Geodesic
Birationally rigid complete intersections of high codimension
Izvestiya. Mathematics , Tome 83 (2019) no. 4, pp. 743-769

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We prove that a Fano complete intersection of codimension $k$ and index $1$ in the complex projective space ${\mathbb P}^{M+k}$ for $k\geqslant 20$ and $M\geqslant 8k\log k$ 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.
Keywords: birational rigidity, maximal singularity, multiplicity, hypertangent divisor, complete intersection singularity. 
@article{IM2_2019_83_4_a5,
     author = {D. Evans and A. V. Pukhlikov},
     title = {Birationally rigid complete intersections of high codimension},
     journal = {Izvestiya. Mathematics },
     pages = {743--769},
     publisher = {mathdoc},
     volume = {83},
     number = {4},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2019_83_4_a5/}
}
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D. Evans; A. V. Pukhlikov. Birationally rigid complete intersections of high codimension. Izvestiya. Mathematics , Tome 83 (2019) no. 4, pp. 743-769. http://geodesic.mathdoc.fr/item/IM2_2019_83_4_a5/