On the variety of plane curves of degree $d$ with $\delta$ nodes and $\kappa$ cusps
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Abstract:
Let ${{\mathbf {P}}^N}$ be the projective space which parametrizes all plane curves of degree $d$ and $V(d,\delta ,\kappa )$ the subvariety of ${{\mathbf {P}}^N}$ consisting of all reduced and irreducible plane curves of degree $d$ with $\delta$ nodes and $\kappa$ cusps as their only singularities. In this paper we prove that $V(d,\delta ,\kappa )$ is irreducible if $\kappa \leqslant 3$, except possibly when $\kappa = 3$ and $d = 5$ or $6$.References
- E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
- Steven Diaz and Joe Harris, Ideals associated to deformations of singular plane curves, Trans. Amer. Math. Soc. 309 (1988), no. 2, 433–468. MR 961600, DOI 10.1090/S0002-9947-1988-0961600-2
- Steven Diaz and Joe Harris, Geometry of the Severi variety, Trans. Amer. Math. Soc. 309 (1988), no. 1, 1–34. MR 957060, DOI 10.1090/S0002-9947-1988-0957060-8
- D. Eisenbud and J. Harris, Divisors on general curves and cuspidal rational curves, Invent. Math. 74 (1983), no. 3, 371–418. MR 724011, DOI 10.1007/BF01394242
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- Joe Harris, On the Severi problem, Invent. Math. 84 (1986), no. 3, 445–461. MR 837522, DOI 10.1007/BF01388741
- Joe Harris, Curves and their moduli, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 99–143. MR 927953
- Joe Harris, The genus of space curves, Math. Ann. 249 (1980), no. 3, 191–204. MR 579101, DOI 10.1007/BF01363895
- Joe Harris, Families of smooth curves, Duke Math. J. 51 (1984), no. 2, 409–419. MR 747873, DOI 10.1215/S0012-7094-84-05120-2
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
- 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, 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. 104 (1982), no. 1, 209–226. MR 648487, DOI 10.2307/2374074
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 316 (1989), 165-192
- MSC: Primary 14H10; Secondary 14H20
- DOI: https://doi.org/10.1090/S0002-9947-1989-1014468-3
- MathSciNet review: 1014468