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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:
 1.4e+08 Birational automorphisms, Cremona group and generalizations
##### Keywords:
special birational transformation
Full Text:
##### References:
 [1] Alzati, Alberto; Sierra, Jos\'e Carlos, Quadro-quadric special birational transformations of projective spaces, Int. Math. Res. Not. IMRN, 1, 55-77, (2015) · Zbl 1316.14030 [2] Ein, Lawrence; Shepherd-Barron, Nicholas, Some special Cremona transformations, Amer. J. Math., 111, 5, 783-800, (1989) · Zbl 0708.14009 [3] Fu, Baohua; Hwang, Jun-Muk, Classification of non-degenerate projective varieties with non-zero prolongation and application to target rigidity, Invent. Math., 189, 2, 457-513, (2012) · Zbl 1260.14050 [4] \bibGHbook label=GH, author=Griffiths, Phillip, author=Harris, Joseph, title=Principles of algebraic geometry, series=Pure and Applied Mathematics, pages=xii+813, publisher=Wiley-Interscience [John Wiley & Sons], New York, date=1978, isbn=0-471-32792-1, review=\MR 507725, [5] Hartshorne, Robin, Algebraic geometry, Graduate Texts in Mathematics, No. 52, xvi+496 pp., (1977), Springer-Verlag, New York-Heidelberg · Zbl 0367.14001 [6] Hwang, Jun-Muk; Mok, Ngaiming, Prolongations of infinitesimal linear automorphisms of projective varieties and rigidity of rational homogeneous spaces of Picard number 1 under K\"ahler deformation, Invent. Math., 160, 3, 591-645, (2005) · Zbl 1071.32022 [7] Ionescu, Paltin; Russo, Francesco, Conic-connected manifolds, J. Reine Angew. Math., 644, 145-157, (2010) · Zbl 1200.14078 [8] Ionescu, Paltin; Russo, Francesco, Varieties with quadratic entry locus. II, Compos. Math., 144, 4, 949-962, (2008) · Zbl 1149.14041 [9] Iskovskikh, V. A.; Prokhorov, Yu. G., Fano varieties. Algebraic geometry, V, Encyclopaedia Math. Sci. 47, 1-247, (1999), Springer, Berlin · Zbl 0912.14013 [10] Ivey, Thomas A.; Landsberg, J. M., Cartan for beginners: differential geometry via moving frames and exterior differential systems, Graduate Studies in Mathematics 61, xiv+378 pp., (2003), American Mathematical Society, Providence, RI · Zbl 1105.53001 [11] Landsberg, J. M., On second fundamental forms of projective varieties, Invent. Math., 117, 2, 303-315, (1994) · Zbl 0840.14025 [12] Mok, Ngaiming, Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents. Third International Congress of Chinese Mathematicians. Part 1, 2, AMS/IP Stud. Adv. Math., 42, pt.  1 2, 41-61, (2008), Amer. Math. Soc., Providence, RI · Zbl 1182.14042 [13] Mukai, Shigeru, Biregular classification of Fano $$3$$-folds and Fano manifolds of coindex $$3$$, Proc. Nat. Acad. Sci. U.S.A., 86, 9, 3000-3002, (1989) · Zbl 0679.14020 [14] Novelli, Carla; Occhetta, Gianluca, Projective manifolds containing a large linear subspace with nef normal bundle, Michigan Math. J., 60, 2, 441-462, (2011) · Zbl 1229.14015 [15] Pasquier, Boris, On some smooth projective two-orbit varieties with Picard number 1, Math. Ann., 344, 4, 963-987, (2009) · Zbl 1173.14028 [16] Russo, Francesco, On a theorem of Severi, Math. Ann., 316, 1, 1-17, (2000) · Zbl 0993.14019 [17] Russo, Francesco, Varieties with quadratic entry locus. I, Math. Ann., 344, 3, 597-617, (2009) · Zbl 1170.14040 [18] Sato, Eiichi, Projective manifolds swept out by large-dimensional linear spaces, Tohoku Math. J. (2), 49, 3, 299-321, (1997) · Zbl 0917.14026 [19] Zak, F. L., Tangents and secants of algebraic varieties, Translations of Mathematical Monographs 127, viii+164 pp., (1993), American Mathematical Society, Providence, RI · Zbl 0795.14018
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