2n^2-inequality for cA_1 points and applications to birational rigidity

Abstract

The 4n(2)-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 cA(1), and obtain a 2n(2)-inequality for cA(1) 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 P-1 satisfying the K-2-condition, all of which have at most terminal cA(1 )singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a cA(1) point which is not an ordinary double point.