Skip to main content
SpringerLink
Log in

Equisingular Families of Curves

  • Chapter
  • First Online:
Singular Algebraic Curves

Part of the book series: Springer Monographs in Mathematics ((SMM))

  • 1037 Accesses

Abstract

We are ready to accomplish our main task, that is to answer the two following questions concerning equisingular families (ESF) of curves. whether a family of algebraic curves with a prescribed collection of singularities form a nonempty, T-smooth (i.e. smooth of expected dimension), irreducible stratum in the discriminant in a given linear system |D| on a smooth algebraic surface \({\varSigma }\), and what is the local structure of the discriminant in a neighborhood of the above stratum, in particular, when is a family deformation complete (i.e. when are any simultaneous local deformations of the curve singularities induced by the given linear system).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 79.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    We use the little-o notation, meaning that the correction term o(f(d)) includes only lower powers in d than f(d).

  2. 2.

    The missing cases of \(k=3\) and \(d=5\) or 6 are covered in [AD, Theorem 2.4], [Deg1, Deg2, Section II.7.3].

  3. 3.

    With the capital O-notation, meaning that the correction term O(d) is linear in d.

  4. 4.

    This phenomenon has first been observed by Wahl in 1974 [Wah1], where he presented the first series of irreducible plane curves C (with only nodes and cusps) such that the corresponding ESF is non-smooth, but has a smooth reduction: \(V_{104}^{{ irr}} (3636 \!\!\, \cdot \!\!\, A_1, \,900 \!\cdot \!\!\,A_2)\).

  5. 5.

    A zero-dimensional scheme \(Z\subset {\mathbb P}^n\) is called curvilinear if, for each p in the support of Z, there is a smooth curve germ \((C,p)\subset ({\mathbb P}^n,p)\) and some integer \(\ell \ge 1\) such that \({\mathcal O}_{Z,p}={\mathcal O}_{C,p}/\mathfrak {m}_p^\ell \).

  6. 6.

    Such a curve can easily be constructed by applying the Cremona transformation to the 3-cuspidal quartic curve so that one of the axes is quadratically tangent to the quartic, and the fundamental points of the coordinate system are non-singular points of the quartic (we leave details to the reader).

  7. 7.

    For example, \({\sigma }_1=({\varDelta }_1)_-\).

  8. 8.

    Here \({\mathcal S}_{f_k}\) denotes the collection of contact analytic types of singular points of \(f_k\) in \(({\mathbb C}^*)^n\) in the sense of Sect.  2.3.2.

  9. 9.

    Here “generic” means that the object considered can be chosen arbitrarily in a Zariski open subset of the whole space of objects.

  10. 10.

    Note that, over the reals, the deformation problem is much more delicate than over the complex field.

References

  1. Greuel, G.-M., Karras, U.: Families of varieties with prescribed singularities. Compos. Math. 69, 83–110 (1989)

    Google Scholar 

  2. Greuel, G.-M., Lossen, C.: The geometry of families of singular curves. In: Siersma, D., et al. (eds.) New Developments in Singularity Theory. Proceedings of the NATO Advanced Study Institute on New Developments in Singularity Theory. Cambridge University, Cambridge (2000) (NATO Science Series II, Mathematics, Physics and Chemistry, vol. 21, pp. 159–192. Kluwer, Dordrecht, 2001)

    Chapter  Google Scholar 

  3. Greuel, G.-M., Lossen, C., Shustin, E.: New asymptotics in the geometry of equisingular families of curves. Int. Math. Res. Not. 13, 595–611 (1997)

    Google Scholar 

  4. Greuel, G.-M., Lossen, C., Shustin, E.: Geometry of families of nodal curves on the blown-up projective plane. Trans. Am. Math. Soc. 350, 251–274 (1998)

    Google Scholar 

  5. Greuel, G.-M., Lossen, C., Shustin, E.: Plane curves of minimal degree with prescribed singularities. Invent. Math. 133, 539–580 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Greuel, G.-M., Lossen, C., Shustin, E.: Castelnuovo function, zero-dimensional schemes and singular plane curves. J. Algebr. Geom. 9(4), 663–710 (2000)

    Google Scholar 

  7. Greuel, G.-M., Lossen, C., Shustin, E.: The variety of plane curves with ordinary singularities is not irreducible. Int. Math. Res. Not. 11, 543–550 (2001)

    Google Scholar 

  8. Shustin, E.: Smoothness of equisingular families of plane algebraic curves. Int. Math. Res. Not. 2, 67–82 (1997)

    Google Scholar 

  9. Shustin, E.: On manifolds of singular algebraic curves. Selecta Math. Sov. 10, 27–37 (1991)

    Google Scholar 

  10. Shustin, E.: Real plane algebraic curves with prescribed singularities. Topology 32, 845–856 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Shustin, E.: Analytic order of singular and critical points. Trans. Am. Math. Soc. 356, 953–985 (2004)

    Google Scholar 

  12. Shustin, E., Tyomkin, I.: Versal deformations of algebraic hypersurfaces with isolated singularities. Math. Annalen 313(2), 297–314 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  13. Shustin, E., Westenberger, E.: Projective hypersurfaces with many singularities of prescribed types. J. Lond. Math. Soc. (2) 70(3), 609–624 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Greuel, G.-M., Lossen, C.: Equianalytic and equisingular families of curves on surfaces. Manuscr. math. 91, 323–342 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Shustin, E.: Versal deformation in the space of plane curves of fixed degree. Funct. Anal. Appl. 21, 82–84 (1987)

    Google Scholar 

  16. Hironaka, H.: On the arithmetic genera and the effective genera of algebraic curves. Mem. Coll. Sci. Univ. Kyoto 30, 177–195 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  17. Severi, F.: Vorlesungen über algebraische Geometrie. Teubner (1921), respectively Johnson (1968)

    Google Scholar 

  18. Segre, B.: Dei sistemi lineari tangenti ad un qualunque sistema di forme. Atti Acad. naz. Lincei Rendiconti serie 5(33), 182–185 (1924)

    Google Scholar 

  19. Segre, B.: Esistenza e dimensione di sistemi continui di curve piane algebriche con dati caraterri. Atti Acad. naz. Lincei Rendiconti serie 6(10), 31–38 (1929)

    Google Scholar 

  20. Tannenbaum, A.: Families of curves with nodes on \(K3\)- surfaces. Math. Ann. 260, 239–253 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  21. Vassiliev, V.A.: Stable cohomology of complements to the discriminants of deformations of singularities of smooth functions. J. Sov. Math. 52, 3217–3230 (1990)

    Google Scholar 

  22. Lefschetz, S.: On the existence of loci with given singularities. Trans. AMS 14, 23–41 (1913)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sakai, F.: Singularities of plane curves. Geometry of Complex Projective Varieties, Seminars and Conferences, vol. 9, pp. 257–273. Mediterranean Press, Rende (1993)

    Google Scholar 

  24. Hirzebruch, F.: Singularities of algebraic surfaces and characteristic numbers. Contemp. Math. 58, 141–155 (1986)

    Google Scholar 

  25. Ivinskis, K.: Normale Flächen und die Miyaoka-Kobayashi-Ungleichung. University of Bonn, Diplomarbeit (1985)

    Google Scholar 

  26. Langer, A.: Logarithmic orbifold Euler numbers of surfaces with applications. Proc. Lond. Math. Soc. 3(86), 358–396 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zariski, O.: Algebraic Surfaces, 2nd edn. Springer, Berlin (1971)

    Google Scholar 

  28. Varchenko, A.N.: Asymptotics of integrals and Hodge structures. Mod. Probl. Math. 22 (Itogi nauki i tekhniki VINITI), 130–166 (Russian) (1983)

    Google Scholar 

  29. Varchenko, A.N.: On semicontinuity of the spectrum and an upper estimate for the number of singular points of a projective hypersurface. Sov. Math. Dokl. 27, 735–739 (1983); translation from Dokl. Akad. Nauk SSSR 270, 1294–1297 (1983)

    Google Scholar 

  30. Hirano, A.: Constructions of plane curves with cusps. Saitama Math. J. 10, 21–24 (1992)

    Google Scholar 

  31. Koelman, R.J.: Over de cusp. University of Leiden, Diplomarbeit (1986)

    Google Scholar 

  32. Shustin, E.: Gluing of singular and critical points. Topology 37(1), 195–217 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-6 — A computer algebra system for polynomial computations (2012). http://www.singular.uni-kl.de

  34. Schulze, M.: A singular 3-1-6 library for computing invariants related to the Gauss-Manin system of an isolated hypersurface singularity (2012) (gmssing.lib.)

    Google Scholar 

  35. Gradolato, M.A., Mezzetti, E.: Families of curves with ordinary singular points on regular surfaces. Ann. mat. pura et appl. 150, 281–298 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wall, C.T.C.: Geometry of quartic curves. Math. Proc. Camb. Philos. Soc. 117, 415–423 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wall, C.T.C.: Highly singular quintic curves. Math. Proc. Camb. Philos. Soc. 119, 257–277 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  38. Vik, K.S.: The generalized Chisini conjecture. Proc. Steklov Inst. Math. 241(2), 110–119 (2003)

    Google Scholar 

  39. Calabri, A., Paccagnan, D., Stagnaro, E.: Plane algebraic curves with many cusps with an appendix by Eugenii Shustin. Annali di Matematica Pura ed Applicata (4) 193(3), 909–921 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. Fulton, W.: On the fundamental group of the complement of a node curve. Ann. Math. (2) 111(2), 407–409 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  41. Deligne, P.: Le groupe fondamental du complement d’une courbe plane n’ayant que des points doubles ordinaires est abelien (d’apres W. Fulton). Bourbaki Seminar, vol. 1979/80. Lecture Notes in Mathematics, vol. 842, pp. 1–10. Springer, Berlin (1981)

    Google Scholar 

  42. Harris, J.: On the severi problem. Invent. Math. 84, 445–461 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  43. Albanese, G.: Sulle condizioni perchè una curva algebraica riducible si possa considerare come limite di una curva irreducibile. Rend. Circ. Mat. Palermo 2(52), 105–150 (1928)

    Google Scholar 

  44. Nobile, A.: On specialization of curves I. Trans. Am. Math. Soc. 282(2), 739–748 (1984)

    Google Scholar 

  45. Ran, Z.: On nodal plane curves. Invent. Math. 86, 529–534 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  46. Treger, R.: On the local Severi problem. Bull. Am. Math. Soc. 19(1), 325–327 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ran, Z.: Families of plane curves and their limits: enriques’ conjecture and beyond. Ann. Math. 130(1), 121–157 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  48. Kang, P.-L.: On the variety of plane curves of degree \(d\) with \(\delta \) nodes and \(k\) cusps. Trans. Am. Math. Soc. 316(1), 165–192 (1989)

    Google Scholar 

  49. Kang, P.-L.: A note on the variety of plane curves with nodes and cusps. Proc. Am. Math. Soc. 106(2), 309–312 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  50. Akyol, A., Degtyarev, A.: Geography of irreducible plane sextics. Proc. Lond. Math. Soc. (3) 111(6), 1307–1337 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  51. Degtyarev, A.I.: Isotopic classification of complex plane projective curves of degree 5. Algebra i Analiz 1(4), 78–101 (1989, Russian); English translation Leningrad Math. J. 1(4), 881–904 (1990)

    Google Scholar 

  52. Degtyarev, A.: Topology of Algebraic Curves. An Approach via Dessins D’enfants. De Gruyter Studies in Mathematics. Walter de Gruyter, Berlin (2012)

    Google Scholar 

  53. Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. Springer, Berlin (1998)

    Google Scholar 

  54. Tannenbaum, A.: On the classical characteristic linear series of plane curves with nodes and cuspidal points: two examples of Beniamino Segre. Compos. Math. 51, 169–183 (1984)

    MathSciNet  MATH  Google Scholar 

  55. Wahl, J.: Equisingular deformations of plane algebroid curves. Trans. Am. Math. Soc. 193, 143–170 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  56. Luengo, I.: The \(\mu \)- constant stratum is not smooth. Invent. Math. 90, 139–152 (1987)

    Google Scholar 

  57. Luengo, I.: On the existence of complete families of projective plane curves, which are obstructed. J. Lond. Math. Soc. 2(36), 33–43 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  58. Shustin, E.: Smoothness and irreducibility of varieties of algebraic curves with nodes and cusps. Bull. SMF 122, 235–253 (1994)

    Google Scholar 

  59. Giacinti-Diebolt, C.: Variétés des courbes projectives planes de degré et lieu singulier donnés. Math. Ann. 266, 321–350 (1984)

    Google Scholar 

  60. Flamini, F.: Some results of regularity for severi varieties of projective surfaces. Commun. Algebra 29(6), 2297–2311 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  61. Chiantini, L., Sernesi, E.: Nodal curves on surfaces of general type. Math. Ann. 307, 41–56 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  62. Keilen, T.: Smoothness of equisingular families of curves. Trans. Am. Math. Soc. 357(6), 2467–2481 (2005)

    Google Scholar 

  63. Hartshorne, R.: Algebraic Geometry. Graduate Text in Mathematics, vol. 52. Springer, Berlin (1977)

    Book  MATH  Google Scholar 

  64. Du Plessis, A.A., Wall, C.T.C.: Singular hypersurfaces, versality and Gorenstein algebras. J. Algebraic Geom. 9(2), 309–322 (2000)

    Google Scholar 

  65. Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer, Berlin (2007)

    Google Scholar 

  66. Varchenko, A.N.: A lower bound for the codimension of the stratum \(\mu =\) const in terms of the mixed Hodge structure. Moscow Univ. Math. Bull. 37(2), 28–31 (1982)

    Google Scholar 

  67. Chmutov, S.V.: Examples of projective surfaces with many singularities. J. Algebraic Geom. 1(2), 191–196 (1992)

    Google Scholar 

  68. Shustin, E.: New M-curve of the 8th degree. Math. Notes Acad. Sci. USSR 42, 606–610 (1987)

    Google Scholar 

  69. Shustin, E.: Lower deformations of isolated hypersurface singularities. Algebra i Analiz 10(5), 221–249 (1999) (English translation in St. Petersburg Math. J. 11(5), 883–908 (2000)

    Google Scholar 

  70. Du Plessis, A.A., Wall, C.T.C.: Discriminants and vector fields. In: Arnold, V.I., et al. (eds.) Singularities, the Brieskorn Anniversary Volume. Progress in Mathematics, vol. 162, pp. 119–140. Birkhäuser, Basel (1998)

    Chapter  Google Scholar 

  71. Orevkov, S., Shustin, E.I.: Real algebraic and pseudoholomorphic curves on the quadratic cone and smoothings of singularity \(X_{21}\). St. Petersburg Math. J. 28, 225–257 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  72. Korchagin, A.B., Shustin, E.I.: Affine curves of degree 6 and smoothings of the ordinary 6th order singular point. Math. USSR Izvestia 33(3), 501–520 (1989)

    Google Scholar 

  73. Chevallier, B.: Four \(M\)-curves of degree 8. Funktsional. Anal. i Prilozhen. 36(1), 90–93 (2002) (English translation in Funct. Anal. Appl. 36(1), 76–78 (2002)

    Google Scholar 

  74. Pecker, D.: Note sur la réalité des points doubles des courbes gauches. C. R. Acad. Sci. Paris, Sér. I 324, 807–812 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  75. Pecker, D.: Sur la réalité des points doubles des courbes gauches. Ann. Inst. Fourier (Grenoble) 49, 1439–1452 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  76. Gudkov, D.A.: On the curve of 5th order with \(5\) cusps. Funct. Anal. Appl. 16, 201–202 (1982)

    Google Scholar 

  77. Itenberg, I., Shustin, E.: Real algebraic curves with real cusps. Am. Math. Soc. Transl. 2(173), 97–109 (1996)

    Google Scholar 

  78. Miyaoka, Y.: On the Chern numbers of surfaces of general type. Invent. Math. 42, 225–237 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  79. Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. USA 74, 1798–1799 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  80. Gusein-Zade, S.M., Nekhoroshev, N.N.: Singularities of type \(A_k\) on plane curves of a chosen degree. Funct. Anal. Appl. 34(3), 214–215 (2000); translation from Funkts. Anal. Prilozh. 34(3), 69–70 (2000)

    Google Scholar 

  81. Gusein-Zade, S.M., Nekhoroshev, N.N.: Contiguity of \(A_k\)- singularities at points of the stratum \(\mu =\text{const}\) of a singularity. Funct. Anal. Appl. 17, 312–313 (1983)

    Google Scholar 

  82. Orevkov, S.Y.: Some examples of real algebraic and real pseudoholomorphic curves. Perspectives in Analysis, Geometry, and Topology. Progress in Mathematics, vol. 296, pp. 355–387. Birkhäuser, Basel (2012)

    Google Scholar 

  83. Keilen, T., Tyomkin, I.: Existence of curves with prescribed topological singularities. Trans. Am. Math. Soc. 354(5), 1837–1860 (2002)

    Google Scholar 

  84. Chen, X.: Rational curves on K3 surfaces. J. Algebraic Geom. 8, 245–278 (1999). Thesis, Harvard University (1997)

    Google Scholar 

  85. Chiantini, L., Ciliberto, C.: On the severi varieties of surfaces in \({\mathbb{P}}^3\). J. Algebraic Geom. 8(1), 67–83 (1999)

    Google Scholar 

  86. Göttsche, L., Pandharipande, R.: The quantum cohomology of blow-ups of \({\mathbb{P}}^2\) and enumerative geometry. J. Diff. Geom. 48(1), 61–90 (1998)

    Google Scholar 

  87. Abramovich, D., Bertram, A.: The formula \(12=10+2\times 1\) and its generalizations: counting rational curves on \({\varvec {F}}_2\). Advances in Algebraic Geometry Motivated by Physics (Lowell, MA, 2000). Contemporary Mathematics, vol. 276, pp. 83–88. American Mathematical Society, Providence (2001)

    Google Scholar 

  88. Welschinger, J.-Y.: Courbes algébriques réelles et courbes flexibles sur les surfaces réglées de base \({\mathbb{C}} P^1\). Proc. Lond. Math. Soc. 85(2), 367–392 (2002)

    Google Scholar 

  89. Mikhalkin, G.: Counting curves via the lattice paths in polygons. Comptes Rendus Math. 336(8), 629–634 (2003)

    Google Scholar 

  90. Mikhalkin, G.: Enumerative tropical algebraic geometry in \({\mathbb{R}}^2\). J. Am. Math. Soc. 18, 313–377 (2005)

    Google Scholar 

  91. Yau, S.-T., Zaslow, E.: BPS states, string duality, and nodal curves on K3. Nucl. Phys. B 471, 503–512 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  92. Klemm, A., Maulik, D., Pandharipande, R., Scheidegger, E.: Noether-Lefschetz theory and the Yau-Zaslow conjecture. J. Am. Math. Soc. 23(4), 1013–1040 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  93. Chen, X., Lewis, J.D.: Density of rational curves on K3 surfaces. Math. Ann. 356(1), 331–354 (2013)

    Google Scholar 

  94. Arbarello, E., Cornalba, M.: A few remarks about the variety of irreducible plane curves of given degree and genus. Ann. Sci. École Norm. Sup. 16(3), 467–488 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  95. Bruce, J.W., Gibblin, P.J.: A stratification of the space of plane quartic curves. Proc. Lond. Math. Soc. 442, 270–298 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  96. Deligne, P., Mumford, D.: The irreducibility of the moduli space of curves with given genus. Publ. Math. IHES 36, 75–100 (1969)

    Google Scholar 

  97. Shustin, E.: Smoothness and irreducibility of families of plane algebraic curves with ordinary singularities. Israel Mathematical Conference Proceedings, vol. 9, pp. 393–416. AMS, Providence (1996)

    Google Scholar 

  98. Shustin, E.: Geometry of equisingular families of plane algebraic curves. J. Algebraic Geom. 5, 209–234 (1996)

    Google Scholar 

  99. Nori, M.: Zariski conjecture and related problems. Ann. Sci. Ec. Norm. Sup. 4(16), 305–344 (1983)

    Google Scholar 

  100. Keilen, T.: Irreducibility of equisingular families of curves. Trans. Am. Math. Soc. 355(9), 3485–3512 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  101. Lazarsfeld, R.: Lectures on linear series. In: Kollár, J. (ed.) Complex Algebraic Geometry. AMS, Providence (1997)

    Google Scholar 

  102. Wahl, J.: Deformations of plane curves with nodes and cusps. Am. J. Math. 96, 529–577 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  103. Campillo, A., Greuel, G.-M., Lossen, C.: Equisingular deformations of plane curves in arbitrary characteristic. Compos. Math. 143, 829–882 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  104. Campillo, A., Greuel, G.-M., Lossen, C.: Equisingular calculations for plane curve singularities. J. Symb. Comput. 42(1–2), 89–114 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  105. Boubakri, Y., Greuel, G.-M., Markwig, T.: Invariants of hypersurface singularities in positive characteristic. Rev. Mat. Complut. 25, 61–85 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  106. Greuel, G.-M., Pham, T.H.: On finite determinacy for matrices of power series. Math. Z. (2018). https://doi.org/10.1007/s00209-018-2040-2

    Article  MathSciNet  MATH  Google Scholar 

  107. Zariski, O.: Studies in equisingularity I–III. Am. J. Math. 87, 507–536 and 972–1006 (1965), respectively Am. J. Math. 90, 961–1023 (1968)

    Google Scholar 

  108. Nobile, A.: Families of curves on surfaces. Math. Zeitschrift 187(4), 453–470 (1984)

    Google Scholar 

  109. Tyomkin, I.: On severi varieties on Hirzebruch surfaces. Int. Math. Res. Not. 23, Art. ID rnm109, 31 pp (2007)

    Google Scholar 

  110. Tyomkin, I.: On Zariski’s theorem in positive characteristic. J. Eur. Math. Soc. (JEMS) 15(5), 1783–1803 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  111. Tyomkin, I.: An example of a reducible severi variety. In: Proceedings of the Gökova Geometry-Topology Conference 2013. Gökova Geometry/Topology Conference (GGT), Gökova, pp. 33–40 (2014)

    Google Scholar 

  112. Westenberger, E.: Existence of hypersurfaces with prescribed simple singularities. Commun. Algebra 31(1), 335–356 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  113. Alexander, J., Hirschowitz, A.: An asymptotic vanishing theorem for generic unions of multiple points. Invent. Math. 140(2), 303–325 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  114. Roé, J.: Maximal rank for schemes of small multiplicity by Évain’s differential Horace method. Trans. Am. Math. Soc. 366, 857–874 (2014)

    Article  MATH  Google Scholar 

  115. Bigatti, A., Geramita, A.V., Migliore, J.C.: Geometric consequences of extremal behavior in a theorem of Macaulay. Trans. Am. Math. Soc. 346(1), 203–235 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  116. Nagata, M.: On the 14-th problem of Hilbert. Am. J. Math. 81, 766–772 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  117. Nagata, M.: On rational surfaces II. Mem. Coll. Sci. Univ. Kyoto Ser. A 33, 271–293 (1960)

    Google Scholar 

  118. Dumnicki, M., Harbourne, B., Küronya, A., Roé, J., Szemberg, T.: Very general monomial valuations of \({\mathbb{P}}^2\) and a Nagata type conjecture. Commun. Anal. Geom. 25(1), 125–161 (2017)

    Google Scholar 

  119. Ciliberto, C., Harbourne, B., Miranda, R., Roé, J.: Variations on Nagata’s conjecture. In: Hassett, B., et al. (eds.) A Celebration of Algebraic Geometry. Clay Mathematics Proceedings, vol. 18, pp. 185–203 (2013)

    Google Scholar 

  120. Fulton, W., Pandharipande, R.: Notes on stable maps and quantum cohomology. Algebraic geometry—Santa Cruz 1995. Proceedings of Symposia in Pure Mathematics, vol. 62, Part 2, pp. 45–96. American Mathematical Society, Providence (1997)

    Google Scholar 

  121. Getzler, E.: Intersection theory on \(\overline{\cal{M}}_{1,4}\) and elliptic Gromov-Witten invariants. J. Am. Math. Soc. 10(4), 973–998 (1997)

    Google Scholar 

  122. Kontsevich, M., Manin, Yu.: Gromov-Witten classes, quantum cohomology and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  123. Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. J. Differ. Geom. 42(2), 259–367 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  124. Vakil, R.: Counting curves on rational surfaces. Manuscr. math. 102(1), 53–84 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  125. Pandharipande, R.: Intersections of \({\mathbb{Q}}\)- divisors on Kontsevich’s moduli space \(\overline{\cal{M}}_{0, n}({\mathbb{P}}^r, d)\) and enumerative geometry. Trans. Am. Math. Soc. 351(4), 1481–1505 (1999)

    Google Scholar 

  126. Zinger, A.: Counting rational curves of arbitrary shape in projective spaces. Geom. Topol. 9, 571–697 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  127. Göttsche, L.: A conjectural generating function for numbers of curves on surfaces. Commun. Math. Phys. 196, 523–533 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  128. Göttsche, L., Shende, V.: The \(\chi _{-y}\)- genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces. Algebraic Geom. 2(4), 405–421 (2015)

    Google Scholar 

  129. Göttsche, L., Shende, V.: Refined curve counting on complex surfaces. Geom. Topol. 18, 2245–2307 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  130. Fehér, L., Rimányi, R.: On the structure of Thom polynomials of singularities. Bull. Lond. Math. Soc. 39, 541–549 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  131. Kazarian, M.E.: Multisingularities, cobordisms, and enumerative geometry. Russ. Math. Surv. 58(4), 665–724 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  132. Liu, A.K.: Family blowup formula, admissible graphs and the enumeration of singular curves I. J. Diff. Geom. 56(3), 381–579 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  133. Basu, S., Mukherjee, R.: Enumeration of curves with two singular points. Bull. Sci. Math. 139(6), 667–735 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  134. Basu, S., Mukherjee, R.: Enumeration of curves with one singular point. J. Geom. Phys. 104, 175–203 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  135. Kerner, D.: Enumeration of singular algebraic curves. Isr. Math. J. 155, 1–56 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  136. Kleiman, S., Piene, R.: Enumerating singular curves on surfaces. Algebraic geometry: Hirzebruch 70 (Warsaw, 1998). Contemporary Mathematics, vol. 241, pp. 209–238. American Mathematical Society, Providence (1999)

    Google Scholar 

  137. Vainsencher, I.: Counting divisors with prescribed singularities. Trans. Am. Math. Soc. 267(2), 399–422 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  138. Caporaso, L., Harris, J.: Counting plane curves of any genus. Invent. Math. 131(2), 345–392 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  139. Ran, Z.: On the quantum cohomology of the plane, old and new, and a K3 analogue. Collect. Math. 49(2–3), 519–526 (1998)

    Google Scholar 

  140. Shoval, M., Shustin, E.: On Gromov-Witten invariants of del pezzo surfaces. Int. J. Math. 24(7), 44 pp (2013). https://doi.org/10.1142/S0129167X13500547

    Article  MathSciNet  MATH  Google Scholar 

  141. Brugallé, E.: Floor diagrams relative to a conic, and GW-W invariants of del pezzo surfaces. Adv. Math. 279, 438–500 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  142. Mikhalkin, G.: Tropical geometry and its applications. In: Sanz-Solé, M., et al. (ed.) Proceedings of the ICM, Madrid, Spain, 22–30 August 2006. Volume II: Invited Lectures, pp. 827–852. European Mathematical Society, Zürich (2006)

    Google Scholar 

  143. Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(1), 1–51 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  144. Ganor, Y., Shustin, E.: Enumeration of plane unicuspidal curves of any genus via tropical geometry (2018). arXiv:1807.11443

  145. Kulikov, V.S., Shustin, E.: On rigid plane curves. Eur. J. Math. 2(1), 208–226 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  146. Greuel, G.-M., Shustin, E.: Geometry of equisingular families of curves. In: Bruce, B., Mond, D. (eds.) Singularity Theory. Proceedings of the European Singularities Conference, Liverpool, August 1996. Dedicated to C.T.C. Wall in Occasion of His 60th Birthday (London Mathematical Society Lecture Note Series, vol. 263), pp. 79–108. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  147. Bruce, J.W.: Singularities on a projective hypersurface. Bull. Lond. Math. Soc. 13, 47–50 (1981)

    Google Scholar 

  148. Gudkov, D.A., Utkin, G.A.: The topology of curves of degree 6 and surfaces of degree 4. Uchen. Zap. Gorkov. Univ. 87 (Russian), English translation Transl. AMS 112 (1978)

    Google Scholar 

  149. Shustin, E.: Geometry of discriminant and topology of algebraic curves. Proceedings of the International Congress of Mathematicians, Kyoto 1990, vol. 1. Springer, Berlin (1991)

    Google Scholar 

  150. Barkats, D.: Études des variétés des courbes planes à noeuds et à cusps. Algebraic Geometry (Catania, 1993/Barcelona, 1994). Lecture Notes in Pure and Applied Mathematics, vol. 200, pp. 25–35. Dekker, New York (1998)

    Google Scholar 

  151. Gourevich, A., Gourevich, D.: Geometry of obstructed equisingular families of projective hypersurfaces. J. Pure Appl. Algebra 213(9), 1865–1889 (2009)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fachbereich Mathematik, TU Kaiserslautern, Kaiserslautern, Germany

    Gert-Martin Greuel & Christoph Lossen

  2. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

    Eugenii Shustin

Authors
  1. Gert-Martin Greuel
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christoph Lossen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Eugenii Shustin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gert-Martin Greuel .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Greuel, GM., Lossen, C., Shustin, E. (2018). Equisingular Families of Curves. In: Singular Algebraic Curves. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-03350-7_4

Download citation

Publish with us

Policies and ethics

Search

Navigation