Computing the number of apparent double points of a surface

For a smooth surface S ⊂ P^5 there are well known classical formulas giving the number ρ(S) of secants of S passing through a generic point of P^5. In this paper, for possibly singular surfaces T, a computer assisted computation of ρ(T) from the defining ideal I(T) ⊂ K[x_0,...,x_5] is proposed. It is based on the Stückrad-Vogel self-intersection cycle of T and requires the computation of the normal cone of the ruled join J(T, T) along the diagonal. It is shown that in the case when T ⊂ P^5 arises as the linear projection with center L of a surface S ⊂ P^N (N > 5) (which satisfies some mild assumptions), the computational complexity can be reduced considerably by using the normal cone of Sec S along L ∩ Sec S instead of the former normal cone. Many examples and the relative code for the computer algebra systems REDUCE, CoCoA, Macaulay2 and Singular are given.