  We study several properties of multihomogeneous prime ideals. We show that
the multigraded generic initial ideal of a prime has very special properties,
for instance, its radical is Cohen-Macaulay. We develop a comprehensive study
of multidegrees in arbitrary positive multigraded settings. In these
environments, we extend the notion of Cartwright-Sturmfels ideals by means of a
standardization technique. Furthermore, we recover or extend important results
in the literature, for instance: we provide a multidegree version of
Hartshorne's result stating the upper semicontinuity of arithmetic degree under
flat degenerations, and we give an alternative proof of Brion's result
regarding multiplicity-free varieties.
