On monoidal degree four surfaces of \({\mathbb{P}}^ 3\) singular in codimension one
This paper is the second and last part of the classification of monoidal surfaces \(M_ 4\), of degree 4, in \({\mathbb{P}}^ 3_{{\mathbb{C}}}\), \({\mathbb{C}}\) the complex field, such that for every curve C on \(M_ 4\) there exists a minimum integer \(\mu\), with \(\mu C=complete\) intersection of \(M_ 4\) with another surface in \({\mathbb{P}}^ 3_{{\mathbb{C}}}\) (i.e. C is set- theoretic complete intersection on \(M_ 4).\) In a previous work [cf. Rend. Semin. Math. Univ. Padova 72, 49-68 (1984; Zbl 0557.14029)] the author gave the complete classification of the monoidal surfaces \(M_ 4\) non-singular in codimension one by expliciting the equations and finding for \(\mu\) the values \(\mu =4, 8, 12, 20, 24, 28, 36\). It follows a characterization of the \(M_ 4\) such that the homogeneous coordinate ring \({\mathbb{C}}[M_ 4]\) is semifactorial. In the paper under review it is proved that on every monoidal surface \(M_ 4\), of degree 4, singular in codimension one, there exists a curve which is not set-theoretic complete intersection on \(M_ 4\). It is also quoted a counterexample in characteristic \(p=2\), due to P. C. Craighero, so that the above statement on \(M_ 4\) singular in codimension one, is not extensible in positive characteristic.
curve which is not set-theoretic complete intersection
scientific article; zbMATH DE number 3941687
scientific article