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Special cubic birational transformations of projective spaces

Abstract

We extend our classification of special Cremona transformations whose base locus has dimension at most three to the case when the target space is replaced by a (locally) factorial complete intersection.

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Notes

  1. 1.

    Notice that the difference between the right and left side of the inequality (2.25) coincides with the sum \(\Sigma _{l}\left( {\begin{array}{c}5+l^2\\ 4\end{array}}\right) \), where l runs over all lines contained in the surface \(S\subset {{\mathbb {P}}}^5\) having self-intersection \(\le -2\). Thus (2.25) is a strict inequality if and only if S contains a line with self-intersection \(\le -6\).

  2. 2.

    For further computational details, see the online documentation of the methods abstractRationalMap from Cremona [72], and dualVariety from Resultants [73].

References

  1. 1.

    Aure, A.B., Ranestad, K.: The smooth surfaces of degree 9 in \({\mathbb{P}}^4\). In: Ellingsrud, G., Peskine, C., Sacchiero, G., Stromme, S.A. (eds.) Complex Projective Geometry, pp. 32–46. Cambridge Univ Press, Cambridge (1992)

    Google Scholar 

  2. 2.

    Alzati, A., Sierra, J.C.: Quadro-quadric special birational transformations of projective spaces. Int. Math. Res. Not. IMRN 2015(1), 55–77 (2015). https://doi.org/10.1093/imrn/rnt173

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Alzati, A., Sierra, J.C.: Special birational transformations of projective spaces. Adv. Math. 289, 567–602 (2016)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Addington, N., Thomas, R.: Hodge theory and derived categories of cubic fourfolds. Duke Math. J. 163(10), 1886–1927 (2014)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Besana, G.M., Biancofiore, A.: Degree eleven manifolds of dimension greater or equal to three. Forum Math. 17(5), 711–733 (2005)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Besana, G.M., Biancofiore, A.: Numerical constraints for embedded projective manifolds. Forum Math. 17(4), 613–636 (2005)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Beltrametti, M., Biancofiore, A., Sommese, A.J.: Projective n-folds of log-general type. I. Trans. Am. Math. Soc. 314(2), 825–825 (1989)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Bertram, A., Ein, L., Lazarsfeld, R.: Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. J. Am. Math. Soc. 4(3), 587–602 (1991)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Bolognesi, M., Russo, F., Staglianò, G.: Some loci of rational cubic fourfolds. Math. Ann. (to appear). Preprint arXiv:1504.05863 (2015)

  10. 10.

    Beltrametti, M.C., Sommese, A.J.: The Adjunction Theory of Complex Projective Varieties. de Gruyter Expositions in Mathematics, vol. 16. Walter de Gruyter, Berlin (1995)

    Google Scholar 

  11. 11.

    Beltrametti, M., Schneider, M., Sommese, A.J.: Threefolds of degree 11 in \({\mathbb{P}}^5\). In: Ellingsrud, G., Peskine, C., Sacchiero, G., Stromme, S.A. (eds.) Complex Projective Geometry: Selected Papers. London Mathematical Society Lecture Note series, pp. 59–80. Cambridge Univ. Press, Cambridge (1992)

    Google Scholar 

  12. 12.

    Bertolini, M., Turrini, C.: Threefolds in \({\mathbb{P}}^6\) of degree 12. Adv. Geom. 15(2), 245–262 (2015)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Castelnuovo, G.: Ricerche di geometria sulle curve algebriche. Atti R. Accad. Sci. Torino 24, 346–373 (1889)

    MATH  Google Scholar 

  14. 14.

    Crauder, B., Katz, S.: Cremona transformations with smooth irreducible fundamental locus. Am. J. Math. 111(2), 289–307 (1989)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Crauder, B., Katz, S.: Cremona transformations and Hartshorne’s conjecture. Am. J. Math. 113(2), 269–285 (1991)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Ciliberto, C., Mella, M., Russo, F.: Varieties with one apparent double point. J. Algebr. Geom. 13(3), 475–512 (2004)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Debarre, O.: Higher-Dimensional Algebraic Geometry. Universitext. Springer, New York (2001)

    MATH  Google Scholar 

  18. 18.

    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular—a computer algebra system for polynomial computations (version 4-1-1). (2018). http://www.singular.uni-kl.de. Accessed Feb 2018

  19. 19.

    Dolgachev, I.: Lectures on Cremona transformations, Ann Arbor-Rome. (2010/2011) http://www.math.lsa.umich.edu/~idolga/cremonalect.pdf. Accessed 4 July 2016

  20. 20.

    Edge, W.L.: The number of apparent double points of certain loci. Math. Proc. Camb. Philos. Soc. 28(3), 285–299 (1932)

    MATH  Google Scholar 

  21. 21.

    Ein, L., Shepherd-Barron, N.: Some special Cremona transformations. Am. J. Math. 111(5), 783–800 (1989)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Fano, G.: Sulle forme cubiche dello spazio a cinque dimensioni contenenti rigate razionali del \(4^\circ \) ordine. Comment. Math. Helv. 15(1), 71–80 (1943)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Fu, B., Hwang, J.M.: Special birational transformations of type \((2,1)\). J. Algebr. Geom. 27, 55–89 (2018)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Fania, M.L., Livorni, E.L.: Degree nine manifolds of dimension greater than or equal to 3. Math. Nachr. 169(1), 117–134 (1994)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Fania, M.L., Livorni, E.L.: Degree ten manifolds of dimension \(n\) greater than or equal to 3. Math. Nachr. 188(1), 79–108 (1997)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Fujita, T.: Projective threefolds with small secant varieties. Sci. Pap. Coll. Gen. Ed. Univ. Tokyo 32, 33–46 (1982)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Fujita, T.: Classification Theories of Polarized Varieties. London Mathematical Society Lecture Note Series, vol. 155. Cambridge Univ. Press, Cambridge (1990)

    Google Scholar 

  28. 28.

    Fulton, W.: Intersection Theory. Ergeb. Math. Grenzgeb. (3), vol. 2. Springer, Berlin (1984)

    Google Scholar 

  29. 29.

    Grothendieck, A.: Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. Advanced Studies in Pure Mathematics, vol. 2. North-Holland, Amsterdam (1968). Séminaire de Géométrie Algébrique du Bois-Marie, 1962 (SGA 2)

  30. 30.

    Grayson, D.R., Stillman, M.E.: Macaulay2: a software system for research in algebraic geometry (version 1.13) (2019). http://www.math.uiuc.edu/Macaulay2/. Accessed Jan 2019

  31. 31.

    Hartshorne, R.: Ample Subvarieties of Algebraic Varieties. Lecture Notes in Mathematics, vol. 156. Springer, Berlin (1970)

    MATH  Google Scholar 

  32. 32.

    Hartshorne, R.: Algebraic Geometry. Graduate Texts in Mathematics, vol. 52. Springer, New York (1977)

    Google Scholar 

  33. 33.

    Hartshorne, R.: Generalized divisors on Gorenstein schemes. K-Theory 8(3), 287–339 (1994)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Hassett, B.: Some rational cubic fourfolds. J. Algebr. Geom. 8(1), 103–114 (1999)

    MathSciNet  MATH  Google Scholar 

  35. 35.

    Hassett, B.: Special cubic fourfolds. Compos. Math. 120(1), 1–23 (2000)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Hassett, B.: Cubic fourfolds, K3 surfaces, and rationality questions. In: Pardini, R., Pirola, G.P. (eds.) Rationality Problems in Algebraic Geometry: Levico Terme, Italy 2015, pp. 29–66. Springer, Cham (2016)

    Google Scholar 

  37. 37.

    Hulek, K., Katz, S., Schreyer, F.-O.: Cremona transformations and syzygies. Math. Z. 209(1), 419–443 (1992)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Ionescu, P.: Embedded Projective Varieties of Small Invariants. Algebraic Geometry Bucharest 1982, Lecture Notes in Mathematics, vol. 1056, pp. 142–186. Springer, Berlin (1984)

    Google Scholar 

  39. 39.

    Ionescu, P.: Embedded projective varieties of small invariants, II. Rev. Roum. Math. Pures Appl. 31, 539–544 (1986)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Ionescu, P.: Generalized adjunction and applications. Math. Proc. Camb. Philos. Soc. 99(3), 457–472 (1986)

    MathSciNet  MATH  Google Scholar 

  41. 41.

    Ionescu, P.: Embedded Projective Varieties of Small Invariants, III. Algebraic Geometry, Lecture Notes in Mathematics, vol. 1417, pp. 138–154. Springer, Berlin (1990)

    Google Scholar 

  42. 42.

    Ionescu, P., Russo, F.: Conic-connected manifolds. J. Reine Angew. Math. 644, 145–157 (2010)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Katz, S.: The cubo-cubic transformation of \({\mathbb{P}}^3\) is very special. Math. Z. 195(2), 255–257 (1987)

    MathSciNet  MATH  Google Scholar 

  44. 44.

    Kobayashi, S., Ochiai, T.: Characterizations of complex projective spaces and hyperquadrics. J. Math. Kyoto Univ. 13(1), 31–47 (1973)

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Kuznetsov, A.: Derived Categories of Cubic Fourfolds. Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol. 282, pp. 219–243. Birkhäuser, Boston (2010)

    Google Scholar 

  46. 46.

    Kuznetsov, A.: Derived categories view on rationality problems. In: Pardini, R., Pirola, G.P. (eds.) Rationality Problems in Algebraic Geometry: Levico Terme, Italy 2015, pp. 67–104. Springer, Cham (2016)

    Google Scholar 

  47. 47.

    Lazarsfeld, R.: Positivity in Algebraic Geometry. I. Ergeb. Math. Grenzgeb. (3). Classical Setting: Line Bundles and Linear Series, vol. 48. Springer, Berlin (2004)

    Google Scholar 

  48. 48.

    Le Barz, P.: Quadrisécantes d’une surface de \({\mathbb{P}}^5\). C. R. Acad. Sci. Paris Sér. I Math. 291, 639–642 (1980)

    Google Scholar 

  49. 49.

    Le Barz, P.: Formules pour les multisécantes des surfaces. C. R. Acad. Sci. Paris Sér. I Math. 292, 797–800 (1981)

    MathSciNet  MATH  Google Scholar 

  50. 50.

    Li, Q.: Quadro-quadric special birational transformations from projective spaces to smooth complete intersections. Int. J. Math. 27(1), 1650004 (2016)

    MathSciNet  MATH  Google Scholar 

  51. 51.

    Livorni, E.L., Sommese, A.J.: Threefolds of non negative Kodaira dimension with sectional genus less than or equal to 15. Ann. Sc. Norm. Super. Pisa Cl. Sci. 13(4), 537–558 (1986)

    MATH  Google Scholar 

  52. 52.

    Morin, U.: Sulla razionalità dell’ipersuperficie cubica dello spazio lineare \(S_5\). Rend. Semin. Mat. Univ. Padova 11, 108–112 (1940)

    MATH  Google Scholar 

  53. 53.

    Mukai, S.: Biregular classification of Fano 3-folds and Fano manifolds of coindex 3. Proc. Natl. Acad. Sci. USA 86(9), 3000–3002 (1989)

    MathSciNet  MATH  Google Scholar 

  54. 54.

    Nuer, H.: Unirationality of moduli spaces of special cubic fourfolds and K3 surfaces. Algebr. Geom. 4, 281–289 (2015)

    MathSciNet  MATH  Google Scholar 

  55. 55.

    Okonek, C.: Notes on varieties of codimension 3 in \({\mathbb{P}}^n\). Manuscr. Math. 84(1), 421–442 (1994)

    MathSciNet  MATH  Google Scholar 

  56. 56.

    Ottaviani, G.: On 3-folds in \({\mathbb{P}}^5\) which are scrolls. Ann. Sc. Norm. Super. Pisa 19, 451–471 (1992)

    MATH  Google Scholar 

  57. 57.

    Russo, F., Staglianò, G.: Congruences of 5-secant conics and the rationality of some admissible cubic fourfolds. Duke Math. J. (to appear). Preprint arXiv:1707.00999 (2017)

  58. 58.

    Russo, F., Staglianò, G.: Explicit rationality of some cubic fourfolds. arXiv:1811.03502 (2018)

  59. 59.

    Russo, F.: Varieties with quadratic entry locus, I. Math. Ann. 344(3), 597–617 (2009)

    MathSciNet  MATH  Google Scholar 

  60. 60.

    Semple, J.G.: On representations of the \({S}_k\)’s of \({S}_n\) and of the Grassmann manifolds \({G}(k, n)\). Proc. Lond. Math. Soc. s2–32(1), 200–221 (1931)

    MATH  Google Scholar 

  61. 61.

    Simis, A.: Cremona transformations and some related algebras. J. Algebra 280(1), 162–179 (2004)

    MathSciNet  MATH  Google Scholar 

  62. 62.

    Sommese, A.J.: Hyperplane sections of projective surfaces I. The adjunction mapping. Duke Math. J. 46(2), 377–401 (1979)

    MathSciNet  MATH  Google Scholar 

  63. 63.

    Sommese, A.J.: On the Adjunction Theoretic Structure of Projective Varieties. Complex Analysis and Algebraic Geometry. Lecture Notes in Mathematics, vol. 1194, pp. 175–213. Springer, Berlin (1986)

    Google Scholar 

  64. 64.

    Semple, J.G., Roth, L.: Introduction to Algebraic Geometry. Oxford Univ. Press, New York (1985). Reprint of the 1949 original

    MATH  Google Scholar 

  65. 65.

    Semple, J.G., Tyrrell, J.A.: Specialization of Cremona transformations. Mathematika 15, 171–177 (1968)

    MathSciNet  MATH  Google Scholar 

  66. 66.

    Semple, J.G., Tyrrell, J.A.: The Cremona transformation of \({S}_6\) by quadrics through a normal elliptic septimic scroll \(^1 {R}^7\). Mathematika 16(1), 88–97 (1969)

    MATH  Google Scholar 

  67. 67.

    Semple, J.G., Tyrrell, J.A.: The \({T}_{2,4}\) of \({S}_6\) defined by a rational surface \(^3{F}^8\). Proc. Lond. Math. Soc. s3–20, 205–221 (1970)

    MATH  Google Scholar 

  68. 68.

    Staglianò, G.: On special quadratic birational transformations of a projective space into a hypersurface. Rend. Circ. Mat. Palermo 61(3), 403–429 (2012)

    MathSciNet  MATH  Google Scholar 

  69. 69.

    Staglianò, G.: On special quadratic birational transformations whose base locus has dimension at most three. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 24(3), 409–436 (2013)

    MathSciNet  MATH  Google Scholar 

  70. 70.

    Staglianò, G.: Examples of special quadratic birational transformations into complete intersections of quadrics. J. Symbol. Comput. 74, 635–649 (2015)

    MathSciNet  MATH  Google Scholar 

  71. 71.

    Staglianò, G.: Special cubic Cremona transformations of \({\mathbb{P}}^6\) and \({\mathbb{P}}^7\). Adv. Geom. (in press). arXiv:1509.06028 (2016)

  72. 72.

    Staglianò, G.: A Macaulay2 package for computations with rational maps. J. Softw. Algebra. Geom. 8(1), 61–70 (2018)

    MathSciNet  MATH  Google Scholar 

  73. 73.

    Staglianò, G.: A package for computations with classical resultants. J. Softw. Algebra Geom. 8(1), 21–30 (2018)

    MathSciNet  MATH  Google Scholar 

  74. 74.

    Sommese, A.J., Van de Ven, A.: On the adjunction mapping. Math. Ann. 278(1), 593–603 (1987)

    MathSciNet  MATH  Google Scholar 

  75. 75.

    Vermeire, P.: Some results on secant varieties leading to a geometric flip construction. Compos. Math. 125(3), 263–282 (2001)

    MathSciNet  MATH  Google Scholar 

  76. 76.

    Zak, F.L.: Tangents and Secants of Algebraic Varieties. Translations of Mathematical Monographs. Amer. Math. Soc., Providence (1993)

    MATH  Google Scholar 

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Acknowledgements

The author is grateful to Francesco Russo for useful discussions and for his interest in the work.

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Staglianò, G. Special cubic birational transformations of projective spaces. Collect. Math. 71, 123–150 (2020). https://doi.org/10.1007/s13348-019-00251-8

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Mathematics Subject Classification

  • 14E05
  • 14E07
  • 14J30