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Transactions of the American Mathematical Society

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Geometry of families of nodal curves
on the blown-up projective plane


Authors: Gert-Martin Greuel, Christoph Lossen and Eugenii Shustin
Journal: Trans. Amer. Math. Soc. 350 (1998), 251-274
MSC (1991): Primary 14H10
DOI: https://doi.org/10.1090/S0002-9947-98-02055-8
MathSciNet review: 1443875
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Abstract: Let $\mathbb P^2_r\,$ be the projective plane blown up at $r$ generic points. Denote by $E_0,E_1,\ldots ,E_r$ the strict transform of a generic straight line on $\mathbb P^2$ and the exceptional divisors of the blown-up points on $\mathbb P^2_r$ respectively. We consider the variety $V_{irr}(d;\,d_1,\ldots,d_r;\,k)$ of all irreducible curves $C$ in $|dE_0-\sum _{i=1}^{r} d_iE_i|$ with $k$ nodes as the only singularities and give asymptotically nearly optimal sufficient conditions for its smoothness, irreducibility and non-emptiness. Moreover, we extend our conditions for the smoothness and the irreducibility to families of reducible curves. For $r\leq 9$ we give the complete answer concerning the existence of nodal curves in $V_{irr}(d;\,d_1,\ldots,d_r;\,k)$.


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  • [ArC] Arbarello, E.; Cornalba, M.: Footnotes to a paper of Beniamino Segre. Math. Ann. 256, 341-362 (1981). MR 83d:14016
  • [GrK] Greuel, G.-M.; Karras, U.: Families of varieties with prescribed singularities. Comp. Math. 69, 83-110 (1989). MR 90d:32037
  • [GrL] Greuel, G.-M.; Lossen, C.: Equianalytic and equisingular families of curves on surfaces. Manuscr. Math. 91, 323-342 (1996). CMP 97:3
  • [GrM] Gradolato, M.A.; Mezzetti, E.: Families of curves with ordinary singular points on regular surfaces. Ann. Mat. Pura et Appl. 150, 281-298 (1988). MR 89g:14015
  • [Har] Harris, J.: On the Severi problem. Invent. Math. 84, 445-467 (1985) MR 87f:14012
  • [HiF] Hirzebruch, F.: Singularities of algebraic surfaces and characteristic numbers. Contemp. Math. 58, 141-155 (1986). MR 87j:14057
  • [Hir] Hirschowitz, A.: Une conjecture pour la cohomologie des diviseurs sur les surfaces rationelles génériques. J. Reine Angew. Math. 397, 208-213 (1989). MR 90g:14021
  • [Ivi] Ivinskis, K.: Normale Flächen und die Miyaoka-Kobayashi-Ungleichung, Diplomarbeit, Univ. Bonn (1985).
  • [Nob] Nobile, A.: Families of curves on surfaces. Math. Zeitschrift 187, 453-470 (1984). MR 85i:14017
  • [Ran] Ran, Z.: Families of plane curves and their limits: Enriques' conjecture and beyond. Annals of Math. 130, no.1, 121-157 (1989). MR 90e:14024
  • [Sev] Severi, F.: Vorlesungen über algebraische Geometrie. Teubner (1921) resp. Johnson (1968).
  • [Sh1] Shustin, E.: Real plane algebraic curves with prescribed singularities. Topology 32 845-856 (1993). MR 95f:14049
  • [Sh2] Shustin, E.: Geometry of equisingular families of plane algebraic curves. J. Algebr. Geom. 5, 209-234 (1996). CMP 96:8
  • [Ta1] Tannenbaum, A.: Families of algebraic curves with nodes. Compositio Math. 41, 107-126 (1980). MR 82b:14017
  • [Ta2] Tannenbaum, A.: Families of curves with nodes on K3-surfaces. Math. Ann. 260, 239-253 (1982). MR 84g:14025
  • [Var] Varchenko, A.N.: Asymptotics of integrals and Hodge structures, (Russian), Modern Problems of Math., Vol. 22, 130-166 (1983); English transl. in Soviet Math. Z. 27 (1984), no. 3. MR 85j:32012
  • [Wae] van der Waerden, B.L.: Einführung in die algebraische Geometrie, 2nd edition, Springer (1973). MR 49:8984

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Additional Information

Gert-Martin Greuel
Affiliation: Universität Kaiserslautern, Fachbereich Mathematik, Erwin-Schrödinger-Straße, D – 67663 Kaiserslautern, Germany
Email: greuel@mathematik.uni-kl.de

Christoph Lossen
Affiliation: Universität Kaiserslautern, Fachbereich Mathematik, Erwin-Schrödinger-Straße, D – 67663 Kaiserslautern, German
Email: lossen@mathematik.uni-kl.de

Eugenii Shustin
Affiliation: Tel Aviv University, School of Mathematical Sciences, Ramat Aviv, Tel Aviv 69978, Israel
Email: shustin@math.tau.ac.il

DOI: https://doi.org/10.1090/S0002-9947-98-02055-8
Keywords: Nodal curves, nodes, families of curves, blown--up projective space, existence, equisingular deformation
Received by editor(s): February 1, 1996
Article copyright: © Copyright 1998 American Mathematical Society

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