  We develop a combinatorial approach to the construction of general 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 surfaces over which there is an
elliptically fibered Calabi-Yau threefold $X$ with Hodge number $h^{2, 1} (X)
\geq 150$. These two sets can be used todescribe all 6D F-theory models that
have fewer than seven tensor multiplets or more than 150 neutral scalar fields
respectively in their maximally Higgsed phase. Technical challenges to
constructing the complete list of base surfaces for all Hodge numbers are
discussed.
