The join of algebraic curves (Q1856415)

Let \(G:=G(1, \mathbb{P}^n)\) be the Grassmannian of all projective lines in \(\mathbb{P}^n\) and, for \(P \neq Q\) points in \(\mathbb{P}^n\), \([\overline{PQ}]\) be the point of \(G\) corresponding to the unique line through \(P, \;Q\). Fixing \(X,Y \subset \mathbb{P}^n\) subvarieties, we have two subsets of \(G\): \({\mathcal J}^0={\mathcal J}^0(X,Y):= \{ [\overline{PQ}] \in G \mid P \in X\), \(Q \in Y\), \(P \neq Q \}\), \({\mathcal J}={\mathcal J}(X,Y):= \overline{{\mathcal J}^0(X,Y)}\) (the closure of \({\mathcal J}^0(X,Y)\) in \(G\)) and the corresponding subsets of \(\mathbb{P}^n\): \(J^0(X,Y):= \bigcup_{[L] \in {\mathcal J}^0}L\) and \(J(X,Y):= \bigcup_{[L] \in {\mathcal J}}L\). \({\mathcal J}(X,Y)\) and \(J(X,Y)\) are algebraic subsets of \(G\) and \(\mathbb{P}^n\) respectively, \(J(X,Y)\) is called the ``join'' of \(X\) and \(Y\). If \(X=Y\), \(J(X,Y)\) is called the ``secant variety'' \(\text{Sec}(X)\) of \(X\). If \(X \cap Y \neq \emptyset\), the inclusion \(J^0(X,Y) \subset J(X,Y)\) is in general strict. Thus it is natural to raise the following Problem. Describe the additional projective lines besides those passing through the pairs \((P,Q)\), \(P \in X\), \(Q \in Y\), \(P \neq Q\). The fundamental notion of this paper is the relative tangent cone \(C_P(X,Y)\) to \(X\), \(Y\) in \(P \in X \cap Y\), which, introduced by \textit{R. Achilles, P. Tworzewski} and \textit{T. Winiarski} [Ann. Pol. Math. 51, 21-36 (1990; Zbl 0796.32006)], generalizes one of the Whitney cones and has been used in both algebraic and analytic contexts. Since \(J(X,Y)=J^0(X,Y) \cup \bigcup_{P \in X \cap Y}C_P(X,Y)\) (proposition 4.1), the problem is reduced to study \(C_P(X,Y)\). In case of \(X\), \(Y\) analytic curves, a description is known from \textit{D. Ciesielska} [Ann. Pol. Math. 72, 191-195 (1999; Zbl 0966.32004)] and from \textit{J. Briançon, A. Galligo} and \textit{M. Granger} [``Déformations équisingulieres des germes de courbes gauches reduites'', Mem. Soc. Math. Fr., Nouv. Ser. 1 (1980; Zbl 0447.14004)] (see also proposition 3.2). The author proves an effective analytic formula for \(C_P(X,Y)\) (theorem 3.4), which allows him to obtain a detailed description of the join of algebraic curves (theorem 4.2), solving completely the above problem in the \(1\)-dimensional case.