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Computing invariants of semi-log-canonical surfaces

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Abstract

We describe some general methods to compute fundamental groups, (co)homology, and irregularity of semi-log-canonical surfaces. As an application, we show that there are exactly two irregular Gorenstein stable surfaces with \(K^2=1\), which have \(\chi (X) = 0\) and \({{\mathrm{Pic}}}^0(X)={\mathbb {C}}^*\) but different homotopy type.

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Acknowledgments

The first author is a member of GNSAGA of INDAM. The third author is grateful for support of the DFG through the Emmy Noether program and SFB 701. The collaboration benefited immensely from a visit of the third author in Pisa supported by GNSAGA of INDAM. This project was partially supported by PRIN 2010 “Geometria delle Varietà Algebriche” of italian MIUR. We are indepted to Stefan Bauer, Kai-Uwe Bux, Michael Lönne, Hanno von Bodecker for guidance around the pitfalls of algebraic topology. Kai-Uwe Bux also explained to us how to prove that the fundamental groups computed in Proposition 4.6 are not isomorphic. The third author is grateful to Wenfei Liu for many discussions on stable surfaces and to Filippo Viviani for some helpful email communication.

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Correspondence to Sönke Rollenske.

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Franciosi, M., Pardini, R. & Rollenske, S. Computing invariants of semi-log-canonical surfaces. Math. Z. 280, 1107–1123 (2015). https://doi.org/10.1007/s00209-015-1469-9

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  • DOI: https://doi.org/10.1007/s00209-015-1469-9

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