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Hyperelliptic surfaces with \(K^{2}<4\chi-6\). (English) Zbl 1343.14036

Authors’ abstract: Let \(S\) be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus \(g\). We prove that if \(K_S^2<4\chi(\mathcal{O}_S )-6\), then \(g\) is bounded. The surface \(S\) is determined by the branch locus of the covering \(S\rightarrow S/i\), where \(i\) is the hyperelliptic involution of \(S\). For \(K_S^2<3\chi(\mathcal{O}_S )-6\), we show how to determine the possibilities for this branch curve. As an application, given \(g>4\) and \(K_S^2-3\chi(\mathcal{O}_S )<-6\), we compute the maximum value for \(\chi(\mathcal{O}_S )\). This list of possibilities is sharp.

MSC:

14J29 Surfaces of general type
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References:

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