Extremal $1/2$ Calabi–Yau $3$-folds and six-dimensional F-theory applications

We discuss a method for classifying the singularity types of $1/2$ Calabi–Yau $3$-folds, a family of rational elliptic $3$-folds introduced in a previous study in relation to various U(1) factors in 6D F-theory models. A projective dual pair of del Pezzo manifolds recently studied by Mukai is used to analyze the singularity types. In particular, we studied the maximal rank seven singularity types of 1/2 Calabi–Yau $3$-folds. The structures of the singular fibers are analyzed using blow-ups. Double covers of the 1/2 Calabi–Yau $3$-folds yield elliptic Calabi–Yau $3$-folds and applications to six-dimensional $N=1$ F-theory on the Calabi–Yau $3$-folds are also discussed. The deduced singular fibers have applications in studying the gauge groups formed in 6D F-theory compactifications. The blow-up methods used to analyze the singular fibers and sections utilized in this research might have applications in studying the U(1) factors and hypermultiplets charged under U(1) in 6D F-theory.

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Published 30 March 2023