  Fermat ideals define planar point configurations that are closely related to
the intersection locus of the members of a specific pencil of curves. These
ideals have gained recent popularity as counterexamples to some proposed
containments between symbolic and ordinary powers. We give a systematic
treatment of the family of Fermat ideals, describing explicitly the minimal
generators and the minimal free resolutions of all their ordinary powers as
well as many symbolic powers. We use these to study the ordinary and the
symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbolic
Rees algebras of Fermat ideals are Noetherian. Along the way, we give formulas
for the Castelnuovo-Mumford regularity of the powers of Fermat ideals and we
determine their reduction ideals.
