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Hamburger-Noether matrices over rings. (English) Zbl 0744.14020

The author introduces the Hamburger Noether (HN for short) matrices over rings as a way to study equisingular deformations of curve singularities. In two previous papers A. Campillo and the author in [Algebraic geometry, Proc. Int. Conf., La Rabida 1981, Lect. Notes Math. 961, 22–31 (1982; Zbl 0497.14012)] and the author in [J. Pure Appl. Algebra 43, 119–127 (1986; Zbl 0612.14008)] defined the HN matrices (over the definition field) as a useful tool in order to describe the sequence of infinitely near points related with a resolution procedure of a curve singularity (not necessarily plane). On the other hand A. Campillo [Trans. Am. Math. Soc. 279, 377–388 (1983; Zbl 0559.14020)] extended the HN expansions to the case in which the coefficients are taken over rings and study the applications to the equisingular deformation theory of plane curve singularities.
The relationship between the HN matrices and the HN expansions permits to give a parametrization \(\Phi:A[[X_1,\ldots,X_N]]\to A[[t]]\) with coefficients over the base local ring \(A\). When \(A=k[[V_1,\ldots,V_r]]\), the quotient ring \(R=A[[X_1,\ldots,X_ N]]/(\operatorname{ker} \Phi)\) can be seen as a deformation (HN deformation) of the algebroid curve \(R_0\) induced over the residual field \(k\) of \(A\) with the section \(s\) given by the maximal ideal of \(A\). In characteristic zero the generic fiber is well defined, but in general, the deformation is not flat and \(R_ 0\) is not reduced. The author proves that this deformation is an equisingular deformation along \(s\) in the sense given by O. Zariski [Am. J. Math. 87, 972–1006 (1965; Zbl 0146.42502)] (essentially, the generic fibre and \((R_0)_{red}\) have the same multiplicity sequence) and also in the sense introduced by J. Becker and J. Stutz [Rice. Univ. Stud. 59, No. 2, Proc. Conf. Complex Analysis 1972, part I, 1–9 (1973; Zbl 0286.32009)]. In some restricted cases all the equisingular deformations in the sense of Stutz and Becker can be reached by HN deformations. When one considers the special case of Arf matrices - – his definition is based on the description of HN matrices for the Arf curves introduced by J. Lipman [Am. J. Math. 93, 649–685 (1973; Zbl 0228.13008)] — , the HN deformation becomes flat, with reduced special fibre and has a good support by monoidal transformations. Taking into account that the Arf curve associated to a given space curve reproduces its sequence of multiplicities, the HN deformation of Arf matrices seems to provide a way to construct equisingular flat deformations for reduced space curves.

MSC:

14H20 Singularities of curves, local rings
14E05 Rational and birational maps
14D15 Formal methods and deformations in algebraic geometry
14B10 Infinitesimal methods in algebraic geometry
32S15 Equisingularity (topological and analytic)
14B12 Local deformation theory, Artin approximation, etc.
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[1] Abhyankar, S. S., Note on coefficient field, Amer. J. Math., 90, 347-355 (1968)
[2] Campillo, A., Algebroid curves in positive characteristic, (Lectures Notes in Mathematics, 813 (1980), Springer: Springer Berlin) · Zbl 0451.14010
[3] Campillo, A., Hamburger-Noether expansions over Ring, Trans. Amer. Math. Soc., 279, 1 (1983) · Zbl 0559.14020
[4] Campillo, A.; Castellanos, J., On projections of space curves, (Algebraic Geometry. Algebraic Geometry, Lecture Notes in Mathematics, 961 (1982), Springer: Springer Berlin), 22-31 · Zbl 0497.14012
[5] Castellanos, J., Equisingularidad de curvas algebroides alabeadas, (Tesis doctoral (1982), Publicaciones de la U. Complutense), No. 150/82
[6] Castellanos, J., A relation between the sequence of multiplicities and the semigroups of values of an algebroid curve, J. Pure Appl. Algebra, 43, 119-127 (1986) · Zbl 0612.14008
[7] Lipman, J., Stable ideals and Arf rings, Amer. J. Math., 93, 649-685 (1973) · Zbl 0228.13008
[8] Matsumura, H., Commutative Algebra (1970), Benjamin: Benjamin New York · Zbl 0211.06501
[9] Stuz, J.; Becker, J., Resolving singularities via local transformations, Rice. Univ. Studies, 59, 1-9 (1973)
[10] Teissier, B., The hunting of invariants in the geometry of discriminants, (Sim. in Math. (1976), Nordic Summer School: Nordic Summer School Oslo) · Zbl 0388.32010
[11] Zariski, O., Studies in equisingularity II, Amer. J. Math., 87, 972-1006 (1865) · Zbl 0146.42502
[12] Zariski, O., (Contribution to the problem of equisingularity (1970), C.I.M.E. Varema Edizzione Cremonese: C.I.M.E. Varema Edizzione Cremonese Roma), 261-343 · Zbl 0204.54503
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