Abstract: I will talk about a generalisation of an intersection-theoretic local inequality of Fulton–Lazarsfeld to weighted blowups. As a consequence, we obtain the $4/(k+1) n^2$-inequality for isolated $cA_k$ 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 $cA_k$ singularities, including sextic double solids with $cA_1$ and ordinary $cA_2$ points. This is a joint work with Igor Krylov and Takuzo Okada.