Irreducibility of equisingular families of curves — improved conditions. (English) Zbl 1073.14039
In an earlier paper [Trans. Am. Math. Soc. 355, No. 9, 3485–3512 (2003; Zbl 1032.14005)], the author gave sufficient conditions for the irreducibility of the family \(V^{\text{irr}}\) of irreducible curves which have prescribed arbitrary singularities and belong to a given linear system on smooth projective surfaces \(S\) of several types. The conditions were expressed in the form of an upper bound on the sum of certain singularity invariants by \(\gamma (D-K_{S})^2\), where \(K_{S}\) is the canonical divisor of \(S\) and \(\gamma\) is some constant. In the present paper the author improves this condition, that is the constant \(\gamma\), by a factor of \(9\). The proof proceeds along the same lines of the original one, by showing that some irreducible “regular” subscheme of \(V^{\text{irr}}\) is dense in \(V^{\text{irr}}\), following the approach contained in an unpublished paper of G. M. Greuel, C. Lossen and E. Shustin [“On the irreducibility of families of curves” (1998)], which is based on ideas of L. Chiantini and C. Ciliberto [J. Algebr. Geom. 8, No. 1, 67–83 (1999; Zbl 0973.14017)].
Reviewer: Caterina Cumino (Torino)
MSC:
14H20 | Singularities of curves, local rings |
14J26 | Rational and ruled surfaces |
14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |
14J28 | \(K3\) surfaces and Enriques surfaces |
14J70 | Hypersurfaces and algebraic geometry |
Keywords:
singularity theoryReferences:
[1] | Chiantini L., J. Algebraic Geom. 8 pp 67– (1999) |
[2] | DOI: 10.1007/s002080050021 · Zbl 0870.14027 · doi:10.1007/s002080050021 |
[3] | DOI: 10.1155/S1073792897000391 · Zbl 0905.14014 · doi:10.1155/S1073792897000391 |
[4] | Greuel G.-M., J. Algebraic Geom. 9 (4) pp 663– (2000) |
[5] | DOI: 10.1007/978-1-4757-3849-0 · doi:10.1007/978-1-4757-3849-0 |
[6] | Keilen T., Trans. Amer. Math. Soc. (2003) |
[7] | DOI: 10.1090/S0002-9947-01-02877-X · Zbl 0996.14013 · doi:10.1090/S0002-9947-01-02877-X |
[8] | Lossen , C. ( 1998 ) . The geometry of equisingular and equianalytic families of curves on a surface . Ph.D. thesis, FB Mathematik, Universität Kaiserslautern. http://www.mathematik.uni-kl.de/lossen/download/Lossen002/Lossen002.ps.gz . · Zbl 0949.14018 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.