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We prove the birational rigidity of Fano complete intersections of index 1with a singular point of high multiplicity, which can be close to the degree ofthe variety. In particular, the groups of birational and biregularautomorphisms of these varieties are equal, and they are non-rational. Theproof is based on the techniques of the method of maximal singularities, thegeneralized $4n^2$-inequality for complete intersection singularities and thetechnique of hypertangent divisors.

Birationally rigid complete intersections with a singular point of high multiplicity