We carry out a systematic analysis of Calabi-Yau threefolds that are
elliptically fibered with section ("EFS") and have a large Hodge number h^{2,
1}. EFS Calabi-Yau threefolds live in a single connected space, with regions of
moduli space associated with different topologies connected through transitions
that can be understood in terms of singular Weierstrass models. We determine
the complete set of such threefolds that have h^{2, 1} >= 350 by tuning
coefficients in Weierstrass models over Hirzebruch surfaces. The resulting set
of Hodge numbers includes those of all known Calabi-Yau threefolds with h^{2,
1} >= 350, as well as three apparently new Calabi-Yau threefolds. We speculate
that there are no other Calabi-Yau threefolds (elliptically fibered or not)
with Hodge numbers that exceed this bound. We summarize the theoretical and
practical obstacles to a complete enumeration of all possible EFS Calabi-Yau
threefolds and fourfolds, including those with small Hodge numbers, using this
approach.Comment: 44 pages, 5 tables, 5 figures; v2: minor corrections; v3: minor
corrections, moved figure; v4: typo in Table 2 correcte
