Equisingularity theory for plane curves with embedded points
In this article the author studies different notions of equisingularity for plane curves which are not necessarily reduced, i.e. for generically reduced curves with embedded points. This theory is related with both the theory of equisingularity for plane reduced curves and that for ideals with finite support. In the first part there is a review of known results on the theory for reduced curves. We can associate to an algebroid plane reduced curve \(C\), with branches \(\gamma=(\gamma_1,\ldots,\gamma_s)\) a weighted tree \(T_1(C,\gamma)\) whose vertices are the infinitely near points of \(C\), and each vertex has the weight \((m_1,\ldots,m_r)\) of the multiplicities of the strict transforms of the irreducible components at this point. Then two curves are equisingular if they have the same tree. In the same way we can associate a finite tree \(\tau({\mathcal I})\) to each \({\mathcal M}\)-primary ideal \({\mathcal I}\) of \(kx,y\), each vertex corresponding to a proper transform of \({\mathcal I}\) and its weight being the multiplicity of the corresponding ideal. In the same way we say that two \({\mathcal M}\)-primary ideals are equisingular if they have the same tree. We note that \(\tau({\mathcal I})=\tau(\overline{\mathcal I})\), where \(\overline{\mathcal I}\) is the integral closure of \({\mathcal I}\). An algebroid plane curve \(C\) with embedded point is defined by an ideal \({\mathcal I}\) of \(R=kx,y\), with \({\mathcal I}=(f).{\mathcal N}\), where \(f\in R\) and either the radical \(r({\mathcal N})\) is the maximal ideal \({\mathcal M}\) or \({\mathcal N}=R\). Let \(\pi_1\colon X_1\to{\mathcal A}^2=\text{Spec}R\) be the blowing-up of center \({\mathcal M}\), then the intersection \(E\cap C_1\) of the exceptional divisor \(E\) with the strict transform \(C_1\) of the curve \(C\) is a finite set of points \(\{ P_1,\ldots,P_r\}\), which are by definition the points in the first neighborhood of \(C\). At each point \(P_i\), we have either a curve \(C_i\) with the same conditions as the curve \(C\) in \({\mathcal A}^2\), or we are in the situation of an \({\mathcal M}\)-primary ideal. In any case we can continue the process and we get the infinitely near points of the curve \(C\). Then we can associate to the generically reduced curve \(C\) defined by an ideal \({\mathcal I}\), a bi-weighted tree \(T_2(C,\gamma)=T_2({\mathcal I},\gamma)\), and give the definition of equisingularity: two algebroid, plane, generically reduced curves \(C\) and \(D\) are equisingular if there is an isomorphism of bi-weighted trees \(T_2(C,\gamma)\simeq T_2(D,\delta)\), for suitable orderings \(\gamma\), \(\delta\) of the branches of \(C\) and \(D\). The author wants to study one-dimensional families of generically reduced plane curves: \(X\subset Z@>\pi>>T\), where \(X\) is a surface in a smooth threefold \(Z\), and \(p=\pi_{| X}\) is flat over the regular curve \(T\). In the same way he considers one-dimensional families of ideals \(({\mathcal I};\pi)\), with \({\mathcal I}\subset {\mathcal O}_Z\) and \(Z@>\pi>>T\) smooth. Let \([{\mathcal I}]\) be \(\{ z\in Z:{\mathcal I}_z\text{ is not smooth}\}\), and define by induction \(Z_n\rightarrow Z_{n-1}\) as the blow-up of \(S_n=[{\mathcal I}{\mathcal O}_{Z_{n-1}}]\); then we say that the family of ideals is equisingular if for any \(n \geq 0\) the map \(S_n\to T\) is smooth. Even if a family \(({\mathcal I};\pi)\) is equisingular, it does not follow that the restriction of \(\pi\) at \(Y\subset Z\) defined by \({\mathcal I}\) is flat. To avoid such pathologies it is necessary to consider families of complete ideals. The author studies first a one-dimensional family \(({\mathcal I};\pi)\) which is equisingular and such that the restriction \(\pi_{| Y}\colon Y\to T\) is finite. Then he gets the following result: Theorem: Assume that \({\mathcal I}=\overline{\mathcal I}\), i.e. the ideal \({\mathcal I}\) is complete, then \({\mathcal I}(t) = \overline {{\mathcal I}(t)}\) for all \(t\in T\). To get this result we notice that the trees \(\tau ( {\mathcal I}(t))\) are constant in an equisingular family and that the multiplicity \(e ({\mathcal I}(t))\) of the ideal can be expressed in terms of the weights in \(\tau\), and we use properties of equimultiple families of ideals and the structure of complete ideals in a regular ring of dimension two. From this theorem the author deduces that an equisingular family of complete ideals \(({\mathcal I};\pi)\) is properly flat, i.e. all maps \(Y_n \rightarrow T\) are flat where \(Y_n \subset Z_n\) are defined by the strict transform of \({\mathcal I}\) in \(Z_n\), where \(Z_n\) is obtained by the ``resolution process'' defined previously. In the last part the author introduces three notions of equisingularity for one-dimensional families of generically reduced curves \(X \subset Z@>\pi>>T\), such that the ``singular locus'' \( [{\mathcal I} (X) ]\) is finite over \(T\). The family is ``I-equisingular'' if the family of ideals \(({\mathcal I}(X);\pi)\) is equisingular; the family is ``T-equisingular'' if for any pair of points \(t,t'\in T\) the fibers \(X_t\) and \(X_{t'}\) are equisingular, i.e. they have the same tree \(T_2\); the family is ``\(C\)-equisingular'' if it is ``\(I\)-equisingular'' and properly flat, i.e. the maps \(X_n \rightarrow T\) are flat where \(X_n\) is the proper transform of \(X\) in \(Z_n\). Then we have the following: Theorem: The notions of ``\(I\)-equisingularity'' and of ``\(T\)-equisingularity'' are equivalent; moreover if the ideal \({\mathcal I}\) of \(X\) in \(Z\) is complete they are also equivalent to ``\(C\)-equisingularity''.
scientific article; zbMATH DE number 910242