The join of algebraic curves

Let P be the n-dimensional projective space over C. Denote by G(1,P) the grassmannian of the all projective lines in P. By the Plücker embedding G(1,P) →֒ P( n+1 2 )−1 the grassmannian is an algebraic subset of P( n+1 2 ). For any projective line L ⊂ P we will denote by [L] the corresponding point of G(1,P) and for any P,Q ∈ P, P 6= Q, we will denote by PQ the unique projective line in P spanned by P and Q. Likewise, for any projective subspaces L,K ⊂ P we will denote by Span(L,K) the unique projective subspace in P spanned by L and K. If X is an algebraic subset of P then Sing(X) is the set of singular points of X . For P ∈ X − Sing(X) by TPX ⊂ P we denote the embedded tangent space to X at P . Let X,Y ⊂ P be two varieties in P i.e. irreducible algebraic subsets of P. The definition of the join of X and Y is as follows (see [ H], p.88, [ Z], p.15, [ FOV], Def. 1.3.5). Define the subsets of the grassmannian