Surfaces in \({\mathbb {P}^{4}}\) fibered in cubics
type: Item
instance of: scholarly article
title: Surfaces in \({\mathbb {P}^{4}}\) fibered in cubics
zbMATH Open document ID: 1181.14054
author: Alberto Alzati
author: Alfonso Tortora
published in: Annali dell'Università di Ferrara. Sezione VII. Scienze Matematiche
publication date: 2009-12-02T00:00:00Z
review text: Let \(S\) be a smooth complex projective rational surface in \(\mathbb P^4\) ruled by cubics. \textit{Ph. Ellia} [J. Pure Appl. Algebra 152, No. 1--3, 83--88 (2000; Zbl 0971.14033)] showed that the pair \((d,g)\) consisting of the degree and the sectional genus of \(S\) is confined to 5 possibilities. In the paper under review the authors obtain a complete classification: \((d,g)\) can only be \((5,2)\) or \((6,3)\), \(S\) is \(\mathbb F_1\) blown up at \(7\) or \(9\) points respectively in general position, and the precise description of the linear system providing the embedding is given in both cases. The authors associate to \(S\) a curve \(\Gamma\) in a suitable compactification, \(X\), of the space of smooth rational cubic curves in \(\mathbb P^4\), inspired by a construction of \textit{G. Ellingsrud} and \textit{S. A. Strømme} [Math. Scand. 76, No. 1, 5--34 (1995; Zbl 0863.14033)]. Then they find suitable generators for \(A^1(X) \otimes \mathbb Q\) the degrees of \(\Gamma\) with respect to which have relevant geometric meaning for the surface \(S\). Combining this with the double point formula for surfaces in \(\mathbb P^4\) provides remarkable constraints on the blow-ups a birational morphism \(S \to \mathbb F_e\) factors through. This leads to the result.
Mathematics Subject Classification ID: 14M07
Mathematics Subject Classification ID: 14J26
zbMATH DE Number: 5641479
zbMATH Keywords: rational ruled surface
zbMATH Keywords: low codimension variety
zbMATH Keywords: twisted cubic curve
reviewed by: Antonio Lanteri
MaRDI profile type: Publication
full work available at URL: s11565-009-0059-3
OpenAlex ID: W1974407353
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DOI: 10.1007/S11565-009-0059-3
description: scientific article; zbMATH DE number 5641479