2 n²-inequality for cA₁ points and applications to birational rigidity

The 4 n² -inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type cA₁ , and obtain a 2 n² -inequality for cA₁ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over P¹ satisfying the K² -condition, all of which have at most terminal cA₁ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a cA₁ point which is not an ordinary double point.