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Calabi-Yau threefolds arising from fiber products of rational quasi-elliptic surfaces. II. (English)
Manuscr. Math. 125, No. 3, 325-343 (2008).
The authors construct new projective Calabi-Yau threefolds in characteristic $p=2$. Some of them have Betti number $b_3=0$ and therefore do not lift to characteristic zero. The beautiful construction is analogous to the first part [the authors, Ark. Mat. 45, No. 2, 279‒296 (2007; Zbl 1156.14033)], which dealt with the case $p=3$. The basic idea is to start with two quasielliptic surfaces $Y_1$ and $Y_2$ over the projective line, form the fiber product, and analyse a suitable resolution of singularities $X$. The starting point is {\it H. Ito}’s classification of rational quasielliptic surfaces [Tôhoku Math. J., II. Ser. 46, No. 2, 221‒251 (1994; Zbl 0820.14015)]. The authors show that certain hypersurface singularities in characteristic two admit crepant resolutions, and relate this information to the fiber types and the relative position of the degenerate fibers of the quasielliptic surfaces. Finally, the authors analyse fibrations $X\rightarrow\mathbb{P}^1$ whose geometric fibers are nonsmooth and sometimes even nonnormal surfaces with trivial canonical class.
Reviewer: Stefan Schröer (Düsseldorf)