Perturbing plane curve singularities (Q1884029)

Let \(K\) be a field, let \(R\) be a two-dimensional regular local subring of \(K\) having \(K\) as field of quotients, let \(\mathfrak m\) be the maximal ideal of \(R\), and let \(\Omega(R)\) be the set of all two-dimensional regular local subrings of \(K\) containing \(R\); these rings shall be called points. The local rings of closed points of \(\text{Bl}_{\mathfrak m}(R)\), the blow-up of \(\text{Spec}(R)\) at its closed point, lie in \(\Omega(R)\), and are called the quadratic transforms of \(R\) or the points in the first neighborhood of \(R\). For any \(S\), \(T \in\Omega(R)\) with \(S\subset T\), the ring \(T\) dominates \(S\), there exists a uniquely determined sequence \(S=:S_0\subsetneqq S_1\subsetneqq \cdots\subsetneqq S_n=:T\) such that \(S_i\) is a quadratic transform of \(S_{i-1}\) which is called the quadratic sequence between \(S\) and \(T\), and the residue field of \(T\) is a finite extension of the residue field of \(S\); its degree shall be denoted by \([T:S]\). For \(S\), \(T\in \Omega(R)\) with \(S\subset T\) one says that \(T\) is proximate to \(S\), denoted by \(S\prec T\), if the discrete valuation ring defined by the order function of the maximal ideal of \(S\) contains \(T\). For each \(f\in R\) we have the (strict) transform \((fR)^S\) of the ideal \(fR\) in \(S\) (for these results which do not use the authors' language, one should confer papers of \textit{J. Lipman} [in: Algebraic geometry and commutative algebra, Vol.\ I, 203--231 (1988; Zbl 0693.13011); in: Commutative algebra: syzygies, multiplicities, and birational algebra, Contemp. Math. 159, 293--306 (1994; Zbl 0814.13016)], or Chapter VII of the book ``Resolution of Curve and Surface Singularities in characteristic zero'' by the reviewer and \textit{J. L. Vicente} [Kluwer Academic Publishers, Dordrecht (2004; Zbl 1069.14001)]; for the language used by the authors one should confer to the first author's book [``Singularities of plane curves'' (Cambridge University Press, Cambridge) (2000; Zbl 0967.14018)]). A cluster \({\mathcal K}\) with origin \(R\) is a finite subset of \(\Omega(R)\) with \(R\in {\mathcal K}\) and the following additional property: If \(S\in{\mathcal K}\), then all the rings of the quadratic sequence between \(S\) and \(R\) belong to \({\mathcal K}\), also. A pair \(({\mathcal K},\nu)\) where \({\mathcal K}\) is a cluster and \(\nu\colon {\mathcal K}\to \mathbb Z\) is a map, is called a weighted cluster. A weighted cluster is called consistent if one has \(\nu_S-\sum_{T\in{\mathcal K},S\prec T}[T:S]\nu_T\geq0\) for every \(S\in{\mathcal K}\). To every weighted cluster \(({\mathcal K},\nu)\) one can associate a complete ideal \(H_{\mathcal K}\) of \(R\) of finite colength. Also in this more general context one can define for weighted clusters a process called unloading [cf.\ the first author's book, chapter 4, section 4.6] which associates to a weighted cluster \(({\mathcal K},\nu)\) a new weighted cluster \(({\mathcal K'},\nu')\) in such a way that \(H_{\mathcal K}=H_{\mathcal K'}\), and that after a finite number of steps one gets a consistent cluster. The ideal associated to a consistent cluster is the unique complete finite colength ideal \(\mathfrak a\) of \(R\) with \(\text{ord}_S(\mathfrak a^S)\geq \nu_S\) for all \(S\in {\mathcal K}\). An element \(f\in R\) is said to go through \({\mathcal K}\) if \(f\in H_{\mathcal K}\), and \(f\) is said to go sharply through \({\mathcal K}\) if \(\text{ord}_S((fR)^S)=\nu_S\) for all \(S\in {\mathcal K}\) and \(f\) has no singular points outside \({\mathcal K}\) (a point \(S\in\Omega(R)\) is called a non-singular point of \(f\) if \(\text{ord}_S((fR)^S)=1\) and no \(T\in\Omega(R)\) with \(T\supset S\) and \((fR)^T\neq T\) is a satellite of \(S\), i.e. the quadratic sequence from \(S\) to \(T\) contains only one point which is proximate to \(T\)). The authors of the paper under review start with a ring \(R\) of formal power series in two indeterminates over the complex numbers; all what has been said above remains true if one takes the adic completion of all the rings involved. They consider two non-zero elements \(f\), \(g\in R\) which are not units, which satisfy \(\text{ord}_R(f)\leq \text{ord}_R(g)=:n\) and have no common tangent (equivalently, for any quadratic transform \(S\) of \(R\) not both ideals \((fR)^S\), \((gR)^S\) are proper ideals of \(S\)) and they study the family \(h^\lambda:=f+\lambda g\), \(\lambda\in \mathbb C\). Their main result is theorem \(4.6\) which states: There exists a weighted cluster \({\mathcal T}\) and a finite subset \(M\) of the complex numbers such that, for all \(\lambda\in \mathbb C\setminus M\), , the element \(h^\lambda\) goes sharply through \({\mathcal T}\), and no two such elements have a common point outside \({\mathcal T}\). This result determines -- in the case of convergent power series -- the topological type of \(h^\lambda\) in terms of \(n\) and the singularity of \(f\). For other results in this direction, cf.\ [\textit{T. C. Kuo} and \textit{Y. C. Lu}, Topology 16, 299--310 (1977; Zbl 0378.32001); \textit{H. Maugendre}, C. R. Acad. Sci., Paris, Sér. I 322, No.10, 945--948 (1996; Zbl 0922.32022) and \textit{B. Teissier}, in: Seminaire sur les singularités, 193--221. Publ.\ Math.\ Univ.\ Paris VII 7, Paris (1980; Zbl 0624.32001)].