Birationally rigid complete intersections of high codimension

We prove that a Fano complete intersection of codimension and index in the complex projective space for and 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 . The proof is based on the technique of hypertangent divisors combined with the recently discovered -inequality for complete intersection singularities.

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