Irreducibility of equisingular families of curves

Author:
Thomas Keilen

Journal:
Trans. Amer. Math. Soc. **355** (2003), 3485-3512

MSC (2000):
Primary 14H10, 14H15, 14H20; Secondary 14J26, 14J27, 14J28, 14J70

DOI:
https://doi.org/10.1090/S0002-9947-03-03304-X

Published electronically:
April 25, 2003

MathSciNet review:
1990160

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In 1985 Joe Harris proved the long-standing claim of Severi that equisingular families of plane nodal curves are irreducible 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 irreducible. Considering different surfaces, including general surfaces in and products of curves, we produce a sufficient condition of the type

where is some constant and some zero-dimensional scheme associated to the singularity type. Our results carry the same asymptotics as the best known results in this direction in the plane case, even though the coefficient is worse. For most of the surfaces considered these are the only known results in that direction.

**[AC83]**Enrico Arbarello and Maurizio Cornalba,*A few remarks about the variety of irreducible plane curves of given degree and genus*, Ann. Sci. École Norm. Sup.**16**(1983), no. 3, 467-488. MR**86a:14020****[Bar93]**Didier Barkats,*Irréductibilité des variétés des courbes planes à noeuds et à cusps*, Preprint no. 364, Univ. de Nice-Sophia-Antipolis, 1993.**[Bog79]**Fedor A. Bogomolov,*Holomorphic tensors and vector bundles on projective manifolds*, Math. USSR Izvestija**13**(1979), no. 3, 499-555. MR**80j:14014****[Bru99]**Andrea Bruno,*Limit linear series and families of equisingular plane curves*, Preprint, 1999.**[CC99]**Luca Chiantini and Ciro Ciliberto,*On the Severi variety of surfaces in*, J. Algebraic Geom.**8**(1999), 67-83. MR**2000f:14082****[Che01]**Xi Chen,*Some remarks on the Severi varieties of surfaces in*, Int. J. Math. Math. Sci.**26**(2001), no. 1, 1-5. MR**2002f:14011****[DH88]**Steven Diaz and Joe Harris,*Ideals associated to deformations of singular plane curves*, Trans. Amer. Math. Soc.**309**(1988), 433-468. MR**89m:14003****[Fla01]**Flaminio Flamini,*Some results of regularity for Severi varieties of projective surfaces*, Comm. Algebra**29**(2001), no. 6, 2297-2311. MR**2002c:14043****[GK89]**Gert-Martin Greuel and Ulrich Karras,*Families of varieties with prescribed singularities*, Compositio 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****[GLS98a]**Gert-Martin Greuel, Christoph Lossen, and Eugenii Shustin,*Geometry of families of nodal curves on the blown-up projective plane*, Trans. Amer. Math. Soc.**350**(1998), 251-274. MR**98j:14034****[GLS98b]**Gert-Martin Greuel, Christoph Lossen, and Eugenii Shustin,*On the irreducibility of families of curves*, Unpulished Manuscript, 1998.**[GLS98c]**Gert-Martin Greuel, Christoph Lossen, and Eugenii Shustin,*Plane curves of minimal degree with prescribed singularities*, Inv. math.**133**(1998), 539-580. MR**99g:14035****[GLS00]**Gert-Martin Greuel, Christoph Lossen, and Eugenii Shustin,*Castelnuovo function, zero-dimensional schemes, and singular plane curves*, J. Algebraic Geom.**9**(2000), no. 4, 663-710. MR**2001g:14045****[Har77]**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York, 1977. MR**57:3116****[Har85]**Joe Harris,*On the Severi problem*, Inventiones Math.**84**(1985), 445-461. MR**87f:14012****[Kan89a]**Pyung-Lyun Kang,*A note on the variety of plane curves with nodes and cusps*, Proc. Amer. Math. Soc.**106**(1989), no. 2, 309-312. MR**89k:14046****[Kan89b]**Pyung-Lyun Kang,*On the variety of plane curves of degree with nodes and cusps*, Trans. Amer. Math. Soc.**316**(1989), no. 1, 165-192. MR**90g:14014****[KT02]**Thomas Keilen and Ilya Tyomkin,*Existence of curves with prescribed topological singularities*, Trans. Amer. Math. Soc.**354**(2002), no. 5, 1837-1860, http:// www. mathematik. uni-kl. 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., Providence, RI, 1997, pp. 161-219. MR**98h:14008****[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/ Lossen002.ps.gz.**[Mil68]**John Milnor,*Singular points of complex hypersurfaces*, Princeton University Press, Princeton, NJ, 1968. MR**39:969****[Ran89]**Ziv Ran,*Families of plane curves and their limits: Enriques' conjecture and beyond*, Annals of Math.**130**(1989), no. 1, 121-157. MR**90e:14024****[Sev21]**Francesco Severi,*Vorlesungen über Algebraische Geometrie*, Bibliotheca Mathematica Teubneriana, no. 32, Teubner, 1921.**[Shu91a]**Eugenii Shustin,*Geometry of discriminant and topology of algebraic curves*, Proc. Internat. Congress Math., Kyoto 1990 (Tokyo, Berlin, New York), vol. 1, Springer-Verlag, 1991. MR**93h:14018****[Shu91b]**Eugenii Shustin,*On manifolds of singular algebraic curves*, Selecta Math. Sov.**10**(1991), 27-37. MR**91k:00045****[Shu94]**Eugenii Shustin,*Smoothness and irreducibility of varieties of plane curves with nodes and cusps*, Bull. Soc. Math. France**122**(1994), 235-253. MR**95e:14020****[Shu96a]**Eugenii Shustin,*Geometry of equisingular families of plane algebraic curves*, J. Alg. Geom.**5**(1996), 209-234. MR**97g:14025****[Shu96b]**Eugenii Shustin,*Smoothness and irreducibility of varieties of algebraic curves with ordinary singularities*, Israel Math. Conf. Proc., no. 9, Amer. Math. Soc., 1996, pp. 393-416. MR**97c:14028****[Wal96]**Charles T.C. Wall,*Highly singular quintic curves*, Math. Proc. Cambridge Philos. Soc.**119**(1996), 257-277. MR**97b:14058****[Zar35]**Oscar Zariski,*Algebraic surfaces*, vol. III, Ergebnisse der Mathematik und ihrer Grenzgebiete, no. 5, Springer, 1935. MR**57:9695**

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

Email:
keilen@mathematik.uni-kl.de

DOI:
https://doi.org/10.1090/S0002-9947-03-03304-X

Keywords:
Algebraic geometry,
singularity theory

Received by editor(s):
August 10, 2001

Received by editor(s) in revised form:
February 5, 2002

Published electronically:
April 25, 2003

Additional Notes:
The author was partially supported by the DFG-Schwerpunkt “Globale Methoden in der komplexen Geometrie”. The author would like to thank the referee for pointing out Example \ref{ex:referee}.

Article copyright:
© Copyright 2003
American Mathematical Society