  The Hodge numbers of generic elliptically fibered Calabi-Yau threefolds over
toric base surfaces fill out the "shield" structure previously identified by
Kreuzer and Skarke. The connectivity structure of these spaces and bounds on
the Hodge numbers are illuminated by considerations from F-theory and the
minimal model program. In particular, there is a rigorous bound on the Hodge
number h_{21} <= 491 for any elliptically fibered Calabi-Yau threefold. The
threefolds with the largest known Hodge numbers are associated with a sequence
of blow-ups of toric bases beginning with the Hirzebruch surface F_{12} and
ending with the toric base for the F-theory model with largest known gauge
group.
