DOI QR코드

DOI QR Code

BIRATIONALLY RIGID COMPLETE INTERSECTIONS OF CODIMENSION TWO

  • Evans, Daniel (Department of Mathematical Sciences The University of Liverpool) ;
  • Pukhlikov, Aleksandr (Department of Mathematical Sciences The University of Liverpool)
  • Received : 2016.08.17
  • Accepted : 2016.12.21
  • Published : 2017.09.30

Abstract

We prove that in the parameter space of M-dimensional Fano complete intersections of index one and codimension two the locus of varieties that are not birationally superrigid has codimension at least ${\frac{1}{2}}(M-9)(M-10)-1$.

Keywords

References

  1. H. Ahmadinezhad and T. Okada, Birationally rigid Pfaffian Fano 3-folds, ArXiv: 1508.02974.
  2. F. Call and 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), 15-18, Contemp. Math., 159, Amer. Math. Soc., Providence, RI, 1994.
  3. I. A. Cheltsov, Nonrationality of a four-dimensional smooth complete intersection of a quadric and a quadric not containing a plane, Sb. Math. 194 (2003), no. 11-12, 1679-1699. https://doi.org/10.1070/SM2003v194n11ABEH000782
  4. I. A. Cheltsov, Double cubics and double quartics, Math. Z. 253 (2006), no. 1, 75-86. https://doi.org/10.1007/s00209-005-0879-5
  5. I. A. Cheltsov and M. M. Grinenko, Birational rigidity is not an open property, Bull. Korean Math. Soc. 54 (2017), no. 5, 1485-1526. https://doi.org/10.4134/BKMS.B160677
  6. Th. Eckl and A. V. Pukhlikov, On the locus of non-rigid hypersurfaces, In: Automorphisms in birational and ane geometry, 121-139, Springer Proc. Math. Stat., 79, Springer, Cham, 2014.
  7. T. de Fernex, Birationally rigid hypersurfaces, Invent. Math. 192 (2013), no. 3, 533-566. https://doi.org/10.1007/s00222-012-0417-0
  8. T. de Fernex, Erratum to: Birationally rigid hypersurfaces, ArXiv: 1506:07086.
  9. W. Fulton, Intersection Theory, Springer-Verlag, 1984.
  10. V. A. Iskovskikh and A. V. Pukhlikov, Birational automorphisms of multi-dimensional algebraic varieties, J. Math. Sci. 82 (1996), no. 4, 3528-3613. https://doi.org/10.1007/BF02363913
  11. J. Kollar et al., Flips and Abundance for Algebraic Threefolds, Asterisque 211 (1993).
  12. T. Okada, Birational Mori fiber structures of $\mathbb{Q}$-Fano 3-fold weighted complete intersections. II, ArXiv:1310.5320.
  13. T. Okada, Birational Mori ber structures of $\mathbb{Q}$-Fano 3-fold weighted complete intersections. III, ArXiv:1409.1506.
  14. T. Okada, Birational Mori fiber structures of $\mathbb{Q}$-Fano 3-fold weighted complete intersections, Proc. Lond. Math. Soc. 109 (2014), no. 6, 1549-1600. https://doi.org/10.1112/plms/pdu044
  15. A. V. Pukhlikov, Maximal singularities on the Fano variety $V^3_6$, Moscow Univ. Math. Bull. 44 (1989), no. 2, 70-75.
  16. A. V. Pukhlikov, Birational automorphisms of Fano hypersurfaces, Invent. Math. 134 (1998), no. 2, 401-426. https://doi.org/10.1007/s002220050269
  17. A. V. Pukhlikov, Fiberwise birational correspondences, Math. Notes 68 (2000), no. 1, 103-112. https://doi.org/10.1007/BF02674652
  18. A. V. Pukhlikov, Birationally rigid Fano double hypersurfaces, Sb. Math. 191 (2000), no. 6, 883-908. https://doi.org/10.1070/SM2000v191n06ABEH000485
  19. A. V. Pukhlikov, Birationally rigid Fano complete intersections, J. Reine Angew. Math. 541 (2001), 55-79.
  20. A. V. Pukhlikov, Birational geometry of algebraic varieties with a pencil of Fano cyclic covers, Pure Appl. Math. Q. 5 (2009), no. 2, 641-700. https://doi.org/10.4310/PAMQ.2009.v5.n2.a4
  21. A. V. Pukhlikov, Birationally Rigid Varieties, Mathematical Surveys and Monographs 190, AMS, 2013.
  22. A. V. Pukhlikov, Birationally rigid complete intersections of quadrics and cubics, Izv. Math. 77 (2013), no. 4, 795-845. https://doi.org/10.1070/IM2013v077n04ABEH002661
  23. A. V. Pukhlikov, Birationally rigid Fano complete intersections. II, J. Reine Angew. Math. 688 (2014), 209-218.
  24. F. Suzuki, Birational rigidity of complete intersections, ArXiv:1507.00285.
  25. V. V. Shokurov, Three-dimensional log flips, Izv. Math. 40 (1993), no. 1, 95-202. https://doi.org/10.1070/IM1993v040n01ABEH001862