Conditions imposed by tacnodes and cusps

Roé, Joaquim

doi:10.1090/S0002-9947-01-02740-4

Conditions imposed by tacnodes and cusps
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by Joaquim Roé PDF

Trans. Amer. Math. Soc. 353 (2001), 4925-4948 Request permission

Abstract:

The study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress. The Horace method introduced by A. Hirschowitz has been successfully exploited to prove many $H^1$-vanishing theorems, even in higher dimension. Other specialization techniques, which include degenerations of the plane, are due to Z. Ran and C. Ciliberto and R. Miranda. G. M. Greuel, C. Lossen and E. Shustin use a local specialization procedure together with the Horace method to give the first asymptotically proper general existence criterion for singular curves of low degree. In this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by Greuel, Lossen and Shustin for curves with tacnodes and cusps.

References

J. Alexander and A. Hirschowitz, Interpolation on jets, J. Algebra 192 (1997), no. 1, 412–417. MR 1449967, DOI 10.1006/jabr.1996.6948
Didier Barkats, Études des variétés des courbes planes à nœuds et à cusps, Algebraic geometry (Catania, 1993/Barcelona, 1994) Lecture Notes in Pure and Appl. Math., vol. 200, Dekker, New York, 1998, pp. 25–35 (French, with French summary). MR 1651087
Joël Briançon, Description de $H\textrm {ilb}^{n}C\{x,y\}$, Invent. Math. 41 (1977), no. 1, 45–89. MR 457432, DOI 10.1007/BF01390164
E. Casas-Alvero, Infinitely near imposed singularities and singularities of polar curves, Math. Ann. 287 (1990), no. 3, 429–454. MR 1060685, DOI 10.1007/BF01446904
Casas-Alvero, E., Singularities of plane curves, London Math. Soc. Lecture Notes Series, 276, Cambridge Univ. Press, Cambridge, 2000.
M. V. Catalisano and A. Gimigliano, On curvilinear subschemes of $\textbf {P}^2$, J. Pure Appl. Algebra 93 (1994), no. 1, 1–14. MR 1268779, DOI 10.1016/0022-4049(94)90077-9
Ciro Ciliberto and Rick Miranda, Interpolation on curvilinear schemes, J. Algebra 203 (1998), no. 2, 677–678. MR 1622744, DOI 10.1006/jabr.1997.7241
Ciro Ciliberto and Rick Miranda, Degenerations of planar linear systems, J. Reine Angew. Math. 501 (1998), 191–220. MR 1637857
Ciro Ciliberto and Rick Miranda, Linear systems of plane curves with base points of equal multiplicity, Trans. Amer. Math. Soc. 352 (2000), no. 9, 4037–4050. MR 1637062, DOI 10.1090/S0002-9947-00-02416-8
Federigo Enriques and Oscar Chisini, Lezioni sulla teoria geometrica delle equazioni e delle funzioni algebriche. 1. Vol. I, II, Collana di Matematica [Mathematics Collection], vol. 5, Zanichelli Editore S.p.A., Bologna, 1985 (Italian). Reprint of the 1915 and 1918 editions. MR 966664
Evain, L. Collisions de trois gros points sur une surface algébrique, thesis, Nice 1997.
Evain, L., Une minoration du degré des courbes planes à singularités imposées, Preprint ENS Lyon, 212 (1997), 1-17.
John Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521. MR 237496, DOI 10.2307/2373541
Gert-Martin Greuel, Christoph Lossen, and Eugenii Shustin, Plane curves of minimal degree with prescribed singularities, Invent. Math. 133 (1998), no. 3, 539–580. MR 1645074, DOI 10.1007/s002220050254
Greuel, G.M., Lossen, C., and Shustin, E., Castelnuovo function, zero-dimensional schemes and singular plane curves, J. Alg. Geom. 9 (2000), 663–710.
Grothendieck, A., Dieudonné, J., Eléments de géométrie algébrique. IV, Inst. Hautes Etudes Sci. Publ. Math. 32 (1967).
Brian Harbourne, Complete linear systems on rational surfaces, Trans. Amer. Math. Soc. 289 (1985), no. 1, 213–226. MR 779061, DOI 10.1090/S0002-9947-1985-0779061-2
Brian Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, Proceedings of the 1984 Vancouver conference in algebraic geometry, CMS Conf. Proc., vol. 6, Amer. Math. Soc., Providence, RI, 1986, pp. 95–111. MR 846019
Brian Harbourne, Iterated blow-ups and moduli for rational surfaces, Algebraic geometry (Sundance, UT, 1986) Lecture Notes in Math., vol. 1311, Springer, Berlin, 1988, pp. 101–117. MR 951643, DOI 10.1007/BFb0082911
Brian Harbourne, Rational surfaces with $K^2>0$, Proc. Amer. Math. Soc. 124 (1996), no. 3, 727–733. MR 1307526, DOI 10.1090/S0002-9939-96-03226-1
Brian Harbourne, Anticanonical rational surfaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1191–1208. MR 1373636, DOI 10.1090/S0002-9947-97-01722-4
Hartshorne, B., Algebraic Geometry, GTM 52, Springer 1977.
André Hirschowitz, La méthode d’Horace pour l’interpolation à plusieurs variables, Manuscripta Math. 50 (1985), 337–388 (French, with English summary). MR 784148, DOI 10.1007/BF01168836
André Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles génériques, J. Reine Angew. Math. 397 (1989), 208–213 (French). MR 993223, DOI 10.1515/crll.1989.397.208
Anthony A. Iarrobino, Punctual Hilbert schemes, Mem. Amer. Math. Soc. 10 (1977), no. 188, viii+112. MR 485867, DOI 10.1090/memo/0188
Steven L. Kleiman, Multiple-point formulas. I. Iteration, Acta Math. 147 (1981), no. 1-2, 13–49. MR 631086, DOI 10.1007/BF02392866
Christoph Lossen, New asymptotics for the existence of plane curves with prescribed singularities, Comm. Algebra 27 (1999), no. 7, 3263–3282. MR 1695287, DOI 10.1080/00927879908826626
Rick Miranda, Linear systems of plane curves, Notices Amer. Math. Soc. 46 (1999), no. 2, 192–201. MR 1673756
Augusto Nobile and Orlando E. Villamayor, Equisingular stratifications associated to families of planar ideals, J. Algebra 193 (1997), no. 1, 239–259. MR 1456575, DOI 10.1006/jabr.1997.7022
Giuseppe Paxia, On flat families of fat points, Proc. Amer. Math. Soc. 112 (1991), no. 1, 19–23. MR 1055777, DOI 10.1090/S0002-9939-1991-1055777-6
Ziv Ran, Curvilinear enumerative geometry, Acta Math. 155 (1985), no. 1-2, 81–101. MR 793238, DOI 10.1007/BF02392538
Z. Ran, Enumerative geometry of singular plane curves, Invent. Math. 97 (1989), no. 3, 447–465. MR 1005002, DOI 10.1007/BF01388886
Roé, J., On the existence of plane curves with imposed multiple points, to appear in J. Pure App. Algebra
Eugenii Shustin, Smoothness of equisingular families of plane algebraic curves, Internat. Math. Res. Notices 2 (1997), 67–82. MR 1429954, DOI 10.1155/S1073792897000056
Oscar Zariski, Algebraic surfaces, Second supplemented edition, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 61, Springer-Verlag, New York-Heidelberg, 1971. With appendices by S. S. Abhyankar, J. Lipman, and D. Mumford. MR 0469915
Oscar Zariski, Le problème des modules pour les branches planes, 2nd ed., Hermann, Paris, 1986 (French). Course given at the Centre de Mathématiques de l’École Polytechnique, Paris, October–November 1973; With an appendix by Bernard Teissier. MR 861277