Local inequalities for $c{A}_k$ singularities

We generalize an intersection-theoretic local inequality of Fulton-Lazarsfeld to weighted blowups. As a consequence, we obtain the $4n^2\mathrm{/}(k+1)$-inequality for isolated $c{A}_k$ singularities, an analogue of the $4n^2$-inequality for smooth points. We use this to prove birational rigidity of many families of Fano 3-fold weighted complete intersections with terminal quotient singularities and isolated $c{A}_k$ singularities, including sextic double solids with $c{A}_1$ and ordinary $c{A}_2$ points.