  A computable structure A is x-computably categorical for some Turing degree
x, if for every computable structure B isomorphic to A there is an isomorphism
f:B -> A with f computable in x. A degree x is a degree of categoricity if
there is a computable A such that A is x-computably categorical, and for all y,
if A is y-computably categorical then y computes x.
  We construct a Sigma_2 set whose degree is not a degree of categoricity. We
also demonstrate a large class of degrees that are not degrees of categoricity
by showing that every degree of a set which is 2-generic relative to some
perfect tree is not a degree of categoricity. Finally, we prove that every
noncomputable hyperimmune-free degree is not a degree of categoricity.
