As you probably know, the melting temperature in the mantle is a function of H2O contents. The more H2O you have, the lower the melting point of the rocks (i.e. dry vs wet solidus in the sketch). A less known phenomenon is the decrease of melting point with the addition of CO2.
The CO2 melting curve has a weird shape. At low pressures nothing much happens, but there's a certain pressure where all of the sudden CO2 is soluble in the melt and it greatly depresses the melting point. This kink is known as the "carbonate ledge". The exact depth of the ledge depends on the composition of the rocks and the experimentalist that performed the experiments, but it's somewhere around two to three GPa, corresponding to about 65 to 100 km deep. Low degree melts above the solidus (the yellow area) are proper carbonatites, and higher degree melts are all kinds of carbonated alkaline melts, with more CO2 dissolved in them at higher pressures. True hybrid carbonate and silicate melts are only possible at pressures of let's say 4-5 GPa and higher. Below that an immiscibiliy gap exists (also dependent on the alkali contents of the melts - more alkalis = more miscibility).
Now, for this to happen you obviously need to have CO2 in the mantle. But CO2 is the oxidised form of carbon. In general, the deeper you go in the mantle the more reduced the conditions are. The roots of cratonic lithospheric mantle are very reduced and any carbon in there occurs in the form of reduced carbon: diamonds. Diamonds are no good in facilitating carbonated melting of mantle rocks. However, if a process of metasomatism of fluid flow causes oxidation of the diamonds, they transform to CO2 thus enabling the formation of carbonated mantle melts.