Accessible Unlicensed Requires Authentication Published by De Gruyter March 20, 2018

Special cubic Cremona transformations of ℙ6 and ℙ7

Giovanni Staglianò
From the journal Advances in Geometry

Abstract

A famous result of Crauder and Katz (1989) concerns the classification of special Cremona transformations whose base locus has dimension at most two. They also proved that a special Cremona transformation with base locus of dimension three has to be one of the following: 1) a quinto-quintic transformation of ℙ5; 2) a cubo-quintic transformation of ℙ6; or 3) a quadro-quintic transformation of ℙ8. Special Cremona transformations as in Case 1) have been classified by Ein and Shepherd-Barron (1989), while in our previous work (2013), we classified special quadro-quintic Cremona transformations of ℙ8. Here we consider the problem of classifying special cubo-quintic Cremona transformations of ℙ6, concluding the classification of special Cremona transformations whose base locus has dimension three.

MSC 2010: 14E05; 14E07; 14J30

  1. Communicated by: I. Coskun

Acknowledgements

I wish to thank Francesco Russo for valuable communications and for posing to me the problem studied in this paper.

References

[1] M. Beltrametti, A. Biancofiore, A. J. Sommese, Projective n-folds of log-general type. I.Trans. Amer. Math. Soc. 314 (1989), 825–849. MR1005528 Zbl 0702.14037Search in Google Scholar

[2] M. C. Beltrametti, A. J. Sommese, The adjunction theory of complex projective varieties, volume 16 of De Gruyter Expositions in Mathematics. De Gruyter 1995. MR1318687 Zbl 0845.14003Search in Google Scholar

[3] M. Bertolini, C. Turrini, Threefolds in ℙ6 of degree 12. Adv. Geom. 15 (2015), 245–262. MR3334028 Zbl 1314.14097Search in Google Scholar

[4] A. Bertram, L. Ein, R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. J. Amer. Math. Soc. 4 (1991), 587–602. MR1092845 Zbl 0762.14012Search in Google Scholar

[5] G. M. Besana, A. Biancofiore, Numerical constraints for embedded projective manifolds. Forum Math. 17 (2005), 613–636. MR2154422 Zbl 1096.14043Search in Google Scholar

[6] B. Crauder, S. Katz, Cremona transformations with smooth irreducible fundamental locus. Amer. J. Math. 111 (1989), 289–307. MR987759 Zbl 0699.14015Search in Google Scholar

[7] B. Crauder, S. Katz, Cremona transformations and Hartshorne’s conjecture. Amer. J. Math. 113 (1991), 269–285. MR1099447 Zbl 0754.14009Search in Google Scholar

[8] O. Debarre, Higher-dimensional algebraic geometry. Springer 2001. MR1841091 Zbl 0978.14001Search in Google Scholar

[9] I. Dolgachev, Lectures on Cremona transformations, Ann Arbor-Rome, 2010/2011. Available at Search in Google Scholar

[10] L. Ein, N. Shepherd-Barron, Some special Cremona transformations. Amer. J. Math. 111 (1989), 783–800. MR1020829 Zbl 0708.14009Search in Google Scholar

[11] M. L. Fania, E. L. Livorni, Degree ten manifolds of dimension n greater than or equal to 3. Math. Nachr. 188 (1997), 79–108. MR1484670 Zbl 0922.14027Search in Google Scholar

[12] T. Fujita, Projective threefolds with small secant varieties. Sci. Papers College Gen. Ed. Univ. Tokyo32 (1982), 33–46. MR674447 Zbl 0492.14027Search in Google Scholar

[13] T. Fujita, Classification theories of polarized varieties. Cambridge Univ. Press 1990. MR1162108 Zbl 0743.14004Search in Google Scholar

[14] W. Fulton, Intersection theory. Springer 1984. MR732620 Zbl 0541.14005Search in Google Scholar

[15] D. R. Grayson, M. E. Stillman, MACAULAY2 — A software system for research in algebraic geometry (version 1.9.2), 2016. Search in Google Scholar

[16] P. Griffiths, J. Harris, Principles of algebraic geometry. Wiley-Interscience 1978. MR507725 Zbl 0408.14001Search in Google Scholar

[17] R. Hartshorne, Algebraic geometry. Springer 1977. MR0463157 Zbl 0367.14001Search in Google Scholar

[18] K. Hulek, S. Katz, F.-O. Schreyer, Cremona transformations and syzygies. Math. Z. 209 (1992), 419–443. MR1152265 Zbl 0767.14005Search in Google Scholar

[19] P. Ionescu, Embedded projective varieties of small invariants. In: Algebraic geometry, Bucharest 1982, volume 1056 of Lecture Notes in Math., 142–186, Springer 1984. MR749942 Zbl 0542.14024Search in Google Scholar

[20] P. Ionescu, Generalized adjunction and applications. Math. Proc. Cambridge Philos. Soc. 99 (1986), 457–472. MR830359 Zbl 0619.14004Search in Google Scholar

[21] S. Katz, The cubo-cubic transformation of ℙ3 is very special. Math. Z. 195 (1987), 255–257. MR892055 Zbl 0598.14010Search in Google Scholar

[22] Y. Kawamata, Minimal models and the Kodaira dimension of algebraic fiber spaces. J. Reine Angew. Math. 363 (1985), 1–46. MR814013 Zbl 0589.14014Search in Google Scholar

[23] S. Kleiman, Appendix to Exposé XIII. In: Théorie des intersections et théorème de Riemann–Roch (SGA 6), volume 225 of Lecture Notes in Math., 653–666, Springer 1971. MR0354655 Zbl 0218.14001Search in Google Scholar

[24] R. Lazarsfeld, Positivity in algebraic geometry. I. Springer 2004. MR2095471 Zbl 1093.14501 Zbl 1066.14021Search in Google Scholar

[25] P. Le Barz, Quadrisécantes d’une surface de P5. C. R. Acad. Sci. Paris Sér. A-B291 (1980), A639–A642. MR606452 Zbl 0474.14036Search in Google Scholar

[26] P. Le Barz, Formules pour les multisécantes des surfaces. C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 797–800. MR622422 Zbl 0492.14045Search in Google Scholar

[27] E. L. Livorni, A. J. Sommese, Threefolds of nonnegative Kodaira dimension with sectional genus less than or equal to 15. Ann. Scuola Norm. Sup. Pisa Cl.Sci. (4) 13 (1986), 537–558. MR880398 Zbl 0636.14014Search in Google Scholar

[28] J. C. Migliore, C. Peterson, A construction of codimension three arithmetically Gorenstein subschemes of projective space. Trans. Amer. Math. Soc. 349 (1997), 3803–3821. MR1432204 Zbl 0885.14022Search in Google Scholar

[29] F. Russo, Varieties with quadratic entry locus. I. Math. Ann. 344 (2009), 597–617. MR2501303 Zbl 1170.14040Search in Google Scholar

[30] J. G. Semple, J. A. Tyrrell, The Cremona transformation of S6 by quadrics through a normal elliptic septimic scroll1R7. Mathematika16 (1969), 89–97. MR0249431 Zbl 0176.51001Search in Google Scholar

[31] J. G. Semple, J. A. Tyrrell, The T2,4 of S6 defined by a rational surface 3F8. Proc. London Math. Soc. (3) 20 (1970), 205–221. MR0260744 Zbl 0188.53404Search in Google Scholar

[32] A. J. Sommese, On the adjunction theoretic structure of projective varieties. In: Complex analysis and algebraic geometry (Göttingen, 1985), volume 1194 of Lecture Notes in Math., 175–213, Springer 1986. MR855885 Zbl 0601.14029Search in Google Scholar

[33] A. J. Sommese, A. Van de Ven, On the adjunction mapping. Math. Ann. 278 (1987), 593–603. MR909240 Zbl 0655.14001Search in Google Scholar

[34] G. Staglianò, On special quadratic birational transformations of a projective space into a hypersurface. Rend. Circ. Mat. Palermo (2) 61 (2012), 403–429. MR2996505 Zbl 1261.14005Search in Google Scholar

[35] G. Staglianò, On special quadratic birational transformations whose base locus has dimension at most three. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 24 (2013), 409–436. MR3097021 Zbl 1282.14024Search in Google Scholar

[36] G. Staglianò, Examples of special quadratic birational transformations into complete intersections of quadrics. J. Symbolic Comput. 74 (2016), 635–649. MR3424062 Zbl 1374.14014Search in Google Scholar

[37] P. Vermeire, Some results on secant varieties leading to a geometric flip construction. Compositio Math. 125 (2001), 263–282. MR1818982 Zbl 1056.14016Search in Google Scholar

[38] C. H. Walter, Pfaffian subschemes. J. Algebraic Geom. 5 (1996), 671–704. MR1486985 Zbl 0864.14032Search in Google Scholar

Received: 2016-08-10
Revised: 2017-03-18
Revised: 2017-04-13
Published Online: 2018-03-20
Published in Print: 2019-04-24

© 2019 Walter de Gruyter GmbH Berlin/Boston