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Birational classification of curves on rational surfaces. (English) Zbl 1205.14015
Let \(C \subset \mathbb P^2\) be an irreducible plane curve. It is a natural and classical problem to try to classify Cremona minimal models of the pair \((\mathbb P^2, C)\), i.e. to classify curves \(B\) of minimal degree which are equivalent to \(C\) via Cremona transformations of \(\mathbb P^2\). This paper gives a complete answer to this problem. The Cremona minimal model is one of the following types:
- \(B\) is a line;
- \(B\) has degree \(d\) and points of multiplicities \(m_1, \ldots, m_r\) with \(m_1 \geq \ldots \geq m_r\) that satisfy \(d \geq m_1+m_2+m_3\);
- \(B\) has degree \(d\) and a point \(p\) of maximal multiplicity \(m_0\); all points of multiplicity \(\geq (d-m_0)/2\) are infinitely near to \(p\).
More generally, the authors classify up to birational equivalence pairs \((S, C)\) where \(S\) is a rational surface and \(C\) a smooth irreducible curve on \(S\). The proofs of these results are based on a log-version of the minimal model program which consists in understanding the geometry of the adjoint linear system \(m K_S+C\) where \(m\) is chosen minimal such that \(|m K_S+C|\) is not empty. Note that in a recent preprint [Equivalent birational embeddings. II: Divisors, arXiv:0906.4859] M. Mella and E. Polastri obtained independently similar classification results.
The paper also gives a number of consequences of the main theorem, in particular the authors recover very precise classification results in special cases due to M. De Franchis [Palermo Rend. 13, 1–27 (1899; JFM 30.0511.02)], S. Iitaka [Tokyo J. Math. 22, No. 2, 289–321 (1999; Zbl 0982.14006)], N. Kumar and M. Murthy [J. Math. Kyoto Univ. 22, 767–777 (1983; Zbl 0509.14033)] and M. Nagata [Mem. Coll. Sci., Univ. Kyoto, Ser. A 32, 351–370 (1960; Zbl 0100.16703)].

MSC:
14E05 Rational and birational maps
14J26 Rational and ruled surfaces
14H50 Plane and space curves
14E30 Minimal model program (Mori theory, extremal rays)
14E07 Birational automorphisms, Cremona group and generalizations
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References:
[1] M. Alberich-Carramiñana, Geometry of the plane Cremona maps , Lecture Notes in Math. 1769 , Springer, Heidelberg, 2002. · Zbl 0991.14008
[2] J. W. Alexander, On factorization of Cremona transformations , Trans. Am. Math. Soc. 17 (1916), 295-300.
[3] L. Bădescu, Algebraic Surfaces , Springer, Berlin, 2001.
[4] S. F. Barber and O. Zariski, Reducible exceptional curves of the first kind , Amer. J. Math. 57 (1935), 119-141. JSTOR: · Zbl 0010.37104
[5] A. Calabri, On rational and ruled double planes , Ann. Mat. Pura Appl. (4) 181 (2002), 365-387. · Zbl 1172.14329
[6] G. Castelnuovo, Massima dimensione dei sistemi lineari di curve piane di dato genere , Ann. Mat. (2) 18 (1890), 119-128. · JFM 22.0627.01
[7] G. Castelnuovo, 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. · JFM 23.0653.01
[8] G. Castelnuovo, Le trasformazioni generatrici del gruppo cremoniano nel piano , Atti R. Accad. Sci. Torino 36 (1901), 861-874. · JFM 32.0675.03
[9] O. Chisini, 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. · JFM 48.0706.03
[10] C. Ciliberto, P. Francia, and M. Mendes Lopes, Remarks on the bicanonical map for surfaces of general type , Math. Z. 224 (1997), 137-166. · Zbl 0871.14011
[11] F. Conforto, Le superficie razionali , Zanichelli, Bologna, 1939. · Zbl 0021.05306
[12] J. L. Coolidge, A treatise on algebraic plane curves , Dover, New York, 1959. · Zbl 0085.36403
[13] M. De Franchis, Riduzione dei fasci di curve piane di genere 2 , Rend. Circ. Mat. Palermo 13 (1899), 1-27. · JFM 30.0511.02
[14] P. del Pezzo, Sulle superficie di ordine n immerse nello spazio di n 1 dimensioni , Rend. R. Acc. Sci. Fis. Mat. Napoli 24 (1885), 212-216. · JFM 17.0514.01
[15] D. Dicks, Birational pairs according to S. Iitaka , Math. Proc. Cambridge Philos. Soc. 102 (1987), 59-69. · Zbl 0629.14029
[16] D. Eisenbud and J. Harris, On varieties of minimal degree (a centennial account) , Proc. Sympos. Pure Math. 46 (1987), 3-13. · Zbl 0646.14036
[17] F. Enriques and O. Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche , 4 vols., Zanichelli, Bologna, 1915-1934. · Zbl 0009.15904
[18] A. Franchetta, Sulle curve eccezionali riducibili di prima specie , Boll. Unione Mat. Ital. (2) 4 (1940), 332-341. · Zbl 0023.36901
[19] A. Franchetta, Sulla caratterizzazione delle curve eccezionali riducibili di prima specie , Boll. Unione Mat. Ital. (2) 5 (1941), 372-375. · JFM 67.0606.01
[20] A. Franchetta, Sulle curve riducibili appartenenti ad una superficie algebrica , Rend. Mat. Appl. (5) 8 (1949), 378-398. · Zbl 0039.16304
[21] R. Hartshorne, Algebraic Geometry , Grad. Texts in Math. 52 , Springer, New York, 1977. · Zbl 0367.14001
[22] S. Iitaka, On irreducible plane curves , Saitama Math. J. 1 (1983), 47-63. · Zbl 0517.14009
[23] S. Iitaka, Birational geometry of plane curves , Tokyo J. Math. 22 (1999), 289-321. · Zbl 0982.14006
[24] S. Iitaka, Birational characterization of nonsingular plane curves , preprint, 2006.
[25] S. Iitaka, “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. · Zbl 1140.14306
[26] S. Iitaka, “Relationships between \omega and \sigma ” in Birational Geometry of Algebraic Plane Curves and Related Topics in Algebraic Geometry , Gakushuin University, Tokyo, 2010, 51-142.
[27] S. Iitaka, “On birational invariants A and \Omega of algebraic plane curves” in Birational Geometry of Algebraic Plane Curves and Related Topics in Algebraic Geometry , Gakushuin University, Tokyo, 2010, 143-164.
[28] G. Jung, 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.
[29] S. Kantor, Sur une théorie des courbes et des surfaces admettant des correspondances univoques , C. R. Acad. Sci. Paris Sér. A-B 100 (1885), 343-345. · JFM 17.0605.03
[30] S. Kantor, Premiers fondements pour une théorie des transformations périodiques univoques , Atti Accad. Sci. Fis. Mat. Napoli (2) 4 (1891), 1-335.
[31] N. M. Kumar and M. P. Murthy, Curves with negative self-intersection on rational surfaces , J. Math. Kyoto Univ. 22 (1982/1983), 767-777. · Zbl 0509.14033
[32] O. Matsuda, On numerical types of algebraic curves on rational surfaces , Tokyo J. Math. 24 (2001), 359-367. · Zbl 1078.14527
[33] K. Matsuki, Introduction to the Mori program , Springer, New York, 2001. · Zbl 0988.14007
[34] M. Mella and E. Polastri, Equivalent birational embeddings , Bull. Lond. Math. Soc. 41 (2009), 89-93. · Zbl 1184.14021
[35] M. Mella and E. Polastri, Equivalent birational embeddings II: divisors , preprint, · Zbl 1184.14021
[36] M. Nagata, On rational surfaces, I. Irreducible curves of arithmetic genus 0 or 1, Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 32 (1960), 351-370. · Zbl 0100.16703
[37] M. Nagata, On rational surfaces, II , Mem. Coll. Sci. Univ. Kyoto Ser. A Math. 33 (1960/1961), 271-293. · Zbl 0100.16801
[38] D. Nencini, Sulla classificazione aritmetica di Noether dei sistemi lineari di curve algebriche piane , Ann. Mat. Pura Appl. (3) 27 (1918), 259-292. · JFM 46.0904.03
[39] M. Noether, Rationale Ausführung der Operationen in der Theorie der algebraischen Functionen , Math. Ann. 23 (1883), 311-358.
[40] M. Reid, Surfaces of small degree , Math. Ann. 275 (1986), 71-80. · Zbl 0579.14017
[41] 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 (1900-1901), 645-651. · JFM 32.0675.02
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