Ideals associated to deformations of singular plane curves

Diaz, Steven; Harris, Joe

doi:10.1090/S0002-9947-1988-0961600-2

Ideals associated to deformations of singular plane curves
HTML articles powered by AMS MathViewer

by Steven Diaz and Joe Harris

Trans. Amer. Math. Soc. 309 (1988), 433-468

DOI: https://doi.org/10.1090/S0002-9947-1988-0961600-2

PDF | Request permission

Abstract:

We consider in this paper the geometry of certain loci in deformation spaces of plane curve singularities. These loci are the equisingular locus $ES$ which parametrizes equisingular or topologically trivial deformations, the equigeneric locus $EG$ which parametrizes deformations of constant geometric genus, and the equiclassical locus $EC$ which parametrizes deformations of constant geometric genus and class. (The class of a reduced plane curve is the degree of its dual.) It was previously known that the tangent space to $ES$ corresponds to an ideal called the equisingular ideal and that the support of the tangent cone to $EG$ corresponds to the conductor ideal. We show that the support of the tangent cone to $EC$ corresponds to an ideal which we call the equiclassical ideal. By studying these ideals we are able to obtain information about the geometry and dimensions of $ES$, $EC$, and $EG$. This allows us to prove some theorems about the dimensions of families of plane curves with certain specified singularities.

References

Norbert A’Campo, Le groupe de monodromie du déploiement des singularités isolées de courbes planes. I, Math. Ann. 213 (1975), 1–32 (French). MR 377108, DOI 10.1007/BF01883883
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. (4) 16 (1983), no. 3, 467–488 (1984). MR 740079
E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932, DOI 10.1007/978-1-4757-5323-3
M. Artin, Versal deformations and algebraic stacks, Invent. Math. 27 (1974), 165–189. MR 399094, DOI 10.1007/BF01390174

Deformations of singularities

Edward D. Davis, Anthony V. Geramita, and Paolo Maroscia, Perfect homogeneous ideals: Dubreil’s theorems revisited, Bull. Sci. Math. (2) 108 (1984), no. 2, 143–185 (English, with French summary). MR 769926
Joe Harris, On the Severi problem, Invent. Math. 84 (1986), no. 3, 445–461. MR 837522, DOI 10.1007/BF01388741
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Heisuke Hironaka, On the arithmetic genera and the effective genera of algebraic curves, Mem. Coll. Sci. Univ. Kyoto Ser. A. Math. 30 (1957), 177–195. MR 90850, DOI 10.1215/kjm/1250777055
Arnold Kas and Michael Schlessinger, On the versal deformation of a complex space with an isolated singularity, Math. Ann. 196 (1972), 23–29. MR 294701, DOI 10.1007/BF01419428
Ragni Piene, Polar classes of singular varieties, Ann. Sci. École Norm. Sup. (4) 11 (1978), no. 2, 247–276. MR 510551
Michael Schlessinger, Functors of Artin rings, Trans. Amer. Math. Soc. 130 (1968), 208–222. MR 217093, DOI 10.1090/S0002-9947-1968-0217093-3

Infinitesimal deformations of singularities

Allen Tannenbaum, Families of algebraic curves with nodes, Compositio Math. 41 (1980), no. 1, 107–126. MR 578053
Michel Demazure, Henry Charles Pinkham, and Bernard Teissier (eds.), Séminaire sur les Singularités des Surfaces, Lecture Notes in Mathematics, vol. 777, Springer, Berlin, 1980 (French). Held at the Centre de Mathématiques de l’École Polytechnique, Palaiseau, 1976–1977. MR 579026
Bernard Teissier, The hunting of invariants in the geometry of discriminants, Real and complex singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976) Sijthoff and Noordhoff, Alphen aan den Rijn, 1977, pp. 565–678. MR 0568901
Jonathan M. Wahl, Equisingular deformations of plane algebroid curves, Trans. Amer. Math. Soc. 193 (1974), 143–170. MR 419439, DOI 10.1090/S0002-9947-1974-0419439-2
Oscar Zariski, Dimension-theoretic characterization of maximal irreducible algebraic systems of plane nodal curves of a given order $n$ and with a given number $d$ of nodes, Amer. J. Math. 104 (1982), no. 1, 209–226. MR 648487, DOI 10.2307/2374074
Oscar Zariski, Studies in equisingularity. I. Equivalent singularities of plane algebroid curves, Amer. J. Math. 87 (1965), 507–536. MR 177985, DOI 10.2307/2373019