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.

R. Achilles, M. Manaresi (2013). Computing the number of apparent double points of a surface. RENDICONTI DEL SEMINARIO MATEMATICO, 71(3-4), 277-306.

Computing the number of apparent double points of a surface

ACHILLES, HANS JOACHIM RUDIGER;MANARESI, MIRELLA
2013

Abstract

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.
2013
R. Achilles, M. Manaresi (2013). Computing the number of apparent double points of a surface. RENDICONTI DEL SEMINARIO MATEMATICO, 71(3-4), 277-306.
R. Achilles; M. Manaresi
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/213638
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