Superficie cubiche di $$\mathbb{A}_\mathbb{C}^3$$ a curve sottoinsieme intersezione completa

Abstract

In this paper we classify the algebraic cubic surfaces ℱ of the affine space\(\mathbb{A}_\mathbb{C}^3\) ℂis the complex field, whose algebraic curves are set-theoretic complete intersections of ℱ; in other words surfaces ℱ such that every prime ideal of height 1in the coordinate ring ℂ[ℱ] of ℱ is the radical of a principal ideal; if ℱ is non singular in codimension 1this means that ℂ[ℱ] is semifactorial. We give the equations of such surfaces within linear isomorphisms of\(\mathbb{A}_\mathbb{C}^3\) providing also methods by which one can construct the equations of the surfaces cutting on ℱ its curves as set-theoretic complete intersections. Moreover for each of these surfaces we determine the minimum positive number λ such that every algebraic curve of ℱ with multiplicity of intersection λ, is complete intersection of ℱ itself with another surface\(\mathcal{G}\) § 8where the results are summarized). We tackle also the problem of such a classification over algebraically closed fields k different from ℂ.