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K3 surfaces with an automorphism of order 66, the maximum possible
In each characteristic , it was shown in a previous work that the
order of an automorphism of a K3 surface is bounded by 66, if finite. Here, it
is shown that in each characteristic a K3 surface with a cyclic
action of order 66 is unique up to isomorphism. The equation of the unique
surface is given explicitly in the tame case () and in the wild case
().Comment: 12 papges. arXiv admin note: text overlap with arXiv:1204.171
A vanishing theorem on fake projective planes with enough automorphisms
For every fake projective plane with automorphism group of order 21, we
prove that for all and for every ample line bundle with
. For every fake projective plane with automorphism group of order 9, we
prove the same vanishing for every cubic root (and its twist by a 2-torsion) of
the canonical bundle . As an immediate consequence, there are exceptional
sequences of length 3 on such fake projective planes.Comment: 17 page
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