Canonical surfaces in \(\mathbb P^4\) with \(p_g=p_a=5\) and \(K^2=11\)
type: Item
instance of: scholarly article
title: Canonical surfaces in \(\mathbb P^4\) with \(p_g=p_a=5\) and \(K^2=11\)
zbMATH Open document ID: 1139.14038
DOI: 10.4171/RLM/480
author: Christian Böhning
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 2008-06-24T00:00:00Z
review text: The author uses his results characterising Gorenstein algebras in codimension two [J. Algebra 288, No. 2, 545--555 (2005; Zbl 1099.13040)] to obtain information about the moduli of general type surfaces with \(p_g=5\), \(K^2=11\) and \(q=0\). The study proceeds via images of the surfaces in \({\mathbb P}^4\) by linear subsystems of the canonical system. Suppose then that \(S\) is a smooth regular surface and \(L\subset H^0({\mathcal O}_S(K_S))\) is a \(5\)-dimensional base-point free linear subspace such that the corresponding morphism \(\pi \colon S\to {\mathbb P}^4\) is birational onto its image~\(Y\). Then \(S\) is a minimal surface of general type and its canonical ring \({\mathcal R}\) is a Gorenstein \({\mathcal A}\)-algebra of codimension~\(2\), where \({\mathcal A}\) is the homogeneous coordinate ring of \({\mathbb P}^4\). One can deduce the shape of a minimal resolution of \({\mathcal R}\) as an \({\mathcal A}\)-algebra. In the case \(p_g=5\), \(K^2=11\), \(q=0\) one can be more precise, calculating the twists involved exactly. In this case there is a converse, which is proved here (Theorem~1.6): an \({\mathcal A}\)-algebra \({\mathcal R}\) with resolution of this shape and satisfying mild depth and singularity conditions arises this way. The author then considers this situation and writes down in detail the condition on the resolution imposed by the assumption that \({\mathcal R}\) is a ring. This yields (rather laboriously) a standard form of the matrices involved and hence a parametrisation of the moduli of the surfaces. One obtains a \(38\)-dimensional unirational open subset of the moduli of surfaces of general type with \(p_g=5\), \(K^2=11\) and \(q=0\), consisting of those whose canonical map is a birational morphism that satisfy a genericity assumption. There remains the need to check that this subset is nonempty, which is done by a computer calculation, sketched here but not given in every detail.
Mathematics Subject Classification ID: 14M05
Mathematics Subject Classification ID: 13D02
Mathematics Subject Classification ID: 14C20
Mathematics Subject Classification ID: 14J29
zbMATH DE Number: 5292351
zbMATH Keywords: surfaces of general type
zbMATH Keywords: canonical surfaces
zbMATH Keywords: canonical ring
zbMATH Keywords: Gorenstein algebras
zbMATH Keywords: canonical projections
reviewed by: G. K. Sankaran
describes a project that uses: Macaulay2
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5292351