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Birational classification of curves on rational surfaces

Published online by Cambridge University Press:  11 January 2016

Alberto Calabri  and
Ciro Ciliberto
Alberto Calabri
Affiliation:
Dipartimento di Matematica, Università degli Studi di Ferrara, 44121 Ferrara, Italy, [email protected]
Ciro Ciliberto
Affiliation:
Dipartimento di Matematica, Università degli Studi di Roma “Tor Vergata”, 00133 Roma, Italy, [email protected]
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Abstract

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In this paper we consider the birational classification of pairs (S, ℒ), with S a rational surface and ℒ a linear system on S. We give a classification theorem for such pairs, and we determine, for each irreducible plane curve B, its Cremona minimal models, that is, those plane curves which are equivalent to B via a Cremona transformation and have minimal degree under this condition.

Keywords

14E0514J2614H5014E0714E30
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 199 , September 2010 , pp. 43 - 93
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2010

References

[A] Alberich-Carramiñana, M., Geometry of the plane Cremona maps, Lecture Notes in Math. 1769, Springer, Heidelberg, 2002.Google Scholar
[Al1] Alexander, J. W., On factorization of Cremona transformations, Trans. Am. Math. Soc. 17 (1916), 295–300.Google Scholar
[B] Bădescu, L., Algebraic Surfaces, Springer, Berlin, 2001.Google Scholar
[BZ] Barber, S. F. and Zariski, O., Reducible exceptional curves of the first kind, Amer. J. Math. 57 (1935), 119–141.Google Scholar
[C] Calabri, A., On rational and ruled double planes, Ann. Mat. Pura Appl. (4) 181 (2002), 365–387.Google Scholar
[Ca1] Castelnuovo, G., Massima dimensione dei sistemi lineari di curve piane di dato genere, Ann. Mat. (2) 18 (1890), 119–128.CrossRefGoogle Scholar
[Ca2] Castelnuovo, G., Ricerche generali sopra i sistemi lineari di curve piane, Mem. R. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (2) 42 (1890–1891), 137–188.Google Scholar
[Ca3] Castelnuovo, G., Le trasformazioni generatrici del gruppo cremoniano nel piano, Atti R. Accad. Sci. Torino 36 (1901), 861–874.Google Scholar
[Ch] Chisini, O., Sul teorema di Noether relativo alla decomponibilità di una trasformazione cremoniana in un prodotto di trasformazioni quadratiche, Atti Soc. Nat. Mat. Modena 6 (1921), 7–13.Google Scholar
[CFM] Ciliberto, C., Francia, P., and Lopes, M. Mendes, Remarks on the bicanonical map for surfaces of general type, Math. Z. 224 (1997), 137–166.CrossRefGoogle Scholar
[Con] Conforto, F., Le superficie razionali, Zanichelli, Bologna, 1939.Google Scholar
[Coo] Coolidge, J. L., A treatise on algebraic plane curves, Dover, New York, 1959.Google Scholar
[DF] Franchis, M. De, Riduzione dei fasci di curve piane di genere 2, Rend. Circ. Mat. Palermo 13 (1899), 1–27.Google Scholar
[dP] Pezzo, P. del, Sulle superficie di ordine n immerse nello spazio di n+1 dimensioni, Rend. R. Acc. Sci. Fis. Mat. Napoli 24 (1885), 212–216.Google Scholar
[D] Dicks, D., Birational pairs according to S. Iitaka, Math. Proc. Cambridge Philos. Soc. 102 (1987), 59–69.CrossRefGoogle Scholar
[EH] Eisenbud, D. and Harris, J., On varieties of minimal degree (a centennial account), Proc. Sympos. Pure Math. 46 (1987), 3–13.Google Scholar
[EC] Enriques, F. and Chisini, O., Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche, 4 vols., Zanichelli, Bologna, 1915–1934.Google Scholar
[F1] Franchetta, A., Sulle curve eccezionali riducibili di prima specie, Boll. Unione Mat. Ital. (2) 4 (1940), 332–341.Google Scholar
[F2] Franchetta, A., Sulla caratterizzazione delle curve eccezionali riducibili di prima specie, Boll. Unione Mat. Ital. (2) 5 (1941), 372–375.Google Scholar
[F3] Franchetta, A., Sulle curve riducibili appartenenti ad una superficie algebrica, Rend. Mat. Appl. (5) 8 (1949), 378–398.Google Scholar
[H] Hartshorne, R., Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.Google Scholar
[I1] Iitaka, S., On irreducible plane curves, Saitama Math. J. 1 (1983), 47–63.Google Scholar
[I2] Iitaka, S., Birational geometry of plane curves, Tokyo J. Math. 22 (1999), 289–321.Google Scholar
[I3] Iitaka, S., Birational characterization of nonsingular plane curves, preprint, 2006.Google Scholar
[I4] Iitaka, S., “Birational embedding of algebraic plane curves by mixed pluricanonical maps” in Complex Analysis and Its Applications, OCAMI Stud. 2, Osaka Municipal University Press, Osaka, 2007, 207–211.Google Scholar
[I5] Iitaka, S., “Relationships between ω and σ” in Birational Geometry of Algebraic Plane Curves and Related Topics in Algebraic Geometry, Gakushuin University, Tokyo, 2010, 51–142.Google Scholar
[I6] Iitaka, S., “On birational invariants A and Ω of algebraic plane curves” in Birational Geometry of Algebraic Plane Curves and Related Topics in Algebraic Geometry, Gakushuin University, Tokyo, 2010, 143–164.Google Scholar
[J] Jung, G., Ricerche sui sistemi lineari di curve algebriche di genere qualunque, Ann. Mat. (2) 15 (1888), 277–312; Ricerche sui sistemi lineari di genere qualunque e sulla loro riduzione all’ordine minimo, Ann. Mat. (2) 16 (1889), 291–327.Google Scholar
[K1] Kantor, S., Sur une théorie des courbes et des surfaces admettant des correspondances univoques, C. R. Acad. Sci. Paris Sér. A–B 100 (1885), 343–345.Google Scholar
[K2] Kantor, S., Premiers fondements pour une théorie des transformations périodiques univoques, Atti Accad. Sci. Fis. Mat. Napoli (2) 4 (1891), 1–335.Google Scholar
[KM] Kumar, N. M. and Murthy, M. P., Curves with negative self-intersection on rational surfaces, J. Math. Kyoto Univ. 22 (1982/1983), 767–777.Google Scholar
[M] Matsuda, O., On numerical types of algebraic curves on rational surfaces, Tokyo J. Math. 24 (2001), 359–367.Google Scholar
[Ma] Matsuki, K., Introduction to the Mori program, Springer, New York, 2001.Google Scholar
[MP1] Mella, M. and Polastri, E., Equivalent birational embeddings, Bull. Lond. Math. Soc. 41 (2009), 89–93.Google Scholar
[MP2] Mella, M. and Polastri, E., Equivalent birational embeddings II: divisors, preprint, arXiv:math/0906.4859 [math.AG].Google Scholar
[Na1] Nagata, M., On rational surfaces, I. Irreducible curves of arithmetic genus 0 or 1, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 32 (1960), 351–370.Google Scholar
[Na2] Nagata, M., On rational surfaces, II, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1960/1961), 271–293.Google Scholar
[Ne] Nencini, D., Sulla classificazione aritmetica di Noether dei sistemi lineari di curve algebriche piane, Ann. Mat. Pura Appl. (3) 27 (1918), 259–292.Google Scholar
[No] Noether, M., Rationale Ausführung der Operationen in der Theorie der algebraischen Functionen, Math. Ann. 23 (1883), 311–358.Google Scholar
[R] Reid, M., Surfaces of small degree, Math. Ann. 275 (1986), 71–80.Google Scholar
[S] Segre, C., 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 (1900–1901), 645–651.Google Scholar