On smooth Fano fourfolds of Picard number two

Abstract

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

On smooth Calabi-Yau threefolds of Picard number two

“…According to Remark 5.7 there is an isomorphism ϕ : S µ1 → S µ2 of vector spaces such that g and ϕ(g) arise as Σ i -homogenization of the same Laurent polynomial whenever g ∈ U µ1 . Besides, by [20,Lem. 4.9] the µ i -homogeneous prime polynomials form an open subset of S µi , which is non-empty by assumption.…”

“…The proof of Theorem 1.1 basically uses the combinatorial framework for the classification of smooth Mori dream spaces of Picard number two with hypersurface Cox ring established in [20,Sec. 5]; see also [29].…”

“…This section is devoted to the construction of general hypersurface Cox rings with prescribed specifying data. First, we describe the toolbox for producing general hypersurface Cox rings from [20,Sec. 4] in outline.…”

“…Following [20], we say that R g resp. g is spread if each monomial of degree µ is a convex combination over those monomials showing up in g with non-zero coefficient.…”

“…g is spread if each monomial of degree µ is a convex combination over those monomials showing up in g with non-zero coefficient. Besides, we call R g general (smooth, Calabi-Yau) if g admits an open neighbourhood U in the finite dimensional vector space of all µ-homogeneous polynomials such that every h ∈ U yields a hypersurface Cox ring R h of a normal (smooth, Calabi-Yau) variety X h with divisor class group K. In [20] general hypersurface Cox rings were applied to the classification of smooth Fano fourfolds of Picard number two.…”