Abstract
We use classification of non-symplectic automorphisms of K3 surfaces to obtain a partial classification of log del Pezzo surfaces of index three. We characterize them as those with “Multiple Smooth Divisor Property”, whose definition we will give. Our methods include the construction of right resolutions of quotient singularities of index three and an analysis of automorphism-stable elliptic fibrations on K3 surfaces combined with lattice theory. In particular we find several log del Pezzo surfaces of Picard number one with non-toric singularities of index three. As a byproduct, we also obtain the holomorphic description of all non-symplectic automorphisms of K3 surfaces of order three whose fixed locus contains a smooth curve of genus g ≥ 2 (called of elliptic type).
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Ohashi, H., Taki, S. K3 surfaces and log del Pezzo surfaces of index three. manuscripta math. 139, 443–471 (2012). https://doi.org/10.1007/s00229-011-0524-z
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DOI: https://doi.org/10.1007/s00229-011-0524-z