Abstract:
Let Δ be a canonical Fano 3-polytope, i.e., a 3-dimensional lattice
polytope containing exactly one interior lattice point. Then the affine
surface ZΔ defined by a generic Laurent polynomial fΔ with the Newton
polytope Δ is birational to a smooth projective minimal surface SΔ
with q=0 and pg=1. Using the classification of all 674,688 canonical
Fano 3-polytopes obtained by Kasprzyk, we show that SΔ is a K3-
surface except for exactly 9,089 canonical Fano 3-polytopes Δ. In the
latter case, we obtain 9,040 canonical Fano 3-polytopes Δ defining
minimal elliptic surfaces SΔ of Kodaira dimension 1 and 49 canonical
Fano 3-polytopes Δ defining minimal surfaces SΔ of general type with
|π1(SΔ)|=K2∈{1,2} considered by Kynev and Todorov. This is a
joint work with Kasprzyk and Schaller.