×

Patchworking singular algebraic curves. I. (English) Zbl 1128.14019

This paper is devoted to the presentation of a general patchworking procedure to construct reduced singular complex curves having prescribed singularities and belonging to a given linear system on an algebraic surface. The patchworking procedure to construct a curve \(C\) with singularities of given type on an algebraic surface \(X\) uses a reducible surface \(X_0\) which is a degeneration of \(X\), then construct a curve \(C_0\) on \(X_0\) and prove that \(C_0\) deforms to \(C\) on \(X\).
The paper under review significantly generalizes the preceding works by the authors and others by using weaker assumptions to prove that \(C_0\) deforms to \(C\) on \(X\). In the second part of this paper [Isr. J. Math. 151, 145–166 (2006; Zbl 1128.14020)], the authors apply this general procedure to produce detailed examples.

MSC:

14H20 Singularities of curves, local rings

Citations:

Zbl 1128.14020
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] L. Caporaso and J. Harris,Counting plane curves of any genus, Inventiones Mathematicae131 (1998), 345-392. · Zbl 0934.14040 · doi:10.1007/s002220050208
[2] X. Chen,Rational curves on K3 surfaces, Journal of Algebraic Geometry8 (1999), 245-278. · Zbl 0940.14024
[3] L. Chiantini and C. Ciliberto,On the Severi varieties on surfaces in ℙ3, Journal of Algebraic Geometry8 (1999), 67-83. · Zbl 0973.14017
[4] R. E. Gompf,A new construction of symplectic manifolds, Annals of Mathematics (2)142 (1995), 527-595. · Zbl 0849.53027 · doi:10.2307/2118554
[5] G.-M. Greuel and U. Karras,Families of varieties with prescribed singularities, Compositio Mathematica69 (1989), 83-110. · Zbl 0684.32015
[6] G.-M. Greuel and C. Lossen,Equianalytic and equisingular families of curves on surfaces, Manuscripta Mathematica91 (1996), 323-342. · Zbl 0874.14021 · doi:10.1007/BF02567958
[7] R. Hartschorne,Algebraic Geometry, Springer-Verlag, Berlin, 1977. · Zbl 0367.14001 · doi:10.1007/978-1-4757-3849-0
[8] D. Hilbert,Mathematische Probleme, Archiv der Mathematik und Physik3 (1901), 213-237 (German). · JFM 32.0084.05
[9] I. Itenberg and E. Shustin,Singular points and limit cycles of planar polynomial vector fields, Duke Mathematical Journal102 (2000), 1-37. · Zbl 0953.34021 · doi:10.1215/S0012-7094-00-10211-6
[10] I. Itenberg and E. Shustin,Combinatorial patchworking of real pseudo-holomorphic curves, Turkish Journal of Mathematics26 (2002), 27-51. · Zbl 1047.14047
[11] T. Keilen and I. Tyomkin,Existence of curves with prescribed topological singularities, Transactions of the American Mathematical Society354 (2002), 1837-1860. · Zbl 0996.14013 · doi:10.1090/S0002-9947-01-02877-X
[12] T. Oda,Convex Bodies and Algebraic Geometry, Springer-Verlag, Berlin, 1988. · Zbl 0628.52002
[13] Z. Ran,Enumerative geometry of singular plane curves, Inventiones Mathematicae97 (1989), 447-465. · Zbl 0702.14040 · doi:10.1007/BF01388886
[14] E. Shustin,Real plane algebraic curves with prescribed singularities, Topology32 (1993), 845-856. · Zbl 0845.14017 · doi:10.1016/0040-9383(93)90053-X
[15] E. Shustin,Critical points of real polynomials, subdivisions of Newton polyhedra and topology of real algebraic hypersurfaces, American Mathematical Society Translations (2)173 (1996), 203-223. · Zbl 0883.14032
[16] E. Shustin,Gluing of singular and critical points, Topology37 (1998), 195-217. · Zbl 0905.14008 · doi:10.1016/S0040-9383(97)00008-6
[17] E. Shustin,Lower deformations of isolated hypersurface singularities, Algebra i Analiz10 (1999), 221-249 (English translation: St. Petersburg Mathematical Journal11 (2000), 883-908). · Zbl 0967.14002
[18] E. Shustin,Analytic order of singular and critical points, Transactions of the American Mathematical Society356 (2004), 953-985. · Zbl 1044.14008 · doi:10.1090/S0002-9947-03-03409-3
[19] E. Shustin,Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry, A tropical approach to enumerative geometry, Algebra i Analiz17 (2005), 170-214. · Zbl 1100.14046
[20] E. Shustin and I. Tyomkin,Patchworking singular algebraic curves II, Israel Journal of Mathematics, this volume. · Zbl 1128.14020
[21] E. Shustin and E. Westenberger,Projective hypersurfaces with many singularities of prescribed types, Journal of the London Mathematical Society (2)70 (2004), 609-624. · Zbl 1075.14034 · doi:10.1112/S0024610704005794
[22] O. Viro,Curves of degree 7, curves of degree 8 and the Ragsdale conjecture, Doklady Akademii Nauk SSSR254 (1980), 1305-1310 (Russian); English translation: Soviet Mathematics Doklady22 (1980), 566-570. · Zbl 0462.14020
[23] O. Ya. Viro,Gluing of algebraic hypersurfaces, smoothing of singularities and construction of curves, Proceedings of the Leningrad International Topological Conference, Leningrad, Aug. 1982, Nauka, Leningrad, 1983, pp. 149-197 (Russian). · Zbl 0605.14021
[24] O. Viro,Gluing of plane real algebraic curves and constructions of curves of degrees 6 and 7, Lecture Notes in Mathematics1060, Springer-Verlag, Berlin, 1984, pp. 187-200. · Zbl 0576.14031
[25] O. Ya. Viro,Real algebraic plane curves: constructions with controlled topology, Leningrad Mathemaical Journal1 (1990), 1059-1134. · Zbl 0732.14026
[26] O. Viro,Introduction to Topology of Real Algebraic Varieties, Preprint, appears in the internet at http://www.math.uu.se/ oleg/es/index.html. · Zbl 0879.14031
[27] O. Viro,Patchworking real algebraic varieties, Preprint, appears in the internet at http://www.math.uu.se/ oleg/pw.ps. · Zbl 0876.14017
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.