A class of discriminant varieties in the conformal 3-sphere
Let \(P(x, z)\) be a polynomial in \(z\in\mathbb{P}^1_\mathbb{C}\) whose coefficients are smooth complex-valued functions of \(x\in M^m\), where \(M^m\) is a smooth manifold of dimension \(m\). Then the discriminant variety associated to \(P(x, z)\) is defined as the set \(K\) of points \(x\in M^m\) for which the algebraic equation \(P(x,z) = 0\) has a multiple root in \(\mathbb{P}^1_\mathbb{C}\). In order to determine the discriminant variety \(K\), one is led to solve the resultant equation \(R_{P,P'} = 0\) for \(P\) and its derivative \(P' := \frac{\partial P}{\partial z}\) . In general, this is a difficult task, and one of the objectives of the paper under review is to develop other practical ways to compute \(K\). More precisely, the authors consider the particular case of \(M^3 = S^3\) under the following assumptions: 1. \(S^3\) is endowed with some smooth metric conformal to the standard Euclidean metric; 2. any smooth local solution to \(P(x,z) = 0\) is semi-conformal (in a precisely defined sense); 3. the regular fibres of such a solution are arcs of circles. Under these assumptions reflecting the conformal geometry of the 3-sphere \(S^3\), it is shown how it is possible to determine the associated discriminant variety \(K\) in a purely geometric way. The general study of this and other related examples is based upon a fine analysis of the space and circles in \(S^3\), the concept of semi-conformality, and the link to the geometry of complex tori. Moreover, the authors exhibit further classes of examples, namely their so-called \((p,q)\)-examples, for which the associated discriminant variety \(K\) can be explicitly computed. These results are of special interest in the theory of 3-manifolds and their coverings, where related constructions play an important role.
scientific article; zbMATH DE number 1784602