Order \(1\) congruences of linear with smooth fundamental scheme (Q2809912)

Consider a projective space \(\mathbb P^N=\mathbb P(V)\) over \(\mathbb C\) and a congruence \(S\) of lines, i.e. a variety of dimension \(N-1\) embedded in the Grassmannian of lines \(\mathrm{Gr}(1,N)=\mathrm{Gr}(2,V)\). The \textit{order} of \(S\) is the number of lines of the congruence passing through a general point of \(\mathbb P^N\). The \textit{fundamental locus} \(X(S)\) of \(S\) is the set of points belonging to infinitely many lines of the congruence. \(X(S)\) is endowed with a natural scheme structure.NEWLINENEWLINEThe author considers irreducible Cohen-Macaulay congruences \(S\) of order \(1\), with smooth fundamental locus \(X(S)\). He proves a classification for such congruences, based on the \textit{secant index} \(k\), defined as the degree of intersection of \(X(S)\) with the lines of the congruence.