On monoidal degree four surfaces of \({\mathbb{P}}^ 3\) singular in codimension one
type: Item
instance of: scholarly article
title: On monoidal degree four surfaces of \({\mathbb{P}}^ 3\) singular in codimension one
zbMATH Open document ID: 0587.14021
author: M. Cristina Ronconi
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1985-01-01T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108005
review text: 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.
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J25
Mathematics Subject Classification ID: 14M10
Mathematics Subject Classification ID: 14M07
zbMATH DE Number: 3941687
zbMATH Keywords: classification of monoidal surfaces
zbMATH Keywords: curve which is not set-theoretic complete intersection
reviewed by: Ezio Stagnaro
MaRDI profile type: Publication
cites work: Superficie cubiche di \(A^ 3\) a curve sottoinsieme intersezione completa
cites work: Q4175366
cites work: Q4137299
cites work: Q4063556
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description: scientific article; zbMATH DE number 3941687