Symmetry defect of algebraic varieties (Q461943)

Let \(k\) be a field, and \(Z, W \subset k^m\) affine algebraic varieties. For a point \(P\in k^m\), the number \(\mu(P)\) is defined as the cardinality of pairs \(\{z,w\}\) where \(z\in Z,\, w\in W\), and \(P=\frac{z+w}2.\) The number \(\mu_{Z,W}:=\sup_{P:\,\mu(P)<\infty}\{\mu(P)\}\) is called the \textit{generic symmetry of} \(Z\) and \(W\). This construction appears when studying affine invariant symmetries of varieties.NEWLINENEWLINEThe main result of the paper under review is the following: NEWLINENEWLINE\noindent \textit{If \(k={\mathbb C},\) and \(Z, W\) are smooth algebraic manifolds in general position of dimensions \(r\) and \(s\) respectively, with \(r+s=m,\) and degrees \(p\) and \(q\) respectively, then \(0<\mu_{Z,W}\leq pq,\) and there exists an algebraic hypersurface \(B(Z,W)\subset{\mathbb C}^m\) of degree bounded by \(pq(1+sp+rq-m)\) whose complement is the set of all the points in \({\mathbb C}^m\) being the midpoint of exactly \(\mu_{Z,W}\) pairs \(\{z,w\},\,z\in Z,\,w\in W.\)}NEWLINENEWLINEThe set \(B(Z,W)\) of the claim is called the \textit{symmetry defect} of \(Z\) and \(W\). It is constructed as the \textit{bifurcation set} of the map \(\Phi:Z\times W\to{\mathbb C}^m\) given by \((z,w)\mapsto \frac{z+w}2,\) which is the set of those points where the cardinality of their fibre differs from the geometric degree of the map. The authors study properties of bifurcation sets, and show that in general they are either empty or a hypersurface of bounded degree. These properties are used later to prove the main result of the paper.NEWLINENEWLINEIf \(k={\mathbb R},\) an alternative definition of bifurcation set is given which allows them to show that it is closed and semialgebraic, and to bound the number of connected components of the complement of \(B(Z,W)\) for real varieties.NEWLINENEWLINEThe paper then focuses on the case of curves in the plane where more details are given to the general results already obtained, and also the topological stability of the symmetry defect set is studied. It concludes with an algorithmic section where it is shown how to compute equations for symmetry defect by using Gröbner bases.