  We investigate the delicate interplay between the types of singular fibers in
elliptic fibrations of Calabi-Yau threefolds (used to formulate F-theory) and
the "matter" representation of the associated Lie algebra. The main tool is the
analysis and the appropriate interpretation of the anomaly formula for
six-dimensional supersymmetric theories. We find that this anomaly formula is
geometrically captured by a relation among codimension two cycles on the base
of the elliptic fibration, and that this relation holds for elliptic fibrations
of any dimension. We introduce a "Tate cycle" which efficiently describes this
relationship, and which is remarkably easy to calculate explicitly from the
Weierstrass equation of the fibration. We check the anomaly cancellation
formula in a number of situations and show how this formula constrains the
geometry (and in particular the Euler characteristic) of the Calabi-Yau
threefold.
