Kyoto Journal of Mathematics

On the Cremona contractibility of unions of lines in the plane

Alberto Calabri and Ciro Ciliberto

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We discuss the concept of Cremona contractible plane curves, with a historical account on the development of this subject. Then we classify Cremona contractible unions of d12 lines in the plane.

Article information

Source
Kyoto J. Math., Volume 57, Number 1 (2017), 55-78.

Dates
Received: 30 March 2015
Revised: 30 November 2015
Accepted: 3 December 2015
First available in Project Euclid: 11 March 2017

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1489201230

Digital Object Identifier
doi:10.1215/21562261-3759513

Mathematical Reviews number (MathSciNet)
MR3621779

Zentralblatt MATH identifier
06705667

Subjects
Primary: 14H50: Plane and space curves
Secondary: 14E07: Birational automorphisms, Cremona group and generalizations 14N20: Configurations and arrangements of linear subspaces

Keywords
union of lines contractible plane curves log-Kodaira dimension

Citation

Calabri, Alberto; Ciliberto, Ciro. On the Cremona contractibility of unions of lines in the plane. Kyoto J. Math. 57 (2017), no. 1, 55--78. doi:10.1215/21562261-3759513. https://projecteuclid.org/euclid.kjm/1489201230


Export citation

References

  • [1] M. Alberich-Carramiñana, Geometry of the Plane Cremona Maps, Lecture Notes in Math. 1769, Springer, Berlin, 2002.
  • [2] A. Calabri, On rational and ruled double planes, Ann. Mat. Pura Appl. (4) 181 (2002), 365–387.
  • [3] A. Calabri, Rivestimenti del piano: Sulla razionalità dei piani doppi e tripli ciclici, Centro Studi Enriques, Pisa, 2006.
  • [4] A. Calabri and C. Ciliberto, Birational classification of curves on rational surfaces, Nagoya Math. J. 199 (2010), 43–93.
  • [5] A. Calabri and C. Ciliberto, On Cremona contractibility, Rend. Semin. Mat. Univ. Politec. Torino 71 (2013), 389–400.
  • [6] G. Castelnuovo, Ricerche generali sopra i sistemi lineari di curve piane, Mem. Reale Accad. Sci. Torino (2) 42 (1891), 3–43.
  • [7] G. Castelnuovo, Le trasformazioni generatrici del gruppo cremoniano nel piano, Atti Reale Accad. Sci. Torino 36 (1901), 861–874.
  • [8] G. Castelnuovo and F. Enriques, Sulle condizioni di razionalità dei piani doppi, Rend. Circ. Mat. Palermo 14 (1900), 290–302.
  • [9] W. Clifford, “Analysis of Cremona’s transformations” in Mathematical Papers, Macmillan, London, 1882, 538–542.
  • [10] F. Conforto, Sui piani doppi razionali, Rend. Sem. Mat. Univ. Roma (4) 2 (1938), 156–172.
  • [11] J. L. Coolidge, A Treatise on Algebraic Plane Curves, Dover, New York, 1959.
  • [12] F. Enriques and O. Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche, I, Zanichelli, Bologna, 1915; II, 1918; III, 1924; IV, 1934.
  • [13] F. Enriques and F. Conforto, Le superficie razionali, Zanichelli, Bologna, 1939.
  • [14] G. Ferretti, Sulla riduzione all’ordine minimo dei sistemi lineari di curve piane irriducibili di genere $p$; in particolare per i valori 0, 1, 2 del genere, Rend. Circ. Mat. Palermo 16 (1902), 236–279.
  • [15] H. P. Hudson, Cremona Transformations in Plane and Space, Cambridge Univ. Press, Cambridge, 1927.
  • [16] S. Iitaka, “Characterization of two lines on a projective plane” in Algebraic Geometry (Tokyo/Kyoto, 1982), Lecture Notes in Math. 1016, Springer, Berlin, 1983, 432–448.
  • [17] S. Iitaka, Classification of reducible plane curves, Tokyo J. Math. 11 (1988), 363–379.
  • [18] G. Jung, Ricerche sui sistemi lineari di genere qualunque e sulla loro riduzione all’ordine minimo, Ann. Mat. Pura Appl. (2) 16 (1888), 291–327.
  • [19] S. Kantor, Premiers fondements pour une théorie des transformations périodiques univoques, Imprimerie de la Académie, Naples, 1891.
  • [20] H. Kojima and T. Takahashi, Reducible curves on rational surfaces, Tokyo J. Math. 29 (2006), 301–317.
  • [21] G. Marletta, Sulla identità cremoniana di due curve piane, Rend. Circ. Mat. Palermo 24 (1907), 229–242.
  • [22] G. Marletta, Sui sistemi aggiunti dei varii indici alle curve piane, Rend. R. Ist. Lombardo Sci. Lett. (2) 43 (1910), 781–804.
  • [23] N. Mohan Kumar and M. P. Murthy, Curves with negative self-intersection on rational surfaces, J. Math. Kyoto Univ. 22 (1982/83), 767–777.
  • [24] M. Noether, Ueber die auf Ebenen eideutig abbildbaren algebraischen Flaechen, Ges. Wiss. Univ. Göttingen 1870, 1–6.
  • [25] M. Noether, Zur Theorie der eindeutigen Ebenentransformationen, Math. Ann. 5 (1872), 635–639.
  • [26] M. Noether, Ueber einen Satz aus der Theorie der algebraischen Functionen, Math. Ann. 6 (1873), 351–359.
  • [27] G. Pompilj, Sulle trasformazioni cremoniane del piano che posseggono una curva di punti uniti, Rend. Semin. Mat. Univ. Roma (4) 2 (1938), 47–87.
  • [28] J. Rosanes, Ueber diejenigen rationalen Substitutionen, welche eine rationale Umkehrung zulassen, J. Reine Angew. Math. 73 (1871), 97–110.
  • [29] C. Segre, Un’osservazione relativa alla riducibilità delle trasformazioni Cremoniane e dei sistemi lineari di curve piane per mezzo di trasformazioni quadratiche, Atti R. Accad. Sci. Torino 36 (1901), 645–651.