Abstract
We construct an explicit smooth Fano complex threefold with Picard number 1, index 1, and degree 10 (also known as a Gushel–Mukai threefold) and prove that it is not rational by showing that its intermediate Jacobian has a faithful \({{\,\mathrm{PSL}\,}}(2,\mathbf{F}_{11}) \)-action. Along the way, we construct Gushel–Mukai varieties of various dimensions with rather large (finite) automorphism groups. The starting point of all these constructions is an Eisenbud–Popescu–Walter sextic with a faithful \({{\,\mathrm{PSL}\,}}(2,\mathbf{F}_{11}) \)-action discovered by the second author in 2013.
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