Non-general type surfaces in \(\mathbb{P}^4\): Some remarks on bounds and constructions
type: Item
instance of: scholarly article
title: Non-general type surfaces in \(\mathbb{P}^4\): Some remarks on bounds and constructions
zbMATH Open document ID: 1012.14014
author: W. Decker
author: Frank-Olaf Schreyer
published in: Journal of Symbolic Computation
publication date: 2003-06-15T00:00:00Z
review text: The authors focus on one specific problem of classification of non-general type surfaces in \(\mathbb{P}^4\). Via linear projection of a general point outside the secant variety of a surface \( S \subset \mathbb{P}^n \), \(S\) can be embedded in \(\mathbb{P}^5\). Due to the classical result of \textit{F. Severi} [Rend. Circ. Math. Palermo 15, 33-51 (1901; JFM 32.0648.04)] which states that if the secant variety of \(S\) is not all \(\mathbb{P}^5\) then \(S\) is the Veronese surface in \(\mathbb{P}^5\), one expects constraints on the numerical invariants of a smooth surface in \(\mathbb{P}^4\), which satisfy the double point formula. In this direction \textit{G. Ellingsrund} and \textit{C. Peskine} [Invent. Math. 95, No. 1, 1-11 (1989; Zbl 0676.14009)] have shown that there exists an integer \(d_0\) such that every smooth surface in \(\mathbb{P}^4\) of degree \(d>d_0\) is of general type. The authors discuss various bounds for \(d_0\) already known. For example Ellingsrud and Peskine [loc.cit] have shown that \(d_0 \leq 10.000\), with arguments by \textit{R. Braun} and \textit{G. Fløystad} [Compos. Math. 93, No. 2, 211-229 (1994; Zbl 0823.14021)] the authors show that \(d_0 \leq 105\), and \textit{M. Cook} [Compos. Math. 102, No. 2, 141-145 (1996; Zbl 0871.14033)] showed that \(d_0 \leq 76\). The authors prove the following theorem: Let \(S \subset \mathbb{P}^4\) be a smooth surface which is not of general type. Then the degree of \(S\) is smaller or equal to 52. In the first part of their paper they discuss the degree bound. In the second part of the paper the authors explain the method by \textit{W. Decker, L. Ein} and \textit{F.-O. Schreyer} [J. Algebr. Geom. 2, No. 2, 185-237 (1993; Zbl 0795.14019)] which relies on symbolic computation, adjunction theory and the approach of \textit{S. Popescu} and \textit{K. Ranestad} [J. Algebr. Geom. 5, No. 1, 13-76 (1996; Zbl 0861.14041)] of classification.
reviewed by: Isidro Nieto Baños
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 14N30
zbMATH DE Number: 1471046
zbMATH Keywords: classification of non-general type surfaces
zbMATH Keywords: computational aspects in algebraic geometry
zbMATH Keywords: degree bound
zbMATH Keywords: symbolic computation
zbMATH Keywords: adjunction
describes a project that uses: Macaulay2
MaRDI profile type: Publication
full work available at URL: jsco.1999.0323
OpenAlex ID: W2056098719
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DOI: 10.1006/JSCO.1999.0323
description: scientific article; zbMATH DE number 1471046