Tuned and Non-Higgsable U(1)s in F-theory

We study the tuning of U(1) gauge fields in F-theory models on a base of
general dimension. We construct a formula that computes the change in
Weierstrass moduli when such a U(1) is tuned, based on the Morrison-Park form
of a Weierstrass model with an additional rational section. Using this formula,
we propose the form of "minimal tuning" on any base, which corresponds to the
case where the decrease in the number of Weierstrass moduli is minimal.
Applying this result, we discover some universal features of bases with
non-Higgsable U(1)s. Mathematically, a generic elliptic fibration over such a
base has additional rational sections. Physically, this condition implies the
existence of U(1) gauge group in the low-energy supergravity theory after
compactification that cannot be Higgsed away. In particular, we show that the
elliptic Calabi-Yau manifold over such a base has a small number of complex
structure moduli. We also suggest that non-Higgsable U(1)s can never appear on
any toric bases. Finally, we construct the first example of a threefold base
with non-Higgsable U(1)s.