  We study the genuine tripartite nonlocality (GTN) and the genuine tripartite
entanglement (GTE) of Dirac fields in the background of a Schwarzschild black
hole. We find that the Hawking radiation degrades both the physically
accessible GTN and the physically accessible GTE. The former suffers from
"sudden death" at some critical Hawking temperature, and the latter approaches
to the nonzero asymptotic value in the limit of infinite Hawking temperature.
We also find that the Hawking effect cannot generate the physically
inaccessible GTN, but can generate the physically inaccessible GTE for fermion
fields in curved spacetime. These results show that on the one hand the GTN
cannot pass through the event horizon of black hole, but the GTE do can, and on
the other hand the surviving physically accessible GTE and the generated
physically inaccessible GTE for fermions in curved spacetime are all not
nonlocal. Some monogamy relations between the physically accessible GTE and the
physically inaccessible GTE are found.
