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Equisingular Families of Curves

  • Gert-Martin GreuelEmail author
  • Christoph Lossen
  • Eugenii Shustin
Chapter
Part of the Springer Monographs in Mathematics book series (SMM)

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).

References

  1. [GrK]
    Greuel, G.-M., Karras, U.: Families of varieties with prescribed singularities. Compos. Math. 69, 83–110 (1989)Google Scholar
  2. [GrL2]
    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)Google Scholar
  3. [GLS1]
    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. [GLS2]
    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. [GLS3]
    Greuel, G.-M., Lossen, C., Shustin, E.: Plane curves of minimal degree with prescribed singularities. Invent. Math. 133, 539–580 (1998)MathSciNetzbMATHGoogle Scholar
  6. [GLS4]
    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. [GLS5]
    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. [Shu10]
    Shustin, E.: Smoothness of equisingular families of plane algebraic curves. Int. Math. Res. Not. 2, 67–82 (1997)Google Scholar
  9. [Shu4]
    Shustin, E.: On manifolds of singular algebraic curves. Selecta Math. Sov. 10, 27–37 (1991)Google Scholar
  10. [Shu5]
    Shustin, E.: Real plane algebraic curves with prescribed singularities. Topology 32, 845–856 (1993)MathSciNetzbMATHGoogle Scholar
  11. [Shu13]
    Shustin, E.: Analytic order of singular and critical points. Trans. Am. Math. Soc. 356, 953–985 (2004)Google Scholar
  12. [ShT2]
    Shustin, E., Tyomkin, I.: Versal deformations of algebraic hypersurfaces with isolated singularities. Math. Annalen 313(2), 297–314 (1999)MathSciNetzbMATHGoogle Scholar
  13. [ShW]
    Shustin, E., Westenberger, E.: Projective hypersurfaces with many singularities of prescribed types. J. Lond. Math. Soc. (2) 70(3), 609–624 (2004)MathSciNetzbMATHGoogle Scholar
  14. [GrL1]
    Greuel, G.-M., Lossen, C.: Equianalytic and equisingular families of curves on surfaces. Manuscr. math. 91, 323–342 (1996)MathSciNetzbMATHGoogle Scholar
  15. [Shu1]
    Shustin, E.: Versal deformation in the space of plane curves of fixed degree. Funct. Anal. Appl. 21, 82–84 (1987)Google Scholar
  16. [HiH]
    Hironaka, H.: On the arithmetic genera and the effective genera of algebraic curves. Mem. Coll. Sci. Univ. Kyoto 30, 177–195 (1957)MathSciNetzbMATHGoogle Scholar
  17. [Sev]
    Severi, F.: Vorlesungen über algebraische Geometrie. Teubner (1921), respectively Johnson (1968)Google Scholar
  18. [Seg1]
    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. [Seg2]
    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. [Tan2]
    Tannenbaum, A.: Families of curves with nodes on \(K3\)- surfaces. Math. Ann. 260, 239–253 (1982)MathSciNetzbMATHGoogle Scholar
  21. [Vas]
    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. [Lef]
    Lefschetz, S.: On the existence of loci with given singularities. Trans. AMS 14, 23–41 (1913)MathSciNetzbMATHGoogle Scholar
  23. [Sak]
    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. [HiF]
    Hirzebruch, F.: Singularities of algebraic surfaces and characteristic numbers. Contemp. Math. 58, 141–155 (1986)Google Scholar
  25. [Ivi]
    Ivinskis, K.: Normale Flächen und die Miyaoka-Kobayashi-Ungleichung. University of Bonn, Diplomarbeit (1985)Google Scholar
  26. [Lan]
    Langer, A.: Logarithmic orbifold Euler numbers of surfaces with applications. Proc. Lond. Math. Soc. 3(86), 358–396 (2003)MathSciNetzbMATHGoogle Scholar
  27. [Zar3]
    Zariski, O.: Algebraic Surfaces, 2nd edn. Springer, Berlin (1971)Google Scholar
  28. [Var2]
    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. [Var3]
    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. [Hi]
    Hirano, A.: Constructions of plane curves with cusps. Saitama Math. J. 10, 21–24 (1992)Google Scholar
  31. [Koe]
    Koelman, R.J.: Over de cusp. University of Leiden, Diplomarbeit (1986)Google Scholar
  32. [Shu11]
    Shustin, E.: Gluing of singular and critical points. Topology 37(1), 195–217 (1998)MathSciNetzbMATHGoogle Scholar
  33. [DGPS]
    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. [Schu]
    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. [GrM]
    Gradolato, M.A., Mezzetti, E.: Families of curves with ordinary singular points on regular surfaces. Ann. mat. pura et appl. 150, 281–298 (1988)MathSciNetzbMATHGoogle Scholar
  36. [Wal1]
    Wall, C.T.C.: Geometry of quartic curves. Math. Proc. Camb. Philos. Soc. 117, 415–423 (1995)MathSciNetzbMATHGoogle Scholar
  37. [Wal2]
    Wall, C.T.C.: Highly singular quintic curves. Math. Proc. Camb. Philos. Soc. 119, 257–277 (1996)MathSciNetzbMATHGoogle Scholar
  38. [Ku]
    Vik, K.S.: The generalized Chisini conjecture. Proc. Steklov Inst. Math. 241(2), 110–119 (2003)Google Scholar
  39. [CPS]
    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)MathSciNetzbMATHGoogle Scholar
  40. [Ful1]
    Fulton, W.: On the fundamental group of the complement of a node curve. Ann. Math. (2) 111(2), 407–409 (1980)MathSciNetzbMATHGoogle Scholar
  41. [Del]
    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. [HaJ]
    Harris, J.: On the severi problem. Invent. Math. 84, 445–461 (1985)MathSciNetzbMATHGoogle Scholar
  43. [Alb]
    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. [Nob2]
    Nobile, A.: On specialization of curves I. Trans. Am. Math. Soc. 282(2), 739–748 (1984)Google Scholar
  45. [Ran1]
    Ran, Z.: On nodal plane curves. Invent. Math. 86, 529–534 (1986)MathSciNetzbMATHGoogle Scholar
  46. [Tre]
    Treger, R.: On the local Severi problem. Bull. Am. Math. Soc. 19(1), 325–327 (1988)MathSciNetzbMATHGoogle Scholar
  47. [Ran2]
    Ran, Z.: Families of plane curves and their limits: enriques’ conjecture and beyond. Ann. Math. 130(1), 121–157 (1989)MathSciNetzbMATHGoogle Scholar
  48. [Kan1]
    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. [Kan2]
    Kang, P.-L.: A note on the variety of plane curves with nodes and cusps. Proc. Am. Math. Soc. 106(2), 309–312 (1989)MathSciNetzbMATHGoogle Scholar
  50. [AD]
    Akyol, A., Degtyarev, A.: Geography of irreducible plane sextics. Proc. Lond. Math. Soc. (3) 111(6), 1307–1337 (2015)MathSciNetzbMATHGoogle Scholar
  51. [Deg1]
    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. [Deg2]
    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. [HaM]
    Harris, J., Morrison, I.: Moduli of Curves. Graduate Texts in Mathematics, vol. 187. Springer, Berlin (1998)Google Scholar
  54. [Tan3]
    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)MathSciNetzbMATHGoogle Scholar
  55. [Wah1]
    Wahl, J.: Equisingular deformations of plane algebroid curves. Trans. Am. Math. Soc. 193, 143–170 (1974)MathSciNetzbMATHGoogle Scholar
  56. [Lue1]
    Luengo, I.: The \(\mu \)- constant stratum is not smooth. Invent. Math. 90, 139–152 (1987)Google Scholar
  57. [Lue2]
    Luengo, I.: On the existence of complete families of projective plane curves, which are obstructed. J. Lond. Math. Soc. 2(36), 33–43 (1987)MathSciNetzbMATHGoogle Scholar
  58. [Shu6]
    Shustin, E.: Smoothness and irreducibility of varieties of algebraic curves with nodes and cusps. Bull. SMF 122, 235–253 (1994)Google Scholar
  59. [Gia]
    Giacinti-Diebolt, C.: Variétés des courbes projectives planes de degré et lieu singulier donnés. Math. Ann. 266, 321–350 (1984)Google Scholar
  60. [Fla]
    Flamini, F.: Some results of regularity for severi varieties of projective surfaces. Commun. Algebra 29(6), 2297–2311 (2001)MathSciNetzbMATHGoogle Scholar
  61. [ChS]
    Chiantini, L., Sernesi, E.: Nodal curves on surfaces of general type. Math. Ann. 307, 41–56 (1997)MathSciNetzbMATHGoogle Scholar
  62. [Kei2]
    Keilen, T.: Smoothness of equisingular families of curves. Trans. Am. Math. Soc. 357(6), 2467–2481 (2005)Google Scholar
  63. [HaR2]
    Hartshorne, R.: Algebraic Geometry. Graduate Text in Mathematics, vol. 52. Springer, Berlin (1977)zbMATHGoogle Scholar
  64. [DPW2]
    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. [GLS6]
    Greuel, G.-M., Lossen, C., Shustin, E.: Introduction to Singularities and Deformations. Springer, Berlin (2007)Google Scholar
  66. [Var1]
    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. [Chm]
    Chmutov, S.V.: Examples of projective surfaces with many singularities. J. Algebraic Geom. 1(2), 191–196 (1992)Google Scholar
  68. [Shu2]
    Shustin, E.: New M-curve of the 8th degree. Math. Notes Acad. Sci. USSR 42, 606–610 (1987)Google Scholar
  69. [Shu12]
    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. [DPW1]
    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)Google Scholar
  71. [OrS]
    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)MathSciNetzbMATHGoogle Scholar
  72. [KoS]
    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. [Ch2]
    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. [Pec1]
    Pecker, D.: Note sur la réalité des points doubles des courbes gauches. C. R. Acad. Sci. Paris, Sér. I 324, 807–812 (1997)MathSciNetzbMATHGoogle Scholar
  75. [Pec2]
    Pecker, D.: Sur la réalité des points doubles des courbes gauches. Ann. Inst. Fourier (Grenoble) 49, 1439–1452 (1999)MathSciNetzbMATHGoogle Scholar
  76. [Gud4]
    Gudkov, D.A.: On the curve of 5th order with \(5\) cusps. Funct. Anal. Appl. 16, 201–202 (1982)Google Scholar
  77. [ItS2]
    Itenberg, I., Shustin, E.: Real algebraic curves with real cusps. Am. Math. Soc. Transl. 2(173), 97–109 (1996)Google Scholar
  78. [Miy]
    Miyaoka, Y.: On the Chern numbers of surfaces of general type. Invent. Math. 42, 225–237 (1977)MathSciNetzbMATHGoogle Scholar
  79. [Yau]
    Yau, S.-T.: Calabi’s conjecture and some new results in algebraic geometry. Proc. Natl. Acad. Sci. USA 74, 1798–1799 (1977)MathSciNetzbMATHGoogle Scholar
  80. [GuN2]
    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. [GuN1]
    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. [Ore]
    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. [KeT]
    Keilen, T., Tyomkin, I.: Existence of curves with prescribed topological singularities. Trans. Am. Math. Soc. 354(5), 1837–1860 (2002)Google Scholar
  84. [Che]
    Chen, X.: Rational curves on K3 surfaces. J. Algebraic Geom. 8, 245–278 (1999). Thesis, Harvard University (1997)Google Scholar
  85. [ChC]
    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. [GoP]
    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. [AB]
    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. [Wel]
    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. [Mik1]
    Mikhalkin, G.: Counting curves via the lattice paths in polygons. Comptes Rendus Math. 336(8), 629–634 (2003)Google Scholar
  90. [Mik2]
    Mikhalkin, G.: Enumerative tropical algebraic geometry in \({\mathbb{R}}^2\). J. Am. Math. Soc. 18, 313–377 (2005)Google Scholar
  91. [YauZ]
    Yau, S.-T., Zaslow, E.: BPS states, string duality, and nodal curves on K3. Nucl. Phys. B 471, 503–512 (1996)MathSciNetzbMATHGoogle Scholar
  92. [KMPS]
    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)MathSciNetzbMATHGoogle Scholar
  93. [ChL]
    Chen, X., Lewis, J.D.: Density of rational curves on K3 surfaces. Math. Ann. 356(1), 331–354 (2013)Google Scholar
  94. [ArC]
    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)MathSciNetzbMATHGoogle Scholar
  95. [BrG]
    Bruce, J.W., Gibblin, P.J.: A stratification of the space of plane quartic curves. Proc. Lond. Math. Soc. 442, 270–298 (1981)MathSciNetzbMATHGoogle Scholar
  96. [DeM]
    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. [Shu7]
    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. [Shu8]
    Shustin, E.: Geometry of equisingular families of plane algebraic curves. J. Algebraic Geom. 5, 209–234 (1996)Google Scholar
  99. [Nor]
    Nori, M.: Zariski conjecture and related problems. Ann. Sci. Ec. Norm. Sup. 4(16), 305–344 (1983)Google Scholar
  100. [Kei1]
    Keilen, T.: Irreducibility of equisingular families of curves. Trans. Am. Math. Soc. 355(9), 3485–3512 (2003)MathSciNetzbMATHGoogle Scholar
  101. [LaR]
    Lazarsfeld, R.: Lectures on linear series. In: Kollár, J. (ed.) Complex Algebraic Geometry. AMS, Providence (1997)Google Scholar
  102. [Wah2]
    Wahl, J.: Deformations of plane curves with nodes and cusps. Am. J. Math. 96, 529–577 (1974)MathSciNetzbMATHGoogle Scholar
  103. [CGL1]
    Campillo, A., Greuel, G.-M., Lossen, C.: Equisingular deformations of plane curves in arbitrary characteristic. Compos. Math. 143, 829–882 (2007)MathSciNetzbMATHGoogle Scholar
  104. [CGL2]
    Campillo, A., Greuel, G.-M., Lossen, C.: Equisingular calculations for plane curve singularities. J. Symb. Comput. 42(1–2), 89–114 (2007)MathSciNetzbMATHGoogle Scholar
  105. [BoGM]
    Boubakri, Y., Greuel, G.-M., Markwig, T.: Invariants of hypersurface singularities in positive characteristic. Rev. Mat. Complut. 25, 61–85 (2012)MathSciNetzbMATHGoogle Scholar
  106. [GrPh]
    Greuel, G.-M., Pham, T.H.: On finite determinacy for matrices of power series. Math. Z. (2018).  https://doi.org/10.1007/s00209-018-2040-2MathSciNetzbMATHGoogle Scholar
  107. [Zar2]
    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. [Nob1]
    Nobile, A.: Families of curves on surfaces. Math. Zeitschrift 187(4), 453–470 (1984)Google Scholar
  109. [Tyo1]
    Tyomkin, I.: On severi varieties on Hirzebruch surfaces. Int. Math. Res. Not. 23, Art. ID rnm109, 31 pp (2007)Google Scholar
  110. [Tyo3]
    Tyomkin, I.: On Zariski’s theorem in positive characteristic. J. Eur. Math. Soc. (JEMS) 15(5), 1783–1803 (2013)MathSciNetzbMATHGoogle Scholar
  111. [Tyo4]
    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. [Wes]
    Westenberger, E.: Existence of hypersurfaces with prescribed simple singularities. Commun. Algebra 31(1), 335–356 (2003)MathSciNetzbMATHGoogle Scholar
  113. [AH2]
    Alexander, J., Hirschowitz, A.: An asymptotic vanishing theorem for generic unions of multiple points. Invent. Math. 140(2), 303–325 (2000)MathSciNetzbMATHGoogle Scholar
  114. [Roe1]
    Roé, J.: Maximal rank for schemes of small multiplicity by Évain’s differential Horace method. Trans. Am. Math. Soc. 366, 857–874 (2014)zbMATHGoogle Scholar
  115. [BGM]
    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)MathSciNetzbMATHGoogle Scholar
  116. [Nag1]
    Nagata, M.: On the 14-th problem of Hilbert. Am. J. Math. 81, 766–772 (1959)MathSciNetzbMATHGoogle Scholar
  117. [Nag2]
    Nagata, M.: On rational surfaces II. Mem. Coll. Sci. Univ. Kyoto Ser. A 33, 271–293 (1960)Google Scholar
  118. [DHKRS]
    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. [CHMR]
    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. [FuP]
    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. [Get]
    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. [KoM]
    Kontsevich, M., Manin, Yu.: Gromov-Witten classes, quantum cohomology and enumerative geometry. Commun. Math. Phys. 164, 525–562 (1994)MathSciNetzbMATHGoogle Scholar
  123. [RuT]
    Ruan, Y., Tian, G.: A mathematical theory of quantum cohomology. J. Differ. Geom. 42(2), 259–367 (1995)MathSciNetzbMATHGoogle Scholar
  124. [Vak]
    Vakil, R.: Counting curves on rational surfaces. Manuscr. math. 102(1), 53–84 (2000)MathSciNetzbMATHGoogle Scholar
  125. [Pan]
    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. [Zin]
    Zinger, A.: Counting rational curves of arbitrary shape in projective spaces. Geom. Topol. 9, 571–697 (2005)MathSciNetzbMATHGoogle Scholar
  127. [Goet]
    Göttsche, L.: A conjectural generating function for numbers of curves on surfaces. Commun. Math. Phys. 196, 523–533 (1998)MathSciNetzbMATHGoogle Scholar
  128. [GoS1]
    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. [GoS2]
    Göttsche, L., Shende, V.: Refined curve counting on complex surfaces. Geom. Topol. 18, 2245–2307 (2014)MathSciNetzbMATHGoogle Scholar
  130. [FeR]
    Fehér, L., Rimányi, R.: On the structure of Thom polynomials of singularities. Bull. Lond. Math. Soc. 39, 541–549 (2007)MathSciNetzbMATHGoogle Scholar
  131. [Kaz]
    Kazarian, M.E.: Multisingularities, cobordisms, and enumerative geometry. Russ. Math. Surv. 58(4), 665–724 (2003)MathSciNetzbMATHGoogle Scholar
  132. [Liu]
    Liu, A.K.: Family blowup formula, admissible graphs and the enumeration of singular curves I. J. Diff. Geom. 56(3), 381–579 (2000)MathSciNetzbMATHGoogle Scholar
  133. [BM1]
    Basu, S., Mukherjee, R.: Enumeration of curves with two singular points. Bull. Sci. Math. 139(6), 667–735 (2015)MathSciNetzbMATHGoogle Scholar
  134. [BM2]
    Basu, S., Mukherjee, R.: Enumeration of curves with one singular point. J. Geom. Phys. 104, 175–203 (2016)MathSciNetzbMATHGoogle Scholar
  135. [Ker]
    Kerner, D.: Enumeration of singular algebraic curves. Isr. Math. J. 155, 1–56 (2006)MathSciNetzbMATHGoogle Scholar
  136. [KlP]
    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. [Vai]
    Vainsencher, I.: Counting divisors with prescribed singularities. Trans. Am. Math. Soc. 267(2), 399–422 (1981)MathSciNetzbMATHGoogle Scholar
  138. [CaH2]
    Caporaso, L., Harris, J.: Counting plane curves of any genus. Invent. Math. 131(2), 345–392 (1998)MathSciNetzbMATHGoogle Scholar
  139. [Ran4]
    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. [ShSh]
    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/S0129167X13500547MathSciNetzbMATHGoogle Scholar
  141. [Br]
    Brugallé, E.: Floor diagrams relative to a conic, and GW-W invariants of del pezzo surfaces. Adv. Math. 279, 438–500 (2015)MathSciNetzbMATHGoogle Scholar
  142. [Mik3]
    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. [NiS]
    Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(1), 1–51 (2006)MathSciNetzbMATHGoogle Scholar
  144. [GaS]
    Ganor, Y., Shustin, E.: Enumeration of plane unicuspidal curves of any genus via tropical geometry (2018). arXiv:1807.11443
  145. [KuS]
    Kulikov, V.S., Shustin, E.: On rigid plane curves. Eur. J. Math. 2(1), 208–226 (2016)MathSciNetzbMATHGoogle Scholar
  146. [GrS]
    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. [Bru]
    Bruce, J.W.: Singularities on a projective hypersurface. Bull. Lond. Math. Soc. 13, 47–50 (1981)Google Scholar
  148. [GuU]
    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. [Shu3]
    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. [Bar]
    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. [GG]
    Gourevich, A., Gourevich, D.: Geometry of obstructed equisingular families of projective hypersurfaces. J. Pure Appl. Algebra 213(9), 1865–1889 (2009)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Gert-Martin Greuel
    • 1
    Email author
  • Christoph Lossen
    • 1
  • Eugenii Shustin
    • 2
  1. 1.Fachbereich MathematikTU KaiserslauternKaiserslauternGermany
  2. 2.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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