affine line", Illinois J. Math, 22(4), pp.662~671 (1978)
  19. T. Kambayashi, "Flat families of affine lines are affine-line bundles", Illinois J. Math, 29(4), pp.672~681 (1985)
  20. T. Kishimoto, "Affine algebraic geometry", pp.123~163 (2011), Amer. Math. Soc
  21. T. Kishimoto, "Affine cones over Fano threefolds and additive group actions", Osaka J. Math, 51(4), pp.1093~1112 (2014)
  22. E. L. Livorni, "Algebraic geometry", pp.155~179 (1990), Springer
  23. S. Mukai, "Biregular classification of Fano 3-folds and Fano manifolds of coindex 3", Proc. Nat. Acad. Sci. U.S.A, 86(9), pp.3000~3002 (1989)[DOI]
  24. J. Park, "Ga-actions on the complements of hypersurfaces", Transformation Groups (2020)[DOI]
  25. Y. Prokhorov, "Examples of cylindrical Fano fourfolds", Eur. J. Math, 2(1), pp.262~282 (2016)[DOI]
  26. Y. Prokhorov, "Algebraic varieties and automorphism groups", pp.443~463 (2017), Math. Soc. Japan
  27. Y. Prokhorov, "Affine cones over Fano-Mukai fourfolds of genus 10 are flexible", Arxiv:2005.12092, ~ (2020)
  28. V V Przhyjalkowski, "Hyperelliptic and trigonal Fano threefolds", Izvestiya: Mathematics, 69(2), pp.365~421 (2005), IOP Publishing[DOI]
  29. N. I. Shepherd-Barron, "The Hasse principle for 9-nodal cubic threefolds", Arxiv:1210.5178, ~ (2012)
  30. F. Xu, "On the smooth linear section of the Grassmannian Gr(2, n)", PhD, pp.2012

관련 논문AI추천

Birationally rigid complete intersections of codimension twoAleksandr Pukhlikov, Daniel Evans대한수학회 · 대한수학회보2017 · 제54권 제5호Smooth horospherical varieties of Picard number one as linear sections of rational homogeneous varieties홍재현대한수학회 · 대한수학회지2016 · 제53권 제2호FANO MANIFOLDS AND BLOW-UPS OF LOW-DIMENSIONAL SUBVARIETIESElena Chierici, Gianluca Occhetta대한수학회 · 대한수학회지2010 · 제47권 제1호Some irrational quartic threefolds원준영, 홍규식대한수학회 · 대한수학회보2022 · 제59권 제2호On factorial nodal hypersurfaces in $\PP^{4}$홍규식대한수학회 · 대한수학회보2025 · 제62권 제5호A sufficient condition for a toric weak Fano 4-fold to be deformed to a Fano manifoldHiroshi Sato대한수학회 · 대한수학회지2021 · 제58권 제5호

이 간행물의 다른 논문 (11건)

Synchronized components of a subshiftManouchehr Shahamatpp. 1-12Parabolic quaternionic Monge-Amp\`{e}re equation on compact manifolds with a flat hyperK\"ahler metricJiaogen Zhangpp. 13-33Practical FHE parameters against lattice attacks이동건, 천정희, 손용하pp. 35-51The interior gradient estimate for a class of mixed Hessian curvature equationsJundong Zhoupp. 53-69Gorenstein sequences of high socle degrees박정필, 신용수pp. 71-85
원문 보기