Birationally rigid complete intersections of high codimension

摘要

We prove that a Fano complete intersection of codimension k and index 1 in the complex projective space PM+k for k >= 20 and M >= 8k log k with at most multi-quadratic singularities is birationally superrigid. The codimension of the complement of the set of birationally superrigid complete intersections in the natural moduli space is shown to be at least (M - 5k)(M - 6k)/2. The proof is based on the technique of hypertangent divisors combined with the recently discovered 4n(2)-inequality for complete intersection singularities.