Cylinders in del Pezzo fibrations
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We show that a del Pezzo fibration π: V → W of degree d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of π is isomorphic to Z × AK1 for some quasi-projective variety Z defined over the function field K of W, if and only if d ≥ 5 and π: V → W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations π: V → P1 of degree d ≤ 4.
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- A. Corti, A. V. Pukhlikov and M. Reid, Fano 3-fold hypersurfaces, in Explicit Birational Geometry of 3-folds, London Mathematical Society Lecture Notes Series, Vol. 281, Cambridge University Press, Cambridge, 2000.Google Scholar
- A. Dubouloz and T. Kishimoto, Explicit biregular/birational geometry of affine threefolds: completions of A3 into del Pezzo fibrations and Mori conic bundles, in Algebraic Varieties and Automorphism Groups, Advanced Studied in Pure Mathematics, Vol. 75, Mathematical Society of Jaban, Tokyo, 2017, pp. 49–71.Google Scholar
- J. Kollár and S. Mori, Birational Geometry of Algebraic Geometry, Cambridge Tracts in Mathematics, Vol. 134, Cambridge University Press, Cambridge, 1998.Google Scholar
- Y. Prokhorov and M. Zaidenberg, New examples of cylindrical Fano fourfolds, in Algebraic Varieties and Automorphism Groups, Advanced Studied in Pure Mathematics, Vol. 75, Mathematical Society of Jaban, Tokyo, 2017, pp. 443–463.Google Scholar
- H. P. F. Swinnerton-Dyer, Rational points on del Pezzo surfaces of degree 5, in Algebraic Geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.), Wolters-Noordho, Groningen, 1972, pp. 287–290.Google Scholar