Ramification curve of the projection on \(\mathbb{P}^ 2\) of a surface of \(\mathbb{P}^ 3\) (Q1187735)

The aim of this paper is to give necessary and sufficient conditions on a plane curve \(\Gamma\) of degree \(n(n-1)\) in order that \(\Gamma\) is the ramification curve of the general projection of a smooth surface \(S\subseteq\mathbb{P}^ 3\) onto \(\mathbb{P}^ 2\). The complete characterization of \(\Gamma\) is given by its singularities: it was known [see \textit{A. Libgober}, Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, Part 2, Proc., Sympos. Pure Math. 46, No. 2, 29-45 (1987; Zbl 0703.14007)], that the singular points of \(\Gamma\) must be \(n(n-1)(n-2)(n- 3)\) nodes and \(n(n-1)(n-2)\) cusps; here it is shown that if \(\Gamma\) has precisely those singularities the following conditions are what was asked for: -- the smallest degree of a curve \(\mu_ 0\) passing through the singular points of \(\Gamma\) is \(n^ 2-3n+2\); -- there exists a curve \(\mu_ 1\) of degree \(n^ 2-3n+3\) having no common components with \(\mu_ 0\) through the singular points of \(\Gamma\). This is proved by using in a nice way a very classical construction: The (Cayley) monoidal representation of a space curve \(C\) via the cone \(R\) which projects \(C\) to \(\mathbb{P}^ 2\) and a monoid \(\Psi\) (i.e. a surface of degree \(n\) with an \((n-1)\)-ple point) through \(C\) such that \(\Psi\cap R\) is the union of \(C\) and lines through the singular points of \(\Gamma\). What is proved is that the conditions on \(\Gamma\) are also equivalent to the fact that \(\Gamma\) is the projection of a smooth space curve \(C\) which is the complete intersection of two surfaces \(S, B\) of degrees \(n, n-1\). The surface \(S\) can be chosen in such a way that the (unique) surface \(B\) of degree \(n-1\) is the polar of \(S\) with respect to the center of projection. The interest for this problem lies in the fact that there are few methods to construct singular plane curves, and that the fundamental group of \(\mathbb{P}^ 2-\Gamma\) is a discrete invariant on the moduli space of the surface of general type.