## The join of algebraic curves.(English)Zbl 1032.14007

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 R. Achilles, P. Tworzewski and 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 D. Ciesielska [Ann. Pol. Math. 72, 191-195 (1999; Zbl 0966.32004)] and from J. Briançon, A. Galligo and 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.

### MSC:

 14H50 Plane and space curves 14M15 Grassmannians, Schubert varieties, flag manifolds

### Citations:

Zbl 0796.32006; Zbl 0966.32004; Zbl 0447.14004
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