Rational configurations in \(K3\) surfaces and simply-connected \(p_g=1\) surfaces with \(K^2=1,2,3,4,5,6,7,8,9\)
type: Item
instance of: scholarly article
title: Rational configurations in \(K3\) surfaces and simply-connected \(p_g=1\) surfaces with \(K^2=1,2,3,4,5,6,7,8,9\)
zbMATH Open document ID: 1505.14086
author: Giancarlo Urzúa
author: Javier A. Reyes
published in: Mathematische Zeitschrift
publication date: 2022-11-17T00:00:00Z
review text: One by now classical method for constructing simply connected surfaces of general type with given (small) invariants consists in constructing a normal surface \(Y\) with rational singularities with the given invariants and show that it admits a \(\mathbb Q\)-Gorenstein smoothing. This technique has been applied mainly to the case \(p_g=0\), in which the surface \(Y\) is obtained by blowing down configurations of rational curves on a rational or Enriques surface. To apply the same technique to the case \(p_g=1\) one can construct the surface \(Y\) by blowing down configurations of rational curves on a \(K3\) surface, as in [\textit{H. Park} et al., J. Korean Math. Soc. 50, No. 3, 493--507 (2013; Zbl 1293.14005)] where examples are constructed for \(K^2=3,4,5,6,8\). Here the authors make a theoretical analysis of the above situation, obtaining numerical restrictions on the singularities of \(Y\) and construct examples for all values of \(K^2=1, \dots 9\). It is worth noticing that no example of simply connected surface with \(p_g=1\) and \(K^2=7,9\) was previously known and that for these values of \(K^2\) the construction of the singular surface \(Y\) is highly non trivial.
reviewed by: Rita Pardini
Mathematics Subject Classification ID: 14J29
Mathematics Subject Classification ID: 14J28
zbMATH DE Number: 7619455
zbMATH Keywords: surfaces of general type with \(p_g=1\)
zbMATH Keywords: \(K3\) surfaces
zbMATH Keywords: \(Q\)-Gorenstein smoothing
MaRDI profile type: Publication
full work available at URL: s00209-022-03144-y
OpenAlex ID: W3205524284
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DOI: 10.1007/S00209-022-03144-Y
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description: scientific article; zbMATH DE number 7619455