  We show how to construct non-isotrivial families of supersingular K3 surfaces
over rational curves using a relative form of the Artin-Tate isomorphism and
twisted analogues of Bridgeland's results on moduli spaces of stable sheaves on
elliptic surfaces. As a consequence, we show that every point of Artin
invariant 10 in the Ogus space of marked supersingular K3 surfaces lies on
infinitely many pairwise distinct rational curves canonically associated to
elliptic structures on the underlying K3 surface.
