On smooth Fano fourfolds of Picard number two

We classify the smooth Fano 4-folds of Picard number two that have a general hypersurface Cox ring.

[1]  S. Kleiman.  Bertini and his two fundamental theorems , 1997, alg-geom/9704018.

[2]  James McKernan,et al.  Existence of minimal models for varieties of log general type , 2006, 0808.1929.

[3]  Shigeru Mukai,et al.  Classification of Fano 3-folds with B2≥2 , 1981 .

[4]  Christoff Hische,et al.  ON TORUS ACTIONS OF HIGHER COMPLEXITY , 2018, Forum of Mathematics, Sigma.

[5]  Askold Khovanskii,et al.  Newton polyhedra and toroidal varieties , 1977 .

[6]  J. Murre,et al.  The Hodge conjecture for fourfolds admitting a covering by rational curves , 1978 .

[7]  Juergen Hausen,et al.  GIT-Equivalence beyond the ample cone , 2005 .

[8]  C. Shramov,et al.  Bounds for smooth Fano weighted complete intersections , 2016, Communications in Number Theory and Physics.

[9]  C. Casagrande.  On the birational geometry of Fano 4-folds , 2010, 1010.4129.

[10]  A. Laface,et al.  Cox rings of cubic surfaces and Fano threefolds , 2014, 1409.0799.

[11]  F. Oort,et al.  Representability of group functors, and automorphisms of algebraic schemes , 1967 .

[12]  Yuri Prokhorov.  On -Fano threefolds , 2015, 1506.07873.

[13]  F. Berchtold,et al.  Cox rings and combinatorics , 2003, math/0311105.

[14]  I. Dolgachev.  Newton polyhedra and factorial rings , 1980 .

[15]  J. Hausen,et al.  Smooth projective varieties with a torus action of complexity 1 and Picard number 2 , 2016, 1602.04360.

[17]  J. Hausen,et al.  A software package for Mori dream spaces , 2014, 1409.6929.

[18]  Simon Keicher,et al.  A software package to compute automorphisms of graded algebras , 2017, Journal of Software for Algebra and Geometry.

[19]  A. Blanchard.  Sur les variétés analytiques complexes , 1956 .

[20]  A. Laface,et al.  Hypersurfaces in Mori dream spaces , 2011, 1109.0566.

[21]  On Intrinsic Quadrics , 2017, Canadian Journal of Mathematics.

[22]  David A. Cox.  The homogeneous coordinate ring of a toric variety , 2013 .

[23]  Victor V. Batyrev.  On the classification of toric Fano 4-folds , 1998 .

[24]  A. Kasprzyk,et al.  Four-dimensional Fano toric complete intersections , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[25]  S. Mukai,et al.  Classification of Fano 3-folds with B2≥2 , 1981 .

[26]  John J. Cannon,et al.  The Magma Algebra System I: The User Language , 1997, J. Symb. Comput..