Abstract
We develop a combinatorial approach to the construction of gen- eral smooth compact base surfaces that support elliptic Calabi-Yau threefolds. This extends previous analyses that have relied on toric or semi-toric structure. The resulting algorithm is used to construct all classes of such base surfaces S with h 1,1 (S) < 8 and all base sur- faces over which there is an elliptically fibered Calabi-Yau threefold X with Hodge number h 2,1 (X) ≥ 150. These two sets can be used to describe all 6D F-theory models that have fewer than seven ten- sor multiplets or more than 150 neutral scalar fields respectively in their maximally Higgsed phase. Technical challenges to construct- ing the complete list of base surfaces for all Hodge numbers are discussed.
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