$\mathrm{K}3$ surfaces with an order 60 automorphism and a characterization of supersingular $\mathrm{K}3$ surfaces with Artin invariant $1$

In characteristic $p = 0$ or $p \gt 5$, we show that a $\mathrm{K}3$ surface with an order 60 automorphism is unique up to isomorphism. As a consequence, we characterize the supersingular $\mathrm{K}3$ surface with Artin invariant $1$ in characteristic $p \equiv 11$ (mod $12$) by a cyclic symmetry of order 60.

2010 Mathematics Subject Classification

14J27, 14J28, 14J50

Full Text (PDF format)

Accepted 24 February 2014

Published 13 October 2014