Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Smoothness of equisingular families of curves

Author: Thomas Keilen
Journal: Trans. Amer. Math. Soc. 357 (2005), 2467-2481
MSC (2000): Primary 14H10, 14H15, 14H20; Secondary 14J26, 14J27, 14J28, 14J70
Published electronically: November 4, 2004
MathSciNet review: 2140446
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Francesco Severi (1921) showed that equisingular families of plane nodal curves are T-smooth, i.e. smooth of the expected dimension, whenever they are non-empty. For families with more complicated singularities this is no longer true. Given a divisor $D$ on a smooth projective surface $\Sigma$ it thus makes sense to look for conditions which ensure that the family $V_{\vert D\vert}^{irr}\big(\mathcal{S}_1,\ldots,\mathcal{S}_r\big)$ of irreducible curves in the linear system $\vert D\vert _l$ with precisely $r$ singular points of types $\mathcal{S}_1,\ldots,\mathcal{S}_r$ is T-smooth. Considering different surfaces including the projective plane, general surfaces in $\mathbb{P} _{\mathbb{C} }^3$, products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type

\begin{displaymath}\sum\limits_{i=1}^r\gamma_\alpha(\mathcal{S}_i) < \gamma\cdot (D- K_\Sigma)^2, \end{displaymath}

where $\gamma_\alpha$ is some invariant of the singularity type and $\gamma$ is some constant. This generalises the results of Greuel, Lossen, and Shustin (2001) for the plane case, combining their methods and the method of Bogomolov instability. For many singularity types the $\gamma_\alpha$-invariant leads to essentially better conditions than the invariants used by Greuel, Lossen, and Shustin (1997), and for most classes of geometrically ruled surfaces our results are the first known for T-smoothness at all.

References [Enhancements On Off] (What's this?)

  • [CC99] Luca Chiantini and Ciro Ciliberto, On the Severi variety of surfaces in $\mathbb P_{\mathbb C}^3$, J. Algebraic Geom. 8 (1999), 67-83. MR 2000f:14082
  • [CS97] Luca Chiantini and Edoardo Sernesi, Nodal curves on surfaces of general type, Math. Ann. 307 (1997), 41-56. MR 98b:14026
  • [Fla01] Flaminio Flamini, Moduli of nodal curves on smooth surfaces of general type, J. Algebraic Geom. 11 (2001), no. 4, 725-760. MR 2003k:14028
  • [FM01] Flaminio Flamini and Carlo Madonna, Geometric linear normality for nodal curves on some projective surfaces, Bollettino UMI 4-B (2001), no. 8, 269-283. MR 2001m:14046
  • [GH85] Philipp Griffiths and Joe Harris, On the Noether-Lefschetz theorem and some remarks on codimension-two cycles, Math. Ann. 271 (1985), no. 1, 109-205. MR 87a:14030
  • [GK89] Gert-Martin Greuel and Ulrich Karras, Families of varieties with prescribed singularities, Comp. Math. 69 (1989), 83-110. MR 90d:32037
  • [GL96] Gert-Martin Greuel and Christoph Lossen, Equianalytic and equisingular families of curves on surfaces, Manuscripta Math. 91 (1996), 323-342. MR 98g:14023
  • [GLS97] Gert-Martin Greuel, Christoph Lossen, and Eugenii Shustin, New asymptotics in the geometry of equisingular families of curves, Internat. Math. Res. Notices 13 (1997), 595-611. MR 98g:14039
  • [GLS98] -, Geometry of families of nodal curves on the blown-up projective plane, Trans. Amer. Math. Soc. 350 (1998), 251-274. MR 98j:14034
  • [GLS00] -, Castelnuovo function, zero-dimensional schemes, and singular plane curves, J. Algebraic Geom. 9 (2000), no. 4, 663-710. MR 2001g:14045
  • [GLS01] -, The variety of plane curves with ordinary singularities is not irreducible, Intern. Math. Res. Notes 11 (2001), 542-550. MR 2002e:14042
  • [Har77] Robin Hartshorne, Algebraic geometry, Springer, 1977. MR 57:3116
  • [Kei01] Thomas Keilen, Families of curves with prescribed singularities, Ph.D. thesis, Universität Kaiserslautern, 2001, http:// www. mathematik. uni-kl. de/ keilen/ download/ Thesis/
  • [KT02] Thomas Keilen and Ilya Tyomkin, Existence of curves with prescribed singularities, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1837-1860, http:// www. mathematik. uni-kl. de/ keilen/ download/ KeilenTyomkin001/ MR 2003a:14041
  • [Laz97] Robert Lazarsfeld, Lectures on linear series, Complex Algebraic Geometry (János Kollár, ed.), IAS/Park City Mathematics Series, no. 3, Amer. Math. Soc., 1997, pp. 161-219. MR 98h:14008
  • [LK03a] Christoph Lossen and Thomas Keilen, The $\gamma_\alpha$-invariant for plane curve singularities, Preprint, 2003.
  • [LK03b] Christoph Lossen and Thomas Keilen, Injective analytic maps - a counterexample to the proof, Preprint, 2003.
  • [Los98] Christoph Lossen, The geometry of equisingular and equianalytic families of curves on a surface, Ph.d. thesis, FB Mathematik, Universität Kaiserslautern, Aug. 1998, http:// www. mathematik. uni-kl. de/ lossen/ download/ Lossen002/
  • [Lue87a] Ignacio Luengo, The $\mu$-constant stratum is not smooth, Inventiones Math. 90 (1987), 139-152. MR 88m:32021
  • [Lue87b] -, On the existence of complete families of plane curves, which are obstructed, J. LMS 36 (1987), 33-43. MR 88f:14028
  • [Sev21] Francesco Severi, Vorlesungen über Algebraische Geometrie, Bibliotheca Mathematica Teubneriana, no. 32, Teubner, 1921.
  • [Shu87] Eugenii Shustin, Versal deformation in the space of plane curves of fixed degree, Funct. An. Appl. 21 (1987), 82-84. MR 89a:14034
  • [Shu91] -, On manifolds of singular algebraic curves, Selecta Math. Sov. 10 (1991), 27-37.
  • [Shu94] -, Smoothness and irreducibility of varieties of algebraic curves with nodes and cusps, Bull. SMF 122 (1994), 235-253. MR 95e:14020
  • [Shu96] -, Smoothness and irreducibility of varieties of algebraic curves with ordinary singularities, Israel Math. Conf. Proc., no. 9, Amer. Math. Soc., 1996, pp. 393-416.
  • [Shu97] -, Smoothness of equisingular families of plane algebraic curves, Math. Res. Not. 2 (1997), 67-82. MR 97j:14031
  • [Tan80] Allen Tannenbaum, Families of algebraic curves with nodes, Compositio Math. 41 (1980), 107-126. MR 82b:14017
  • [Tan82] -, Families of curves with nodes on K3-surfaces, Math. Ann. 260 (1982), 239-253. MR 84g:14025
  • [Vas90] Victor A. Vassiliev, Stable cohomology of complements to the discriminants of deformations of singularities of smooth functions, J. Sov. Math. 52 (1990), 3217-3230.
  • [Wah74] Jonathan M. Wahl, Deformations of plane curves with nodes and cusps, Amer. J. Math. 96 (1974), 529-577. MR 52:8130

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14H10, 14H15, 14H20, 14J26, 14J27, 14J28, 14J70

Retrieve articles in all journals with MSC (2000): 14H10, 14H15, 14H20, 14J26, 14J27, 14J28, 14J70

Additional Information

Thomas Keilen
Affiliation: Fachbereich Mathematik, Universität Kaiserslautern, Erwin-Schrödinger-Straße, D – 67663 Kaiserslautern, Germany

Keywords: Algebraic geometry, singularity theory
Received by editor(s): September 1, 2003
Received by editor(s) in revised form: December 12, 2003
Published electronically: November 4, 2004
Additional Notes: The author was partially supported by the German Israeli Foundation for Research and Development, by the Hermann Minkowski – Minerva Center for Geometry at Tel Aviv University and by EAGER
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society