Minimal models of surfaces with $p_g = 1; q = 0$ associated with canonical Fano $3$-polytopes

Let $\Delta$ be a canonical Fano $3$-polytope, i.e., a $3$-dimensional lattice polytope containing exactly one interior lattice point. Then the affine surface $Z_{\Delta}$ defined by a generic Laurent polynomial $f_{\Delta}$ with the Newton polytope $\Delta$ is birational to a smooth projective minimal surface $S_{\Delta}$ with $q = 0$ and $p_g = 1$. Using the classification of all $674,688$ canonical Fano 3-polytopes obtained by Kasprzyk, we show that $S_{\Delta}$ is a $K3$- surface except for exactly $9,089$ canonical Fano $3$-polytopes $\Delta$. In the latter case, we obtain $9,040$ canonical Fano $3$-polytopes $\Delta$ defining minimal elliptic surfaces $S_{\Delta}$ of Kodaira dimension $1$ and $49$ canonical Fano $3$-polytopes $\Delta$ defining minimal surfaces $S_{\Delta}$ of general type with $|\pi_1(S_{\Delta})| = K^2 \in \{1, 2\}$ considered by Kynev and Todorov. This is a joint work with Kasprzyk and Schaller.