A birational involution
Résumé
Given a general K3 surface S of degree 18, lattice theoretic considerations allow to predict the existence of an anti-symplectic birational involution \phi of the Hilbert cube S^[3]. We describe this involution in terms of the Mukai model of S, with the help of the famous transitive action of the exceptional group G_2(R) on the six dimensional sphere. We make a connection with Homological Projective Duality by showing that the indeterminacy locus of the involution is birational to a P^2-bundle over the dual K3 surface of degree two. We deduce that \phi is an instance of a Mukai flop.
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