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Special birational transformations of type \((2,1)\). (English) Zbl 1391.14028
Let \(\Phi:\mathbb P ^n \dashrightarrow Z\subset \mathbb P ^N\) be a dominant bi-rational map to a smooth variety \(Z\) such that \(\rho (Z)=1\) then \(\Phi\) is a special birational transformation of type \((a,b)\) if \(\Phi\) is given by a linear system belonging to \(\mathcal O _{\mathbb P ^n}(a)\), \(\Phi ^{-1}\) is is given by a linear system belonging to \(\mathcal O _Z(b)\) and the base locus of \(\Phi\) is an irreducible nonsingular subvariety \(S\subset \mathbb P ^n\). In this paper the authors classify special birational transformations of type \((2,1)\).

MSC:
14E07 Birational automorphisms, Cremona group and generalizations
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