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Special cubic birational transformations of projective spaces


We extend our classification of special Cremona transformations whose base locus has dimension at most three to the case when the target space is replaced by a (locally) factorial complete intersection.

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  1. 1.

    Notice that the difference between the right and left side of the inequality (2.25) coincides with the sum \(\Sigma _{l}\left( {\begin{array}{c}5+l^2\\ 4\end{array}}\right) \), where l runs over all lines contained in the surface \(S\subset {{\mathbb {P}}}^5\) having self-intersection \(\le -2\). Thus (2.25) is a strict inequality if and only if S contains a line with self-intersection \(\le -6\).

  2. 2.

    For further computational details, see the online documentation of the methods abstractRationalMap from Cremona [72], and dualVariety from Resultants [73].


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The author is grateful to Francesco Russo for useful discussions and for his interest in the work.

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Staglianò, G. Special cubic birational transformations of projective spaces. Collect. Math. 71, 123–150 (2020).

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Mathematics Subject Classification

  • 14E05
  • 14E07
  • 14J30