Gorenstein toric threefolds with isolated singularities and cyclic divisor class group (Q1078277)

The author investigates the question of what components of an algebraic K3 surfaces moduli space are realizable as a family of divisors on a toric 3-fold. The most natural case is that of an anticanonical family \(| -K_ X|\) on a projective toric 3-fold X. The main result of the paper (corollary 1.2) states that if \(| -K_ X|\) contains a nonsingular K3 surface S with Pic \(S\cong {\mathbb{Z}}\), then X is isomorphic to either \({\mathbb{P}}^ 3\) or the cone in \({\mathbb{P}}^{10}\) over the triple Veronese surface \(V={\mathbb{P}}^ 2\) embedded into \({\mathbb{P}}^ 9\) via cubics \((={\mathbb{P}}(1,1,1,3))\). So he obtains two well-known families. In the first case the anticanonical divisors are the quartic K3 surfaces, and in the second case they are the double covers of \({\mathbb{P}}^ 2\). The proof is based on the classification of complete Gorenstein toric 3-folds X with isolated singularities and Pic \(X\cong {\mathbb{Z}}\). Theorem 1.1 of the paper under review establishes that any such 3-fold is isomorphic either to \({\mathbb{P}}^ 3\) or \({\mathbb{P}}(1,1,1,3).\) Closely related to these results in methods are the following papers on the classification of toric Fano 3-folds: \textit{V. V. Batyrev}, Izv. Akad. Nauk SSSR, Ser. Mat. 45, 704-717 (1981; Zbl 0478.14032); translation in Math. USSR, Izv. 19, 13-25 (1982); \textit{K. Watanabe} and \textit{M. Watanabe}, Tokyo J. Math. 5, 37-48 (1982; Zbl 0581.14028)].