Plane Curves With Nodes

Treger, Robert

Canadian journal of mathematics, Tome 41 (1989) no. 2, pp. 193-212

Résumé

A smooth algebraic curve is birationally equivalent to a nodal plane curve. One of the main problems in the theory of plane curves is to describe the situation of nodes of an irreducible nodal plane curve (see [16, Art. 45], [10], [7, Book IV, Chapter I, §5], [12, p. 584], and [3]).Let n denote the degree of a nodal curve and d the number of nodes. The case (AZ, d) — (6,9) has been analyzed by Halphen [10]. It follows from Lemma 3.5 and Proposition 3.6 that this is an exceptional case. The case d ≦n(n + 3)/6, d ≦(n — 1)(n — 2)/2, and (n, d) ≠ (6,9) was investigated by Arbarello and Cornalba [3]. We present a simpler proof (Corollary 3.8).We consider the main case which is particularly important due to its applications to the moduli variety of curves, compare [19, Chapter VIII, Section 4]. Let Vn,d be the variety of irreducible curves of degree n with d nodes and no other singularities such that each curve of Vn,d can be degenerated into n lines in general position (see [17]).

Détail

DOI : 10.4153/CJM-1989-010-x

Bibliographie

[1] 1. Albanese, G., Sulle condizioni perché una curva algebrica riducibile si possa considerare come limite di una curva algebrica irreducible, Rend. Circ. Mat. Palermo 52 (1928), 105–150. Google Scholar

[2] 2. Arbarello, E. and Comalba, M., Su una proprieta notevole dei morfismi di una curva a moduli generali in uno spazio proiettivo, Rend. Sem. Mat. Univers. Politecn. Torino 38 (1980), 87–99. Google Scholar

[3] 3. Arbarello, E. and Comalba, M., Footnotes to a paper of Beniamino Segre, Math. Ann. 256 (1981), 341–362. Google Scholar

[4] 4. Arbarello, E., Cornalba, M., Griffiths, P.A. and Harris, J., Geometry of algebraic curves, vol. I (Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984); vol II (to appear). Google Scholar

[5] 5. Briançon, J. and Galligo, A., Deformations distinguées d'un point de C2 ou R2, Astérisque 7-8 (1973), 129–138. Google Scholar

[6] 6. Briançon, J., Granger, M. and Speder, J.P., Sur le schéma de Hilbert d'une courbe plane, Ann. scient. Éc. Norm. Sup. 14 (1981), 1–25. Google Scholar

[7] 7. Coolidge, J.L., A treatise on algebraic plane curves (Oxford University Press, Oxford, 1931). Google Scholar

[8] 8. Fogarty, J., Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521. Google Scholar

[9] 9. Griffiths, P. and Harris, J., Principles of algebraic geometry (Wiley-Interscience, New York, 1978). Google Scholar

[10] 10. Halphen, G.H., Sur les courbes planes du sixième degré à neuf points doubles, Bull. Soc. Math. France 10 (1882), 162–172. Google Scholar

[11] 11. Hironaka, H., On the arithmetic genera and the effective genera of algebraic curves, Mem. Kyoto Univ., ser. A, 30 (1957) 177–195. Google Scholar

[12] 12. Loria, G. Spezielle ebene algebriasche Kurven von hôherer Ordnung, Enzikl. der math. Wissenschaften 2 (Teubner, Leipzig, 1903–1915). Google Scholar

[13] 13. Mumford, D., Lectures on curves on an algebraic surface, Annals of Math. Studies 59 (Princeton, 1966). Google Scholar

[14] 14. Nobile, A., On specializations of curves. I, Trans. Amer. Math. Soc. 282 (1984), 739–748. Google Scholar

[15] 15. Piene, R., Polar classes of singular varieties, Ann. Scient. Éc. Norm. Sup. 11 (1978), 247–276. Google Scholar

[16] 16. Salmon, G., A treatise on the higher plane curves, 2nd ed. (Hodges, Foster, and Co., Dublin, 1873). Google Scholar

[17] 17. Severi, F., Vorlesungen tiber algebraische Géométrie (Teubner, Leipzig, 1921) — cf. Anhang F, 307–353. Google Scholar

[18] 18. Wahl, J., Deformations of plane curves with nodes and cusps, Amer. J. Math. 96 (1974), 529- 577. Google Scholar

[19] 19. Zariski, O., Algebraic surfaces, 2nd suppl. éd., Ergebnisse 61 (Springer-Verlag, Heidelberg, 1971). Google Scholar

[20] 20. Zariski, O., Dimension-theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order n and with a given number d of nodes, Amer. J. Math. J04 (1982), 209–226. Google Scholar