Configurations of Kodaira fibers on rational elliptic surfaces. (English) Zbl 0722.14021
All possible configurations (about 400) of Kodaira fibers on elliptic rational surfaces are listed, together with their realisations (as Weierstrass models), the rank and torsion of their Mordell-Weil groups of sections. - The classification is first reduced to that of plane quartics with only simple singularities and one special point.
[See also R. Miranda, ibid. 205, No.2, 191-211 (1990; see the following review).]
[See also R. Miranda, ibid. 205, No.2, 191-211 (1990; see the following review).]
Reviewer: F.Campana (Vandoeuvre-les-Nancy)
MSC:
14J26 | Rational and ruled surfaces |
14J27 | Elliptic surfaces, elliptic or Calabi-Yau fibrations |
14J10 | Families, moduli, classification: algebraic theory |
14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |
Keywords:
configurations of Kodaira fibers on elliptic rational surfaces; Weierstrass models; Mordell-Weil groupsCitations:
Zbl 0722.14022References:
[1] | Dynkin, E.B.: Semisimple subalgebras of semisimple Lie algebras. Mat. Sb. Nov. Ser.30, 349–462 (1952) · Zbl 0048.01701 |
[2] | Looijenga, E.: On the semi-universal deformation of a simple elliptic hypersurface singularity II. Topology17, 23–40 (1978) · Zbl 0392.57013 · doi:10.1016/0040-9383(78)90010-1 |
[3] | Miranda, R., Persson, U.: On Extremal Rational Elliptic Surfaces. Math. Z.193, 537–558 (1986) · Zbl 0652.14003 · doi:10.1007/BF01160474 |
[4] | Miranda, R., Persson, U.: Configurations ofI n fibers on EllipticK-3 surfaces. Math. Z.201, 339–361 (1989) · Zbl 0694.14019 · doi:10.1007/BF01214900 |
[5] | Miranda, R., Persson, U.: Torsion groups of Elliptic Surfaces. Compos. Math.72, 249–267 (1989) · Zbl 0717.14019 |
[6] | Persson, U.: Double Sextics and SingularK3 surfaces. In: Casas-Severo, E., Welters, G.E., Xambo-Descamps, S. (eds.) Algebraic Geometry. Proceedings, Sitges 1983. (Lect. Notes Math., vol. 1124, pp 262–328) Berlin Heidelberg New York: Springer 1985 |
[7] | Timms, G.: The nodal cubic surfaces and the surfaces from which they are derived by projection. Proc. R. Soc. Lond., Ser A119, 213–248 (1928) · JFM 54.0709.01 · doi:10.1098/rspa.1928.0095 |
[8] | Urabe, T.: On singularities on degenerate Del Pezzo surfaces of degree 1,2. Proc. Symp. Pure Math.40, 587–591 (1983) · Zbl 0515.14021 |
[9] | Du Val, P.: On isolated singularities which do not affect the conditions of adjunctionI, II, III Proc. Cambridge Philos. Soc.30, 453–465, 483–491 (1934) · JFM 60.0599.01 · doi:10.1017/S030500410001269X |
[10] | Wall, C.T.C.: Nets of Conics. Proc. Cambridge Philos. Soc.81, 351–365 (1977) · Zbl 0351.14032 · doi:10.1017/S0305004100053421 |
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