Smoothness of equisingular families of curves
Author:
Thomas Keilen
Journal:
Trans. Amer. Math. Soc. 357 (2005), 24672481
MSC (2000):
Primary 14H10, 14H15, 14H20; Secondary 14J26, 14J27, 14J28, 14J70
Published electronically:
November 4, 2004
MathSciNet review:
2140446
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Francesco Severi (1921) showed that equisingular families of plane nodal curves are Tsmooth, i.e. smooth of the expected dimension, whenever they are nonempty. For families with more complicated singularities this is no longer true. Given a divisor on a smooth projective surface it thus makes sense to look for conditions which ensure that the family of irreducible curves in the linear system with precisely singular points of types is Tsmooth. Considering different surfaces including the projective plane, general surfaces in , products of curves and geometrically ruled surfaces, we produce a sufficient condition of the type
where is some invariant of the singularity type and 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 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 Tsmoothness at all.
 [CC99]
L.
Chiantini and C.
Ciliberto, On the Severi varieties on surfaces in
𝐏³, J. Algebraic Geom. 8 (1999),
no. 1, 67–83. MR 1658208
(2000f:14082)
 [CS97]
L.
Chiantini and E.
Sernesi, Nodal curves on surfaces of general type, Math. Ann.
307 (1997), no. 1, 41–56. MR 1427674
(98b:14026), http://dx.doi.org/10.1007/s002080050021
 [Fla01]
F.
Flamini, Moduli of nodal curves on smooth
surfaces of general type, J. Algebraic Geom.
11 (2002), no. 4,
725–760. MR 1910265
(2003k:14028), http://dx.doi.org/10.1090/S1056391102003223
 [FM01]
F.
Flamini and C.
Madonna, Geometric linear normality for nodal curves on some
projective surfaces, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat.
(8) 4 (2001), no. 1, 269–283 (English, with
Italian summary). MR 1821407
(2001m:14046)
 [GH85]
Phillip
Griffiths and Joe
Harris, On the NoetherLefschetz theorem and some remarks on
codimensiontwo cycles, Math. Ann. 271 (1985),
no. 1, 31–51. MR 779603
(87a:14030), http://dx.doi.org/10.1007/BF01455794
 [GK89]
GertMartin
Greuel and Ulrich
Karras, Families of varieties with prescribed singularities,
Compositio Math. 69 (1989), no. 1, 83–110. MR 986814
(90d:32037)
 [GL96]
GertMartin
Greuel and Christoph
Lossen, Equianalytic and equisingular families of curves on
surfaces, Manuscripta Math. 91 (1996), no. 3,
323–342. MR 1416715
(98g:14023), http://dx.doi.org/10.1007/BF02567958
 [GLS97]
GertMartin
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 1459626
(98g:14039), http://dx.doi.org/10.1155/S1073792897000391
 [GLS98]
GertMartin
Greuel, Christoph
Lossen, and Eugenii
Shustin, Geometry of families of nodal curves
on the blownup projective plane, Trans. Amer.
Math. Soc. 350 (1998), no. 1, 251–274. MR 1443875
(98j:14034), http://dx.doi.org/10.1090/S0002994798020558
 [GLS00]
GertMartin
Greuel, Christoph
Lossen, and Eugenii
Shustin, Castelnuovo function, zerodimensional schemes and
singular plane curves, J. Algebraic Geom. 9 (2000),
no. 4, 663–710. MR 1775310
(2001g:14045)
 [GLS01]
GertMartin
Greuel, Christoph
Lossen, and Eugenii
Shustin, The variety of plane curves with ordinary singularities is
not irreducible, Internat. Math. Res. Notices 11
(2001), 543–550. MR 1836729
(2002e:14042), http://dx.doi.org/10.1155/S1073792801000289
 [Har77]
Robin
Hartshorne, Algebraic geometry, SpringerVerlag, New
YorkHeidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR 0463157
(57 #3116)
 [Kei01]
Thomas Keilen, Families of curves with prescribed singularities, Ph.D. thesis, Universität Kaiserslautern, 2001, http:// www. mathematik. unikl. de/ keilen/ download/ Thesis/ thesis.ps.gz.
 [KT02]
Thomas
Keilen and Ilya
Tyomkin, Existence of curves with prescribed
topological singularities, Trans. Amer. Math.
Soc. 354 (2002), no. 5, 1837–1860 (electronic). MR 1881019
(2003a:14041), http://dx.doi.org/10.1090/S000299470102877X
 [Laz97]
Robert
Lazarsfeld, Lectures on linear series, Complex algebraic
geometry (Park City, UT, 1993) IAS/Park City Math. Ser., vol. 3,
Amer. Math. Soc., Providence, RI, 1997, pp. 161–219. With the
assistance of Guillermo Fernández del Busto. MR 1442523
(98h:14008)
 [LK03a]
Christoph Lossen and Thomas Keilen, The 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. unikl. de/ lossen/ download/ Lossen002/ Lossen002.ps.gz.
 [Lue87a]
Ignacio
Luengo, The 𝜇constant stratum is not smooth, Invent.
Math. 90 (1987), no. 1, 139–152. MR 906582
(88m:32021), http://dx.doi.org/10.1007/BF01389034
 [Lue87b]
Ignacio
Luengo, On the existence of complete families of projective plane
curves, which are obstructed, J. London Math. Soc. (2)
36 (1987), no. 1, 33–43. MR 897672
(88f:14028), http://dx.doi.org/10.1112/jlms/s236.1.33
 [Sev21]
Francesco Severi, Vorlesungen über Algebraische Geometrie, Bibliotheca Mathematica Teubneriana, no. 32, Teubner, 1921.
 [Shu87]
E.
I. Shustin, Versal deformations in a space of plane curves of fixed
degree, Funktsional. Anal. i Prilozhen. 21 (1987),
no. 1, 90–91 (Russian). MR 888028
(89a:14034)
 [Shu91]
, On manifolds of singular algebraic curves, Selecta Math. Sov. 10 (1991), 2737.
 [Shu94]
Eugenii
Shustin, Smoothness and irreducibility of varieties of plane curves
with nodes and cusps, Bull. Soc. Math. France 122
(1994), no. 2, 235–253 (English, with English and French
summaries). MR
1273903 (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. 393416.
 [Shu97]
Eugenii
Shustin, Smoothness of equisingular families of plane algebraic
curves, Internat. Math. Res. Notices 2 (1997),
67–82. MR
1429954 (97j:14031), http://dx.doi.org/10.1155/S1073792897000056
 [Tan80]
Allen
Tannenbaum, Families of algebraic curves with nodes,
Compositio Math. 41 (1980), no. 1, 107–126. MR 578053
(82b:14017)
 [Tan82]
Allen
Tannenbaum, Families of curves with nodes on 𝐾3
surfaces, Math. Ann. 260 (1982), no. 2,
239–253. MR
664378 (84g:14025), http://dx.doi.org/10.1007/BF01457238
 [Vas90]
Victor A. Vassiliev, Stable cohomology of complements to the discriminants of deformations of singularities of smooth functions, J. Sov. Math. 52 (1990), 32173230.
 [Wah74]
Jonathan
M. Wahl, Deformations of planes curves with nodes and cusps,
Amer. J. Math. 96 (1974), 529–577. MR 0387287
(52 #8130)
 [CC99]
 Luca Chiantini and Ciro Ciliberto, On the Severi variety of surfaces in , J. Algebraic Geom. 8 (1999), 6783. MR 2000f:14082
 [CS97]
 Luca Chiantini and Edoardo Sernesi, Nodal curves on surfaces of general type, Math. Ann. 307 (1997), 4156. MR 98b:14026
 [Fla01]
 Flaminio Flamini, Moduli of nodal curves on smooth surfaces of general type, J. Algebraic Geom. 11 (2001), no. 4, 725760. MR 2003k:14028
 [FM01]
 Flaminio Flamini and Carlo Madonna, Geometric linear normality for nodal curves on some projective surfaces, Bollettino UMI 4B (2001), no. 8, 269283. MR 2001m:14046
 [GH85]
 Philipp Griffiths and Joe Harris, On the NoetherLefschetz theorem and some remarks on codimensiontwo cycles, Math. Ann. 271 (1985), no. 1, 109205. MR 87a:14030
 [GK89]
 GertMartin Greuel and Ulrich Karras, Families of varieties with prescribed singularities, Comp. Math. 69 (1989), 83110. MR 90d:32037
 [GL96]
 GertMartin Greuel and Christoph Lossen, Equianalytic and equisingular families of curves on surfaces, Manuscripta Math. 91 (1996), 323342. MR 98g:14023
 [GLS97]
 GertMartin Greuel, Christoph Lossen, and Eugenii Shustin, New asymptotics in the geometry of equisingular families of curves, Internat. Math. Res. Notices 13 (1997), 595611. MR 98g:14039
 [GLS98]
 , Geometry of families of nodal curves on the blownup projective plane, Trans. Amer. Math. Soc. 350 (1998), 251274. MR 98j:14034
 [GLS00]
 , Castelnuovo function, zerodimensional schemes, and singular plane curves, J. Algebraic Geom. 9 (2000), no. 4, 663710. MR 2001g:14045
 [GLS01]
 , The variety of plane curves with ordinary singularities is not irreducible, Intern. Math. Res. Notes 11 (2001), 542550. 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. unikl. de/ keilen/ download/ Thesis/ thesis.ps.gz.
 [KT02]
 Thomas Keilen and Ilya Tyomkin, Existence of curves with prescribed singularities, Trans. Amer. Math. Soc. 354 (2002), no. 5, 18371860, http:// www. mathematik. unikl. de/ keilen/ download/ KeilenTyomkin001/ KeilenTyomkin001.ps.gz. 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. 161219. MR 98h:14008
 [LK03a]
 Christoph Lossen and Thomas Keilen, The 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. unikl. de/ lossen/ download/ Lossen002/ Lossen002.ps.gz.
 [Lue87a]
 Ignacio Luengo, The constant stratum is not smooth, Inventiones Math. 90 (1987), 139152. MR 88m:32021
 [Lue87b]
 , On the existence of complete families of plane curves, which are obstructed, J. LMS 36 (1987), 3343. 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), 8284. MR 89a:14034
 [Shu91]
 , On manifolds of singular algebraic curves, Selecta Math. Sov. 10 (1991), 2737.
 [Shu94]
 , Smoothness and irreducibility of varieties of algebraic curves with nodes and cusps, Bull. SMF 122 (1994), 235253. 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. 393416.
 [Shu97]
 , Smoothness of equisingular families of plane algebraic curves, Math. Res. Not. 2 (1997), 6782. MR 97j:14031
 [Tan80]
 Allen Tannenbaum, Families of algebraic curves with nodes, Compositio Math. 41 (1980), 107126. MR 82b:14017
 [Tan82]
 , Families of curves with nodes on K3surfaces, Math. Ann. 260 (1982), 239253. 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), 32173230.
 [Wah74]
 Jonathan M. Wahl, Deformations of plane curves with nodes and cusps, Amer. J. Math. 96 (1974), 529577. 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, ErwinSchrödingerStraße, D – 67663 Kaiserslautern, Germany
Email:
keilen@mathematik.unikl.de
DOI:
http://dx.doi.org/10.1090/S0002994704035883
PII:
S 00029947(04)035883
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
