  The fragmentation mode of high-mass molecular clumps and the accretion
processes that form the most massive stars ($M\gtrsim 8M_\odot$) are still not
well understood. To this end, we have undertaken a large observational program
(CORE) making use of interferometric observations from the Northern Extended
Millimetre Array (NOEMA) for a sample of 20 luminous ($L>10^4L_\odot$)
protostellar objects in the 1.37 mm wavelength regime in both continuum and
line emission, reaching $\sim$0.4" resolution (800 au at 2 kpc). Using the
dense gas tracer CH$_3$CN, we find velocity gradients across 13 cores
perpendicular to the directions of bipolar molecular outflows, making them
excellent disk candidates. Specific angular momentum ($j$) radial profiles are
on average $\sim10^{-3}$ km /s pc and follow $j \propto r^{1.7}$, consistent
with a poorly resolved rotating and infalling envelope/disk model. Fitting the
velocity profiles with a Keplerian model, we find protostellar masses in the
range of $\sim 10-25$ $M_\odot$. Modelling the level population of CH$_3$CN
lines, we present temperature maps and find median gas temperatures in the
range $70-210$ K. We create Toomre $Q$ maps to study the stability of the disks
and find almost all (11 of 13) disk candidates to be prone to fragmentation due
to gravitational instabilities at the scales probed by our observations. In
particular, disks with masses greater than $\sim10-20\%$ of the mass of their
host (proto)stars are Toomre unstable, and more luminous protostellar objects
tend to have disks that are more massive and hence more prone to fragmentation.
Our finings show that most disks around high-mass protostars are prone to disk
fragmentation early in their formation due to their high disk to stellar mass
ratio. This impacts the accretion evolution of high-mass protostars which will
have significant implications for the formation of the most massive stars.
