Abstract
We obtain local inequalities for log canonical thresholds and multiplicities of movable log pairs. We prove the non-rationality and birational superrigidity of the following Fano varieties: a double covering of a smooth cubic hypersurface in branched over a nodal divisor that is cut out by a hypersurface of degree ; a cyclic triple covering of a smooth quadric hypersurface in branched over a nodal divisor that is cut out by a hypersurface of degree ; a double covering of a smooth complete intersection of two quadric hypersurfaces in branched over a smooth divisor that is cut out by a hypersurface of degree .
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